Cosmic Inflation in an Extended Non-Commutative Foliated Quantum Gravity: The Wave Function of the Universe

The chronological evolution of the Universe comprises different phases, of which the recombination and decoupling periods stand out, as well as a ‘primordial’ era characterized by inflation, followed by the dominant phases of radiation, matter and, currently, a dark energy dominated period, which is widely assumed to be the main cause of the accelerated cosmic expansion.

In this work, we focus mainly on the inflationary period, proposing a novel extended approach to the recently developed non-commutative Riemannian foliated branch-cut quantum gravity (BCQG) [1].

We investigate the effects of imposing an extended enhanced non-commutative symplectic deformation of the conventional Poisson algebra on a topological manifold that conveys a natural isomorphic setting of a canonically conjugate triad of dual vector spaces. The extended triad approach encompasses the cosmic scale factor BCQG and two complementary dual quantum counterparts, outlined in the Hermann Weyl’s perfect fluid domain and the inflaton theory of exponential expansion of the early Universe [2]. This scenario allows the coexistence of the quantum domain of branch-cut gravity with an inflationary mechanism that, although controversial for assuming an entire Universe starting essentially from nothing, has had remarkable success in explaining relevant qualitative and quantitative properties of the Universe [2,3]. Among them, the approach to the origin of the large-scale structure of the cosmos stands out, based on quantum fluctuations in the microscopic inflationary region, amplified to cosmic size, becoming the seeds for the structure growth of the Universe.

Furthermore, based on this extended scenario, additional conceptual elements, such as density perturbations, grounded on heuristic derivation of the scale invariance of the primordial spectrum of the Universe, as well as predictions of an eternal inflation and even primordial chaotic mechanisms associated with the origin of the Universe, contribute to broadening the descriptive basis of the original BCQG formulation in order to incorporate elements of standard inflation conceptions.

Moreover, in view of the peculiar characteristics of BCQG in overcoming the primordial singularity, this formulation makes it possible, - due to the complex character of the non-commutative symplectic extended composition -, to make predictions about the evolutionary scenarios of the mirror Universe by means of complex- and complex-conjugated Friedmann-type BCQG equations materialized in a super-Hamiltonian that describes the wave function of an evolving Universe contemplating two unfolding phases, from negative to positive cosmological time. This brings unique features to the main BCQG scenario of cosmic evolution, with a parallel, mirror-like, evolutionary Universe adjacent to ours, nested in the fabric of space and time, with its evolutionary process in the negative sector of cosmological thermal time corresponding to a contraction process followed by a continuous expansion in its positive temporal sector. These characteristics open up a range of opportunities involving a reverse process to cosmic inflation in the mirror sector.

Returning to the aforementioned scenario, the assumed continuous contraction of the mirror Universe is accompanied by an increase in temperature and a decrease in entropy before reaching the cosmic transition region. In the subsequent domain to the transition region, the continuous expansion of the Universe is accompanied by a systematic decrease in temperature and an increase in entropy corresponding to the positive sector of the complex cosmological time. Using this formulation, we describe in the present investigation the dynamical evolution of the wave function of the Universe, and then, in another contribution, in the final stages of preparation [4], we describe the evolution of the cosmic scale factor and its complementary dual quantum counterparts, with unprecedented results.

The proposed theoretical approach follows the branch-cut quantum gravity (BCQG) framework [1,5,6], based on the WheelerDeWitt [7] and the Hořava-Lifshitz [8] formulations, incorporating elements of symplectic geometry and non-commutative algebra [1] to investigate the interplay between small-scale quantum effects and large-scale cosmic evolution. BCQG provides moreover an alternative descriptive approach for cosmic evolution in comparison to conventional inflationary theories by building a quantum gravity environment through a fundamental restructuring of the conventional topological spacetime geometry rather than relying on specific initial conditions and ad hoc mechanisms.

Additionally, as previously stated [1], this approach offers a fresh perspective on the generation of relic gravitational waves and a possible solution to unresolved cosmological puzzles, such as the horizon and flatness problems. As previously mentioned [1], the realization of a non-commutative algebra structure induces the capture of short and long spatio-temporal scales, driving not only the evolutionary dynamics of the Universe’s wave function and the cosmic scale factor but also a reconfiguration of matter on small and intermediate scales, inducing the generation of relic gravitational waves, a topic to be addressed in a future investigation.

The symplectic non-commutative BCQG approach, as previously emphasized [1], was assembled by means of a deformation of the conventional Poisson algebra, and enhanced with a symplectic metric, based on the Faddeev-Jackiw two-fields quantization formulation [9,10] and extended in this contribution to a triad of dual and complementary quantum fields. Motivated by the compactification study of the properties of topological spaces, the non-commutative symplectic algebra is reflected in the validity of differential graded algebra with finite dimensional cohomology (compactification) and a rational Poincaré duality [11,12], which comprises harmonic foliations with minimal leaves on a Riemannian manifold [13], a crucial topic to embrace BCGQ. In other words, the Poisson brackets have the form $\{q_i,p_j\}=g_{ij}$, and because of the antisymmetry of the Poisson brackets, the $g_{ij}$ automatically correspond to a symplectic block structured metric, no further assumptions required (for the details see [1]).

As previously stressed [1], BCQG comprises two foliation levels. The first level may be understood by assuming a D-dimensional Euclidean manifold $\mathcal{M}$ associated to a given analytically continuation to the complex plane metric $g^{\left[ac\right]}\left(x\right)$, carrying coordinates $x_{\alpha }$. Following canonical steps, a preferred time-direction may be set up by defining a time function $\tau \left(x\right)$ and assign a specific time $\tau$ to each spacetime coordinate $x\left(\tau \right)$. Then we may decompose the manifold $\mathcal{M}$ into spatial slices ${\Sigma }_{{\tau }_i}\equiv \{x:\tau \left(x\right)={\tau }_i\}$ encompassing all points $x\left(\tau \right)$ with the same ‘time-coordinate’. At this point, caution should be taken. This is because there is no time variable in the WDW equation in its original formulation, an intriguing feature for an equation with the ambition of describing the dynamics of quantum spacetime. Regardless of the long debates raised by the WDW equation about the nature of time, it is important to emphasize that $\tau$ in this conception should be highlighted as a non-dynamical parameter. The gradient of the time function can be used to define a vector normal to the spatial slices, with the lapse function ensuring normalization with respect to the metric, and the lapse defined in terms of the time coordinates $\tau$ and ${\tau }_{\Sigma }$ related to ${\Sigma }_{{\tau }_i}$. In the projectable Hořava-Lifshitz gravity, N is restricted to a Euclidean time function only (see [14,15]). In our proposal, this restriction is overcome. In short, similarly to what occurs in general relativity based on the ADM-foliation tecnique [14], foliation in BCQG consists of slicing up the spacetime into three-dimensional spacelike hypersurfaces of constant time, relative to the global time function. For a coordinate (or stationary) observer at ${\Sigma }_{{\tau }_i}$, his spatial coordinates remain constant after a time ‘lapse’ at ${\Sigma }_{{\tau }_i+d{\tau }_i}$. The second foliation level of BCQG is characterized by an analytically continued Riemannian foliation which corresponds to the reciprocal of a complex multi-valued function, the natural complex logarithm function ${ln}^{-1}\left[\beta \left(t\right)\right]$, a helix-like superposition of cut-planes, which correlates Riemann sheets, with an upper edge cut in the n-th plane joined with a lower edge of cut in the (n + 1)-th plane. The BCQG Universe’s scale factor ${ln}^{-1}\left[\beta \left(t\right)\right]$ maps an infinite number of Riemann sheets onto horizontal strips, which represent in the branch-cut cosmology the evolution of the time-parameter dependence on horizon sizes. The patch sizes in turn map progressively the various branches of the ${ln}^{-1}\left[\beta \left(t\right)\right]$ function which are glued along the copies of each upper-half plane with their copies on the corresponding lower-half planes. In the branch-cut cosmology, the cosmic singularity is replaced by a family of Riemann sheets in which the scale factor shrinks to a finite critical size, - the range of ${ln}^{-1}\left[\beta \left(t\right)\right]$, associated to the cuts in the branches, shaped by the $\beta \left(t\right)$ function —, well above the Planck length. In the contraction phase, as the patch size decreases with a linear dependence on ${ln}^{-1}\left[\beta \left(t\right)\right]$, light travels through geodesics on each Riemann sheet, circumventing continuously the branch-cut, and although the horizon size scales with ${ln}^{-1}{\left[\beta \left(t\right)\right]}^{ϵ}$, the length of the path to be traveled by light compensates for the scaling difference between the patch and horizon sizes. Here, $ϵ\left(t\right)$ represents the dimensionless thermodynamical connection between the energy density E and the pressure P of a perfect fluid thus enabling the fully description of the equation of state (EoS) of the system. Under these conditions, causality between the horizon size and the patch size may be achieved through the accumulation of branches in the transition region between the present state of the Universe and the past events. Conventionally, the theory deals exclusively with finite-dimensional real symplectic spaces. The BCQG in turn extends the ontological domain of general relativity to the complex plane. In BCQG, the presence of a branch-cut and a branch point defines the domain of the scale factor ${ln}^{-1}\left[\beta \left(t\right)\right]$ of the analytically continuous Universe for the complex sector, overcoming the presence of a cosmic singularity. The BCQG offers additionally a theoretical alternative to inflation models [2,3], based on the mathematical augmentation technique and notions of closure and existential completeness [16], which have proved highly useful in both quantum mechanics [17,18,19] and pseudo-complex general relativity (pc-GR) [20], with direct physical and cosmological manifestations. For the sake of completeness, we present in the following foundational elements of the BCQG formalism [1,5,6].

The complete line element of the BCQG quantum gravity, resulting from the complexification of the FLRW metric [21,22,23,24], in line with the ADM-foliation [14], may be expressed as [25,26,27]In this expression, $\left[\mathrm{ac}\right]$ denotes analytical continuation to the complex plane, where r and t represent real and complex space-time parameters, respectively, and k denotes the spatial curvature of the multiverse, corresponding to negative curvature ($k=-1$), flat ($k=0$), or positively curved spatial hypersurfaces ($k=1$). ${ln}^{-1}\left[\beta \left(t\right)\right]$ represents the foliated scale factor, and $N\left(t\right)$ denotes the lapse function. The gauge invariance of the action in general relativity yields a Hamiltonian constraint that requires a gauge-fixing condition on the lapse function (see [28]). We extend the gauge fixing constraints further to the algebraic structure of the BCQG action.

At this point it is important to emphasize that BCQG does not require any external mechanism for its realization, nor even the assumption of a Universe created from nothing. BCQG is based on a peculiar kind of topological quantization. In simplified terms, topology involves the study of the preserved properties of a geometric object under continuous deformations, such as stretching, twisting, crumpling and bending. In this sense, the topology and geometry of spacetime play a fundamental role in our understanding of quantum gravity. In general, a topological quantum field theory comprises topological invariants, properties that are insensitive to changes in the shape of spacetime, where the correlation or partition functions of the system, — computed by the path integral of metric-independent action functionals —, stand out. A relevant aspect of quantum gravity approaches, a topic of intense debate, is their subdivision into two large classes, depending on whether they are spacetime background-dependent or background-independent. Spacetime in general relativity, unlike the Newtonian and special relativity Minkowski approaches, is not fixed but it is dynamical, in the sense that its shape depends on its matter and energy content, which implies no privileged spacetime background [29].

Background independence and covariance, concepts intertwined with the diffeomorphism invariance of gravity, elevated to the status of a fundamental principle, represent basic building blocks of general relativity [30] and in the quantization of gravity [31]. As highlighted by Lee Smolin [31], general relativity was conceived as a principle theory instead of a constructive theory. Similarly, BCQG follows the same logical conception developed by Albert Einstein, that is, instead of following the fundamentals of a constructive theory, based on the description of particular phenomena, fields or particles that constitute nature, specified in terms of dynamical equations of motion that the constituents obey, BCQG in turn, was conceived as a principle theory, guided this way by universal principles. Although this statement should imply future advanced studies, BCQG should obey in consequence the same fundamental principles as general relativity. More precisely, BCQG should fit into the context of a background-independent covariant theory that obeys diffeomorphism invariance. Following the logical principles of a principled theory, the concept of quantum topology would therefore apply to BCQG with respect to the presence of topological invariants, properties that are insensitive to changes in the shape of spacetime, a topic that requires further study. In this context, future studies should focus on identifying candidates for topological invariants of BCQG, among which the torsion tensor stands out as the most relevant. This is because results presented in this contribution indicate that spacetime torsions would represent essential ingredients to drive the acceleration of the Universe as well as the generation of relic gravitational waves.

As pointed out [32], Einstein’s gedenken elevator experiment broughted to the Equivalence Principle that asserts the local equivalence of inertial and gravitational effects, leading him to associate the gravitational field with the space-time metric. And since the gravitational field is a dynamical object, the metric must therefore also be dynamic, that is, should also depend on the composition of matter and energy in the Universe. At this point, conventional topological quantization finds, in BCQG, a peculiar aspect. The evolution of the scale factor in the convencional cosmology is governed by the Friedmann equations which depends on the the total energy density and pressure of the universe. In this sense, the scale factor characterizes the global properties of the Universe, representing a crucial element in determining the overall size and shape of the metric configuration of the universe. Assuming the FLRW metric, in order do not violate the cosmological principle, preserving the homogeneity and isotropicity of the Universe, in conventional cosmology, the dimensionless scale factor is a real scalar quantity and has only a temporal dependence.

In BCQG in turn, the ${ln}^{-1}\left[\beta \left(t\right)\right]$ function corresponds to a helix-like superposition of cut-planes, the Riemann sheets, with an upper edge cut in the n-th plane joined with a lower edge of cut in the (n + 1)-th plane. ${ln}^{-1}\left[\beta \left(t\right)\right]$ maps an infinite number of Riemann sheets onto horizontal strips, which represent in the branch-cut cosmology the time evolution of the time-dependent horizon sizes. The patch sizes in turn maps progressively the various branches of the ${ln}^{-1}\left[\beta \left(t\right)\right]$ function which are glued along the copies of each upper-half plane with their copies on the corresponding lower-half planes. In the branch-cut cosmology, the cosmic singularity is replaced by a family of Riemann sheets in which the scale factor shrinks to a finite critical size, — the range of ${ln}^{-1}\left[\beta \left(t\right)\right]$, associated to the cuts in the branch cut, shaped by the $\beta \left(t\right)$ function —, well above the Planck length.

In a formulation based on the multiverse concept [33] and the analytic continuation technique in complex analysis applied to the FLRW metric [21,22,23,24], Friedmann-type equations for a complexified version of the $\Lambda$CDM model ($\Lambda \ne 0$) yield (see [25,26,27]): In these expressions, $\Lambda \left(\tau \right)$ represents the BCQG cosmological constant. These equations due to the nature of BCQG allow the construction of a set of complex conjugate Friedmann’s-type equations. These complex and complex-conjugate Friedmann-type equations materialize in a super-Hamiltonian developed below.

The starting point of this study is the commutative BCQG action which depends on the branching scalar curvature of the Universe, $\mathcal{R}$, and on its covariant derivatives, ${\nabla }^2\mathcal{R},{\nabla }_i{\mathcal{R}}_{jk},{\nabla }^i{\mathcal{R}}^{jk}$, in different orders [1,5,6], and whose development was based on the formulation of Hořava-Lifshitz [8]:In this expression, $g_i$ represent running coupling constants, $M_P$ denotes the Planck mass, and by imposing a maximum symmetric surface foliation [1], the branching Ricci components of the three dimensional metrics are given as: where the variable change $u\left(t\right)\equiv {ln}^{-1}\left[\beta \left(t\right)\right]$, with $du\equiv d{ln}^{-1}\left[\beta \left(t\right)\right]$, was introduced. Moreover, the trace of the extrinsic curvature tensor $K=K^{ij}{g}_{ij}$, represented by $K_{ij}$ [1,5,6,34], is given as:Exploring the ADM-type formalism domain, for the metric given by equation (1), adopting the $N_i=0$ gauge, the three-metric $h_{ij}\to g_{ij}$ corresponds to a projectable analytically continued quantum gravity formulation:Applying standard canonical quantization procedures to the BCQG action, the canonical momentum reads and the resulting Hamiltonian density becomesIn this expression, the running coupling constants depict the matter-energy contributions of radiation ($g_r$), baryon matter ($g_m$), curvature ($g_k$), quintessence ($g_q$), cosmological constant dark energy ($g_{\Lambda })$, and stiff matter ($g_s$) (for the details see [35,36] and references therein).

In order to extend the BCQG symplectic non-commutative algebra, we embrace a similar analytical strategy adopted in the elaboration of the non-commutative two-fields BCQG. We base our strategy firstly on the construction of an extended three-field commutative BCQG manifold, composed of an extended mini-super-space of variables ($u\left(t\right),v\left(t\right),\varphi \left(t\right)$). Then, this extended commutative manifold is related to a non-commutative structure, based on an extended Faddeev-Jackiw symplectic-type phase space transformation applied to the mini-super-space of variables $u\left(t\right),v\left(t\right),\varphi \left(t\right)$. The insertion of the field $\varphi \left(t\right)$ in the non-commutative symplectic formulation substantiates this way the extension of the BCQG scope in order to make contact with standard inflation approaches.

As a result of the previously adopted gauge-transformations, from equation (11) we arrive at the following super-Hamiltonian:In this expression, for notation simplicity, we eliminate the tilde identification of the commutative variables. By adopting the reverse mapping path proposal, the above equation materializes the effects of reconfiguration of the original super-Hamiltonian through the imposition of a non-commutative algebra. The resulting equation, although dependent on the original commutative variables, highlights this reconfiguration through the imposition of a structural composition that inserts new dynamic components into the original formalism, modulated by the parameters $\sigma$, $\chi$, $\gamma$, $\alpha$. Unlike a conventional symplectic transformation, this procedure makes it possible to identify, in a comprehensible way, the striking outcome of the non-commutative algebraic transformations when compared to a standard formulation.

From expression (33), making the quadratic terms explicit, we obtainCombining similar terms, from this expression we obtainAssuming the perfect fluid radiation condition for the early Universe matter content, $\omega =1/3$ and naturalness condition, the corresponding expressions for the coefficients ${\mathcal{C}}_i(i=1,...,6)$ are shown in Table 1.

Canonical quantization procedures applied to the Hamiltonian (35) results in the the wave equation where $\psi (u,v,\varphi )$ denotes the wave-function of the Universe. In this equation, the variables $u\left(t\right)$, $v\left(t\right)$, and $\varphi \left(t\right)$ along with their corresponding conjugate momenta $p_u$, $p_v$, and $p_{\varphi }$ acquire the status of quantum linear operators which act on the wave-function of the universe:Following the steps outlined in Appendix B, equation (36) is reduced to the canonical form (for the details see [49]):

Exploiting a relevant feature of the adopted canonical transformation (see Appendix B), which can be summarized in the freedom of choice of canonical variables, we opt for the following associations, $u\to \eta$, $v\to \xi$ and $\varphi \to \phi$. Evidently, the connections imposed by the relations (A7), (A8), (A9) and (A10) attribute to each variable $\eta \left(t\right)$, $\xi \left(t\right)$ and $\phi \left(t\right)$ a new identity when compared with their original counterparts, $u\left(t\right)$, $v\left(t\right)$ and $\varphi \left(t\right)$, forming a triplet of canonical complementary dual-conjugate quantum fields. Complementarity in conventional quantum mechanics is commonly interpreted in terms of duality and opposition, considering conjugate fields as dual and opposite. The formulation of quantum mechanics is shaped by complex, dual, and complementary Hilbert spaces. Quantum complementarity unifies duality and opposition in a consistent way, thus underpinning the physical world. This character of dual Hilbert spaces extends to quantum fields, when they experience sets of transformations in a non-commutative algebra, attributing to them new identities that materialize in the sharing of identities in comparison to fields originally belonging to a commutative algebraic structure. The concept of entanglement, coined in quantum mechanics to designate attributes of correlation related to conjugate variables, establishes that the quantum state of each field of a given group cannot be described independently of the state of the others, even when the particles are separated by large distances. In our view, we identify these aspects of quantum entanglement with the attributes of the present correlation, which fundamentally characterizes a non-commutative symplectic transformation of coordinates.

In the reverse Faddeev-Jackiw formalism proposed, the variables $u\left(t\right)$, $v\left(t\right)$, and $\varphi \left(t\right)$ are non-commutative, and after the symplectic transformation, the new canonical variables $\xi \left(t\right)$, $\eta \left(t\right)$, and $\varphi \left(t\right)$, — and despite conforming a commutative set of variables —, they obey a set of non-commutatively structured equations, shaped by the corresponding parametric attributes of the Poisson-type non-commutative algebra. In summary, the resulting set of variables, $\eta \left(t\right)$, $\xi \left(t\right)$, and $\phi \left(t\right)$ are canonically conjugate dual and complementary variables, which span reciprocal spaces, so the following relation between these variables holds: and

The ideal fluid condition for the radiation era, assuming the $\alpha$ parameter is a real quantity, $\alpha =1/3$, less a constant allows the variables separation of equation (38) in the form with $\psi (\eta ,\xi ,\phi )\equiv \psi \left(\eta \right)\psi \left(\xi \right)\psi \left(\phi \right)$, $g_m^{\prime }\equiv g_m-1/3$ andThe parameters $\gamma$ ans $\varsigma$ may be assumed as real or imaginary quantities. In the first case, equations (41) and () may be cast as In the second case, assuming $\gamma =i\left|\gamma \right|$ and $\varsigma =i\left|\varsigma \right|$, equations (41) and () become:

Quantum field theory in non-commutative spacetime leads to an uncertainty relation for coordinates analogous to the Heisenberg uncertainty principle, whose lower bound has given rise to the minimum scale problem. An unique feature of non-commutative field theory is the realization of a mixing between ultraviolet and infrared radiation (UV/IR), which characterizes an interrelation between short and long-range scales, absent in commutative quantum field theories (see for instance [11]). The theoretical problems above were also identified by Seiberg–Witten [57]. On basis of the weak gravity conjecture, the notion of hierarchical UV/IR mixing was implemented with scalar fields by Lüst and Palti [58]. In addition, the deep-low IR and far-high UV connection, according to Craig and Koren, by studying IR dynamics from UV divergences, in the context of UV/IR mixing, could even solve (part of) the hierarchical naturalness problem [59].

Different hypotheses can be raised to explain the cosmic inflation acceleration. A compelling possibility emulates from the electroweak hierarchy problem, based on an inspiring article [60]. The authors, in their study of perturbative dynamics of a particular set of non-commutative field theories, found an intriguing mixing of the ultraviolet (UV) and the infrared (IR) domains, identified with the observation that high energy UV virtual particles in loops produce non-analyticity at low momentum, interpreted as IR divergences. This finding, a realization of the mixing between the the short-range, high-energy ultraviolet (UV) and long-range, low-energy infrared (IR) phenomena, according to the authors, arises from the underlying non-commutativity, a phenomenon reminiscent of the channel duality of the double twist diagram in open string theory (for the details, see [60]). (For an alternative discussion see [61]).

In short, in the field theory prevails the conception that the origin of the IR/UR mixture results from the capture of large and small scales by the presence of a non-commutative structure. Translating this conception to quantum gravity, the non-commutative symplectic quantum structure of spacetime, through the capture of small-, medium- and large-scales, generates, on the one hand, a topological restructuring of space-time and, on the other, a restructuring of the primordial distribution of matter and energy.

Finally, the presence of the inflaton-type field $\phi$, similarly to the complementary quantum counterpart field $\xi$ of the the canonical BCQG cosmic scale factor $\eta$ enables the materialization of a spatial plenum, a transition space between the present Universe and its mirror counterpart completely filled with matter, instead of a spatial vacuum or a topological quantum transition region, providing a theoretically consistent medium required by seed conceptions.

In particular, the presence of these fields could provide the physics environment which allows, quantitatively, to determine how density fluctuations in the early Universe are imprinted in anisotropic signals indicating the presence of material seeds. In particular, the presence of these fields could provide imprints of gravitational waves generated before the recombination process and about the phase transition involving the early Universe and its mirror counterpart. Generated long before the recombination phase, the anisotropic distribution of these gravitational waves could reveal the effects of density perturbations seeded by the inflation process, providing new clues for a better understanding of the primordial density perturbations of our early Universe and thus improving our understanding of the inflation theory.