Quintessential Inflation with Dynamical Higgs Generation as an Affine Gravity

First, we propose a scale-invariant modified gravity interacting with a neutral scalar"inflaton"and a Higgs-like SU(2)xU(1) iso-doublet scalar field based on the formalism of non-Riemannian (metric-independent) spacetime volume-elements. This model describes in the physical Einstein frame a quintessential inflationary scenario driven by the"inflaton"together with gravity-inflaton assisted dynamical spontaneous SU(2)xU(1) symmetry breaking in the post-inflationary universe, whereas SU(2)xU(1) symmetry remains intact in the inflationary epoch. Next, we find the explicit representation of the latter quintessential inflationary model with a dynamical Higgs effect as an Eddington-type purely affine gravity.


Introduction
Studies in cosmology are dominated by the fundamental concept of "inflation" -a period of exponential expansion, which provides a plausible solution for the "puzzles" of the Big-Bang cosmology (the horizon problem, the flatness problem, the magnetic monopole problem, etc.) [1][2][3][4][5][6][7][8]. For more extensive accounts, see the books [9][10][11][12][13][14][15][16][17][18][19][20][21]. The most widely discussed mechanism for generating a period of accelerated expansion is through the presence of some vacuum energy. In the context of models with scalar field(s)-driven inflation, vacuum energy density appears naturally when the scalar field(s) acquire an effective potential U eff which has flat regions so that the scalar field(s) can "slowly roll" [22][23][24][25][26] and its/their kinetic energy can be neglected resulting in an energy-momentum tensor of the form T µν −g µν U eff .
With the discovery of the accelerating expansion of the present universe [27][28][29][30][31][32][33][34][35] it appears plausible that a small vacuum energy density, usually referred in this case as "dark energy", is also present even today. The two vacuum energy densities -the one of inflation and the other of the dark energy dominated universe nowadays, have however a totally different scale which demans a plausible explanation of how cosmological evolution may naturally interpolate between such two apparently quite distinctive physical situations.
Another parallel groundbreaking development alongside the quintessential inflationary cosmology is the advent of extended modified gravitational theories. The main motivation aims to overcome the limitations of the canonical Einstein's general relativity manifesting themselves in: (i) Cosmology -for solving the problems of dark energy and dark matter and explaining the large scale structure of the Universe [65][66][67]; (ii) Quantum field theory in curved spacetime -because of the non-renormalizabilty of ultraviolet divergences in higher loops [68][69][70][71][72][73]; (iii) Modern string theory -because of the natural appearance of higher-order curvature invariants and scalar-tensor couplings in low-energy effective field theories [74][75][76][77][78].
Various classes of modified gravity theories have been employed to construct plausible inflationary models: f (R)-gravity, scalar-tensor gravity, Gauss-Bonnet gravity (see [79,80] for extensive review); also recent proposals based on non-local gravity ([81] and references therein) or based on brane-world scenarios ( [82] and references therein). Let us recall the first early successful cosmological model based on the extended f (R) = R + R 2 -gravity producing the classical Starobinsky inflationary scalar field potential [2].
For a recent detailed work on quintessential inflation based on f (R)-gravity, where the role of dark matter is being played by axions, see Ref. [83].
A broad class of actively developed modified/extended gravitational theories is based on employing (one or more) alternative non-Riemannian spacetime volume-forms, i.e., metric-independent generally covariant volume-elements in the pertinent Lagrangian actions on spacetime manifolds with an ordinary Riemannian geometry, instead of (or alongside with) the canonical Riemannian volume-element √ −g d 4 x, whose density is given by the square-root of the determinant of the Riemannian metric √ −g ≡ − det g µν . Originally the formalism employing non-Riemannian volume-elements in generally-covariant Lagrangian actions as in Eq.(7) below was proposed in [84][85][86][87][88]. The concise geometric formulation was presented in [89,90]. A brief outline of the basics of the formalism of non-Riemannian volume-elements is given in Section 2 below.
This formalism was used as a basis for constructing a series of modified gravity-matter models describing unified dark energy and dark matter scenario [91,92], quintessential cosmological models with gravity-assisted and inflaton-assisted dynamical suppression (in the "early" universe) or dynamical generation (in the post-inflationary universe) of electroweak spontaneous symmetry breaking and charge confinement [93][94][95], as well as a novel mechanism for the supersymmetric Brout-Englert-Higgs effect (dynamical spontaneous supersymmetry breaking) in supergravity [89].
In the present paper our first principal goal is to analyze (Section 3 below) the close interplay between cosmological dynamics and the pattens of (spontaneous) symmetry breaking along the history of universe, which itself is one of the most important paradigms at the interface of particle physics and cosmology. We will extend our construction, started in [93], of a modified gravity model coupled to (the Higgs part) of the standard electroweak matter content (see, e.g. [96,97] besides the scalar "inflaton" field. The main aim here is to provide an explicit realization from first (Lagrangian action) principles of the remarkable proposal of Bekenstein [98] about the so called gravity-assisted dynamical generation of the Higgs effect -dynamical symmetry breaking of the electroweak SU(2) × U(1) symmetry -without introducing unnatural (according to Bekenstein's opinion) ingredients like negative ("ghost"-like) mass squared and quartic self-interaction for the Higgs field. Here we study the interrelation between the presence or absence of dynamical spontaneous electroweak symmetry breaking and the different stages of universe's evolution driven by the "inflaton" -triggering inflation in the "early" universe as well as representing quintessential variable dark-energy in the "late" universe.
It is shown that during inflation there is no spontaneous electroweak symmetry breaking and the Higgs field resides in its "wrong" vacuum state ("wrong" from the point of view of standard high-energy particle physics). The non-trivial symmetry-breaking Higgs vacuum is dynamically generated in the post-inflationary epoch.
To establish the connection of our non-Riemannian volume-element formalism and the purely affine formalism, our next task (Section 4) will be to represent the above quintessential inflationary model with a dynamical Higgs effect in the form of a no-metric purely affine (Eddington-type) gravity.

The Essence of the Non-Riemannian Volume-Form Formalism
Volume-forms define volume-elements (generally covariant integration measures) over differentiable manifolds M, not necessarily Riemannian ones, so no metric is a priori needed [139]. They are given by nonsingular maximal-rank differential forms ω on M (for definiteness we will consider the case of where: The conventions for the alternating symbols ε µνκλ and ε µνκλ are: ε 0123 = 1 and ε 0123 = −1. The volume-element density (integration measure density) Ω transforms as scalar density under general coordinate reparametrizations. In standard general-relativistic theories the Riemannian spacetime volume-form is defined through the tetrad canonical one-forms e A = e A µ dx µ (A = 0, 1, 2, 3): which yields: Instead of √ −g d 4 x we can employ another alternative non-Riemannian volume-element as in (1)-(2) given by a non-singular exact 4-form ω = dA where: Therefore, the corresponding non-Riemannian volume-element density is defined in terms of the dual field-strength scalar density of an auxiliary rank 3 tensor gauge field A µνκ .
In the next Section we will discuss in some detail the properties of a quintessential inflationary model coupled to a truncated version of the electro-weak particle content carrying the standard electro-weak SU(2) × U(1) symmetry. Namely, for simplicity we retain only a Higgs-like scalar field and discard the electro-weak gauge fields and fermions.
Before proceeding let us note the following important property of Lagrangian action terms involving (one or more) non-Riemannian volume-elements: The equations of motion of (7) w.r.t. the auxiliary tensor gauge fields A (j) µνκ according to (6) imply: where M j are free integration constants not present in the original action (7). This illustrates the significant advantage of the non-Riemannian volume-element formalism over the "Lagrange-multiplier gravity" method [140], which appeared a decade later and which requires picking a priori some ad hoc constant as opposed to the dynamical appearance of the arbitrary integration constants (8). For further advantages of the non-Riemannian volume-element formalism, see the above remarks. A characteristic feature of the modified gravitational theories (7) is that when starting in the first-order (Palatini) formalism all non-Riemannian volume-elements Φ(A (j) ) yield almost pure-gauge degrees of freedom, i.e. they do not introduce any additional physical (field-propagating) gravitational degrees of freedom except for few discrete degrees of freedom with conserved canonical momenta appearing as arbitrary integration constants M j . The reason is that the modified gravity action (7) in Palatini formalism is linear w.r.t. the velocities of some of the components of the auxiliary gauge fields A (j) µνκ defining the non-Riemannian volume-element densities, and does not depend on the velocities of the rest of auxiliary gauge field components. The (almost) pure-gauge nature of the latter is explicitly shown in [90,93] (appendices A) employing the standard canonical Hamiltonian treatment of systems with gauge symmetries, i.e., systems with first-class Hamiltonian constraints a'la Dirac [141,142].

Quintessential Inflationary Model with Dynamical Higgs Effect
Our starting point is the following specific example of the general class of modified gravity models [89,90,[93][94][95]143,144]) involving several non-Riemannian volume-elements (using units with 16πG Newton = 1): Here the following notations are used: • The scalar curvature R(g, Γ) = g µν R µν (Γ) is given in terms of the Ricci tensor R µν (Γ) in the first-order (Palatini) formalism: defined by the affine connection Γ λ µν a priori independent of the metric g µν . • The non-Riemannian volume-element densities Φ 1 (A), Φ 2 (B), Φ 0 (C) are defined as in (6): • φ is a neutral scalar "inflaton" and σ ≡ (σ a ) is a complex SU(2) × U(1) iso-doublet Higgs-like scalar field with the isospinor index a = +, 0 indicating the corresponding U(1) charge. The corresponding kinetic energy terms in (9) read: and is a canonical mass term for the Higgs-like field, i.e., neither negative ("ghost-like") mass-squared term nor quartic self-interaction are introduced unlike the case in the standard electro-weak model [96,97]. • f 1,2 and α are dimensionful coupling constants in the "inflaton" potential. Λ 0 is small dimensionful constant which will be identified in the sequel with the "late" universe cosmological constant in the dark energy dominated accelerated expansion's epoch.
The specific form of the action (9) is fixed by the requirement of global Weyl-scale invariance under: where scaling parameter λ = const. The importance of global scale symmetry within the context of non-Riemannian volume-element formalism has been already stressed in the first original papers (see [86]), where in particular models with spontaneously broken dilatation symmetry have been constructed along these lines, which are free of the Fifth Force Problem [145].
Varying the action (9) w.r.t. g µν , Γ λ µν , A µνλ , B µνλ , C µνλ , φ and σ a , yield the following equations of motion, respectively: Eqs.(18)- (19) are special cases of the general Eq.(8) discussed above. Here M 1,2 and χ 2 are arbitrary (dimensionful and dimensionless, respectively) integration constants, with M 1,2 triggering a spontaneous breaking of the global Weyl-scale symmetry (15). Taking the trace of Eq.(16) and comparing with (18)- (19) we find for the ratio of volume-element densities: On the other hand, following analogous derivation in [86], Eq.(17) yields a solution for Γ µ νλ as a Levi-Civita connection: w.r.t. to a Weyl-conformally rescaled metric:ḡ with χ 1 (φ, σ) as in (22). Conformal transformation g µν →ḡ µν via (24) convert the modified gravity action (9) into the physical Einstein-frame action (objects in the Einstein-frame indicated by a bar): with an effective Einstein-frame scalar field potential: which is entirely dynamically generated due to the appearance of the free integration constants M 1,2 and χ 2 (18)- (19). As discussed in [90,94,95,146] the scalar potential U eff (φ, σ) (26) has a remarkable feature -it possesses two (infinitely) large flat regions as a function of φ at σ a = fixed (see the graphical representation on Fig.1) with the following properties: • (a) (-) flat "inflaton" region for large negative values of φ (and σ a -finite) corresponding to the "slow-roll" inflationary evolution of the "early" universe driven by φ. Here the effective potential (26) reduces to (an almost) constant value independent of the finite value of σ a -this is energy scale of the inflationary epoch: Thus, in the "early" universe the Higgs-like field σ a must be (approximately) either massless or constant with no non-zero vacuum expectation value, therefore there is no spontaneous breaking of SU(2) × U(1) symmetry. Moreover, in fact as shown in the Remark below, σ a does not participate in the "slow-roll" inflationary evolution, so σ stays constant there equal to the "false"vacuum value σ = 0. • (b) (+) flat "inflaton" region for large positive values of φ (and σ a -finite) corresponding to the evolution of the post-inflationary ("late") universe, where: acquires the form of a dynamically induced SU(2) × U(1) spontaneous symmetry breaking Higgs potential with a Higgs "vacuum" at: where the parameters are naturally identified as: in terms of the electro-weak energy scale M EW ∼ 10 −16 M Pl . • Thus, the residual cosmological constant Λ 0 in (28) has to be identified with the current epoch observable cosmological constant (∼ 10 −122 M 4 Pl ) and, therefore, according to (27) the integration constants M 1,2 are naturally identified by orders of magnitude as since the latter case the order of magnitude of the vacuum energy density in the (-) flat region (27) becomes: which conforms to the Planck Collaboration data [147,148] for the "early" universe's energy scale of inflation being of order 10 −2 M Pl .
• Here the order of magnitude for f 2 is determined from the mass term of the Higgs-like field σ in the (+) flat region resulting from (28) upon expansion around the Higgs vacuum (σ = σ vac + σ): which implies that: • Let us specifically note that the viability of the present model (in a slightly simplified form without the Higgs scalar) concerning confrontation with the observational data has already been analyzed and confirmed numerically in Ref. [146]. In particular, a graphical plot of the evolution of r (tensor-to-scalar ratio) vs n s (scalar spectral index) has been provided there.
To conclude this section, we see that thanks to the remarkable dynamically generated scalar potential (26) the "inflaton" φ plays the role both of driving "slow-roll" inflationary dynamics in the "early" universe, as well as it plays the role of a quintessential variable dark-energy field triggering slowly accelerating de Sitter expansion in the "late" universe.
Accordingly, gravity-inflaton dynamics generates dynamically spontaneous SU(2) × U(1) symmetry breaking -Higgs effect -in the post-inflationary epoch, whereas it dynamically suppresses spontaneous symmetry breaking during inflation in the "early" universe. Namely, the dynamical transition from the "false" Higgs vacuum to the genuine electroweak spontaneously broken vacuum is driven by the inflaton evolving from (large) negative values (on the "(-)" flat region (27) of the scalar potential (26)) to (large) positive values (on the "(+)" flat region (28) of the scalar potential (26)).
Thus, our scale invariant modified gravity model (9) in its Einstein-frame representation (25)-(26) turns out to be an explicit implementation of Bekenstein's idea [98] about a gravity-assisted spontaneous symmetry breaking of electro-weak (Higgs) type without invoking negative mass squared and a quartic Higgs field self-interaction unlike the canonical case in the standard particle model [96,97].

Eddinton-type No-Metric Gravity and Quintessential Inflation
Let us now consider a generic model of gravity, with some Riemannian metricḡ µν and with the ordinary Riemannian volume-element √ −ḡ within the first-order (Palatini) formalism, interacting with a multi-component scalar field φ A , A = 1, . . . , N (using again units with 16πG Newton = 1): where the Ricci tensor R µν (Γ) is the same as in (10), and h AB (φ) indicates some "metric" in the scalar field target space (in the present case it will be just a unit matrix). The equations of motion w.r.t g µν , φ A and Γ λ µν read accordingly: Following again the analogous derivation in [86], the solution of Eq.(43) is that Γ λ µν becomes the canonical Levi-Civita connection w.r.t.ḡ µν : Eqs.(40)- (41) can be equivalently written as: that is, the metricḡ µν in (39) is expressed entirely in terms of the affine connection and the matter field. Now, we will show that the gravity-matter theory (39) is equivalent, in a sense of producing the same equations of motion (41)- (44), to the following Eddington-type purely affine gravity theory: i.e. (46) does not involve at all a Riemannian metric.
The above derivation of purely affine gravity interacting with multi-component scalar fields appeared previously in [149]. Historically, this formulation was proposed for the first time in [119] in the special case of a single Klein-Gordon field with U(φ) = 1 2 m 2 φ 2 , see also [150]. Applying the above established equivalence between the models (39) and (46) to the initial modified gravity action (9) and its Einstein-frame representation (25) with U eff (φ, σ) as in (26), analyzed in Section 3 above, we find that the following specific Eddington-type purely affine no-metric gravity model: actually describes a quintessential inflationary dynamics with dynamically generated Higgs effect in the post-iflationary epoch with all the properties discussed in section 3. The metric g µν in the initial modified scale-invariant gravity action (9) with non-Riemannian volume-elements, taking into account relation (45) applied for the Einstein-frame metric (24) where U(φ) = U eff (φ, σ) (26) and using the on-shell relations (19) and (22), is identified as: with R µν (Γ) as in (10), and χ 1 (φ, σ) and U eff (φ, σ) explicitly given in (22) and (26), respectively: 1

Conclusions
In the present paper we have employed two fundamental concepts -of non-Riemannian metric-independent spacetime volume-elements and of (global) scale invariance to construct a self-consistent model of modified gravity coupled to a neutral scalar "inflaton" and to a Higgs-like SU(2) × U(1) iso-boublet scalar possessing the following extraordinary features: (a) In the physical Einstein frame, thanks to a dynamical generation of a remarkable scalar potential with two long flat "inflaton" regions with vastly different heights, the model describes a plausible quintessential inflationary scenario, driven by the "inflaton", with a "slow-roll" inflationary stage in the "early" universe and a slow accelerating de Sitter expansion in the "late" universe; (b) This model provides an explanation of the interplay between cosmological dynamics and the patterns of symmetry breaking during the evolution of the universe. Namely, we find an explicit realization from first (Lagrangian-action) principles of the noteworthy proposal of Bekenstein from 1986 about "gravity-assisted" dynamical Higgs-like spontaneous symmetry breakdown (Higgs effect). We exhibit gravity-"inflaton" suppression of the Higgs effect during inflation,i.e., no electroweak spontaneous breakdown there), whereas in the post-inflationary epoch a Higgs-type symmetry breaking potential is dynamically created.
(c) The coupling constants in the initial modified gravity action are naturally identified as powers of the standard electroweak mass scale.
(d) It is shown how to represent the above quintessential inflationary model with a dynamical Higgs effect in the form of a no-metric purely affine (Eddington-type) gravity.
A next important task is to study in some detai, within the present quintessential inflationary scenario with a dynamical Higgs effect, the numerical solutions for the basic inflationary observables (scalar power spectral index, tensor-to-scalar ratio, etc.) extending the numerical analysis from Ref. [146] (where the Higgs field was absent). Since in the present scenario the Higgs field during (most of the) inflation resides in its "false" vacuum, the significant impact of Higgs field dynamics will occur after end of inflation when the "inflaton" starts to generate the non-trivial Higgs symmetry breaking potential.