Pushing the Limits of Time Beyond the Big Bang Singularity: Scenarios for the Branch Cut Universe

Received 05 april 2021; Revised 11 april 2021; Accepted 17 april 2021

DOI: xxx/xxxx

ARTICLE TYPE Pushing the limits of time beyond the Big Bang singularity: Scenarios for the branch cut universe

1Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil In this contribution we identify two scenarios for the evolutionary branch cut uni- 2International Center for Relativistic verse. In the ﬁrst scenario, the universe evolves continuously from the negative Astrophysics Network (ICRANet), Pescara, Italy complex cosmological time sector, prior to a primordial singularity, to the positive 3Universidad Nacional Autónoma de one, circumventing continuously a branch cut, and no primordial singularity occurs Mexico (UNAM), México City, México in the imaginary sector, only branch points. In the second scenario, the branch cut 4Frankfurt Institute for Advanced Studies (FIAS), J.W. von Goethe University and branch point disappear after the realisation of the imaginary component of the (JWGU), Hessen, Germany complex time by means of a Wick rotation, which is replaced by the thermal time. In 5 Unversidade Federal de Santa Maria the second scenario, the universe has its origin in the Big Bang, but the model con- (UFSM), Santa Maria, Brazil 6Laboratório de Geociências Espaciais e templates simultaneously a mirrored parallel evolutionary universe going backwards Astrofísica (LaGEA), Universidade Federal in the cosmological thermal time negative sector. A quantum formulation based on do PAMPA (UNIPAMPA), Caçapava do the WDW equation is sketched and preliminary conclusions are drawn. Sul, Brazil

Correspondence KEYWORDS: *Av. Bento Gonçalves, 9500 - Agronomia, Big Bang, General Relativity, Friedmann’s Equations, Big Bounce, WdW Equation Porto Alegre - RS, 91501-970. Email: [email protected]

1 THE ILLUSION OF TIME Time in physics is usually considered a fundamental variable, defined by its measurement as the reading of a clock. In Newton’s conception of the clockwork universe, evolving like opposition to this view, John Weeler and Bryce DeWitt devel- a mechanical perfect clock and whose movements of its gears oped in 1967 the so called WdW equation (DeWitt, 1967) arXiv:2105.08108v1 [gr-qc] 17 May 2021 are governed by the laws of physics, with an inherent pre- based on the audacious idea of physics without time, a theo- dictability, prevailed for more than three centuries, until a retical framework that sought to combine quantum mechanics revolutionary concept emerged, thanks to the genius mind of and general relativity, representing a step towards a consistent Hermann Minkowski(Minkowski, 1915), with profound con- theory of quantum gravity. sequences for our current understanding of its structure and More recently, Carlo Rovelli affirmed that the flow of time evolution. is an illusion and that our naive perception of it doesn’t corre- According to his view, instead of being considered sepa- spond to physical reality (Rovelli, 2019), a vision that is in tune rate entities (though intimately related), space and time were with Albert Einstein’s perception of time2. Carlos Rovelli also combined into a single continuum entity, the spacetime1. recently revisited the idea of ‘physics without time’ (Rovelli,

2Albert Einstein in a letter to the family of Michele Besso, his collaborator and closest friend, once wrote: “Now he has departed from this strange world a little 1In Minkowski’s own words: “Henceforth space by itself and time by itself are ahead of me. That means nothing. People like us, who believe in physics, know doomed to fade away into mere shadows, and only a kind of union of the two will that the distinction between past, present, and future is only a stubbornly persistent preserve an independent reality” (Minkowski, 1915). illusion.” (Einstein, 2020). 2 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer

2004, 2015; Rovelli & Smerlak, 2011) bearing in mind that, where ̄̂ denotes the time-ordering operator. In the continuum in accordance with the second law of thermodynamics, for- limit, we write the amplitude (xbtbðxata) as a path integral ward in time is the direction in which entropy increases, and in xb which we gain information, so the flow of time is a subjective i(x)∕` (xbtbðxata) ≡ xe . (3) feature of the universe, not an objective part of physical real- Ê x ity3. In this realm, in which the observable universe does not a show time reversal symmetry, events, rather than particles or This equation is the corresponding Feynman’s formula for fields, are the basic constituents of the universe, and the task of the quantum-mechanical amplitude (1) and represents the sum physics would be to describe the relationship between events. over all paths in configuration space with a phase factor con- These conceptions present some points of contact with a taining the action [x]. line of thought we recently developed (Vasconcellos, Had- jimichef, Razeira, Volkmer, & Bodmann, 2020; Vasconcellos 2.2 Wick rotation in statistical and quantum et al., 2021) where we applied the tools of singular semi- mechanics Riemannian geometry to push the limits of general relativity and time beyond a primordial singularity, giving rise to a In statistical mechanics, the quantum partition function Z(T ), branch cut universe. In this contribution we sought to identify which contains all information about the thermodynamical evolutionary scenarios for the branch cut universe. equilibrium properties of a quantum system, reads −Ĥ ∕k T −H( ̂p,̂x)∕k T Z(T ) ≡ T r e B ≡ T r e B . (4) 2 WICK ROTATION OF In this expression, T r(F̂) denotes the trace of the operator COSMOLOGICAL TIME F̂ = e−H( ̂p,̂x)∕kB T , (5)

Wick rotation is a well known theoretical method that encapsu- and kB is the Boltzmann constant. For a N-particle sys- lates a connection between quantum mechanics and quantum tem described by the Schrödinger equation for instance, the statistical mechanics and in another ground relativistic ﬁeld quantum-statistical system refers to a canonical ensemble. theory in Minkowski spacetime manifolds and Euclidean ﬁeld The quantum statistical partition function Z(T ) may be theory in Riemannian spacetime manifolds. related to the quantum-mechanical time evolution operator ZQM (tb − ta)

i t t Ĥ ` Z t t T r Û t , t T r e− ( b− a) ∕ , 2.1 Path integral formalism QM ( b − a) ≡ ( b a) = (6) The path integral formalism (Feynman & Hibbs, 1965) by making an analytical continuation of the time interval tb −ta describes the quantum transition amplitudes of the unitary time to the negative imaginary value using a Wick rotation: ̂ evolution operator, U(tb, ta) (a representation of the abelian tb − ta → −i`∕kBT. (7) group of time translations), between the localised quantum In quantum mechanics and quantum field theory, the Hamilto- mechanical states of a particle (x , t ) to (x , t ), with x and t a a b b nian density acts as the generator of the Lie group of time denoting space and time Cartesian coordinates. The matrix ele-  translations while in statistical mechanics the Hamiltonian ments of the quantum time evolution amplitudes, using bra’s represents a Boltzmann weight in an ensemble. The contour- ( x ) and ket’s ( x ) notation, read ⟨ bð ð a⟩ rotation from the real to the imaginary time-axis, results in a ̂ (xbtbðxata) = ⟨xbðU(tb, ta)ðxa⟩ tb > ta , (1) correspondence between the imaginary time component and For a system with a time-independent Hamiltonian operator, the inverse of the temperature, T and it can be understood as a Ĥ , the time evolution operator is simply realisation of the imaginary component of time. − i Ĥ (t −t ) ̂ ̂ ` b a U(tb, ta) = ̄e , (2) 2.3 Euclidean quantum gravity Euclidean quantum gravity refers to a quantum theory of Rie- mannian manifolds in which the quantisation of gravity occurs 3In general relativity, the reading of a clock is not given by the time variable t, but is instead expressed by a line integral depending on the gravitational field, in a Euclidean spacetime, generated by means of a Wick computed along the clock’s world-line , given as rotation. The corresponding gravitational path integral in the t presence of a field may be expressed as  = g (x, t) dx dx ; Ê √ d4x g R  = [g][]e∫ ð ð . (8) (see Rovelli (2015) for more details). Ê César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 3

Additional assumptions imposed to the manifolds as com- represents a dimensionless thermodynamical connection pactness, connectivity and boundaryless (no singularities), between the energy density (t) and pressure p(t) of a perfect make this formulation a strong candidate for overcoming the fluid thus enabling the fully description of the equation of state limitations presented by general relativity in the domain of (EoS) of the system. Positive pressure corresponds to > 3∕2, strong gravity, more precisely, the elimination of singulari- negative pressure to < 3∕2 and for a universe dominated by ties in extreme physical conditions. There are other techniques a cosmological constant, → 0. that sought to overcome these limitations of general relativity. In the limit in which the dimensionless thermodynamical Among these, we highlight the pseudo-complex general rel- connection obeys (t) → = constant, the integral (11) ativity, a very powerful technique based on pseudo-complex reduces to spacetime coordinates (Hess, 2017; Hess & Boller, 2020; −1 ln[(t)∕0] = −2 lim (t) dln(ln [ (t)]) Hess, Schäfer, & Greiner, 2015) with observational predictions (t)→ Ê given in (Schönenbach et al., 2014). ≃ −2 ln(ln−1[ (t)]) ⇒ ln (ln−1[ (t)])−2 ⇒ (t) ≃ 0 , (13) ln−2[ (t)] 3 COSMOGRAPHY IN AN UNIVERSE which corresponds to an analytically continued expression for WITH A BRANCH CUT the density of the branch cut universe. Applying complex conjugation to this expression we get The tracking of the analytically continued scale factor ∗ ln−1[ (t)] and the background cosmological Hubble rate ∗(t∗) = 0 . (14) ln−2( ∗(t∗)) Hac(t), analytically continued to the complex plane enable us to trace the evolutive paths of the branch cut universe from its initial stages to the present days (for the details see Vasconcel- 3.2 Horizons and cosmological curvatures los et al. (2021)). An event’s causality is limited to its frontal light cone since The scale factor ln−1[ (t)], a dimensionless quantity, information cannot travel faster than the speed of light. Light describes the change in sizes of portions of space (or patches) rays travel in null geodesics, so the following expressions due to the expansion or contraction of the branch cut universe. results for the analytically continued (ac) co-moving (cm), The Hubble parameter H (t) in turn measures the expansion ac cm(t), and proper (p), p (t), distances to the horizon: rate of the branch cut universe. We assume the observable uni- ac ac t t verse corresponds today to a patch of space with radius R(t0), ¨ cm cdt p −1 cdt and that the patch size of the branch cut universe at any other  (t) = ;  (t) = ln [ (t)] . ac Ê ln−1[ (t)] ac Ê ln−1[ (t¨)] period of time is given by tP tP (15) ln−1[ (t)] ln( ) R(t ) = 0 R(t ) . (9) We also develop expressions for the analytically continued −1 0 ln[ (t)] 0 ln ( 0) time-dependent and dimensionless cosmic curvature factor ccf (ccf), Ωac (t) (apparent spatial curvature), and the cosmic anisotropy factor (caf), Ωcaf (t) (apparent anisotropy): 3.1 Cosmological parameters ac kc2 2 The analytically continued energy-stress conservation law in Ωccf (t)=− H−2(t);Ωcaf (t)= H−2(t) . ac −2 ac ac −6 ac the expanding universe may be written as (for the details ln [ (t)] ln [ (t)] (16) see Vasconcellos et al. (2021)): Combining these expressions with the definition of Hac (Vas- 1 d p(t) 1 d ln−1[ (t)] (t)+3 1 + (10) concellos et al., 2021), we get (t) dt c2(t) ln−1[ (t)] dt 2 2 2 ccf k (t) caf (t) d p(t) d Ωac (t) = − ;Ωac (t) = . ⇒ ln((t))+3 1 + ln[ln−1[ (t)]=0. ̇2(t) ln−4[ (t)] ̇2(t) ln−8[ (t)] dt c2(t) dt (17) From this equation it results Applying complex conjugation to these expressions we obtain H I k ∗2(t∗) −1 Ω∗ccf (t∗) = − , (18) (t) = 0exp −2 (t)dln (ln [ (t)]) , (11) ac ̇∗2 ∗ −4 ∗ ∗ Ê (t ) ln [ (t )] where ∗2 ∗2(t∗) and Ω∗caf (t∗) = . (19) 3 p(t) ac −8 (t) ≡ 1 + , (12) ̇∗2(t∗) ln [ ∗(t∗)] 2 c2(t) In appendix A, we present solutions for the cosmography parameters in a branch cut universe. 4 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 3.3 Cosmological Redshift represents the analytically continued deceleration parameter. On small scales and at small redshifts we obtain the analyti- Light emitted by distant objects from our galaxy travels from cally continued Hubble’s law, the point of emission at t = te, r = re to the observation point today t = to, r = ro along the geodesic curves of a czac = Hac0d . (26) manifold, which correspond essentially to local straight lines 2 (d ∼ d ∼ 0), satisfying ds = 0. The line element of the modified FLWR metric of a three- 4 ON THE ROAD OF A QUANTUM dimensional spatial slice of an analytically continued space- APPROACH time, in co-moving coordinates may be written as H 2 I The challenge of building a quantum theory of gravitation dr ds2 = c2dt2 − a2(t) + r2 d2 + sin2 d2 . based on the simple combination of quantum mechanics and k r2 1 − general relativity, due to their so distinct characteristics, is sig- (20) nificant. In the following we present a few remarks about the 2 The conditions d ∼ d ∼ 0 and ds = 0 applied to this −1 physical and geometric meaning of ln [ (t)] and (t) and we 4 equation allows the mapping (Vasconcellos et al., 2020, 2021) sketch first steps on the road of a quantum approach for the dr2 −1 1 ln [ (t)] branch cut universe. → c2dt2 − a2(t) = 0 ,→ , 2 1 + z ≡ −1 1 − k r ac ln ( 0) ln[ (t)] − ln( ) ,→ z 0 ; (21) 4.1 The problem of time ac ≡ ln( ) 0 It is believed that general relativity and quantum mechanics z represents the analytically continued cosmological corre- ac should be reconciled in a theory of quantum gravity, merg- sponding to the metric (20), t denotes the proper time measured ing at the Planck scale. It is also believed that the spacetime by a co-moving observer and the radial and angular coordinates geometry cannot be measured below the Planck scale (Calmet, in the co-moving frame are represented by r, and . From Graesser, & Hsu, (2004; Garay, 1999), since quantum space- equation (21), variations of z , more specifically Δz obey ac time fluctuations would spoil at this scale its description as a ln[ (t)]∕ 0 smooth spacetime manifold (Garay, 1999). Δzac = . (22) ln( 0) The essential difference between quantum mechanics and A Taylor expansion of the analytically continued Hubble’s general relativity that makes their reconciliation extremely law for two objects at a distance d apart gives: difficult is the interpretation and the role of time. In quan- d tum mechanics, time corresponds to a universal and absolute ln−1[ (t)] = ln−1( ) + ln−1[ (t)] ó t − t 0 ót t 0 dt ó = 0 parameter. As a consequence, the formal treatment of time in 1 d2 2 quantum mechanics differs from the other coordinates that may + ln−1[ (t)] ó t − t + ⋅ ⋅ ⋅ (23) 2 ót t 0 2 dt ó = 0 be raised to the category of quantum operators and observ- On small scales, the distance to an emitter, d is approximately ables. In general relativity, in turn, time is said to be malleable related to the time of emission, t, so we can then rewrite (21) and relative (Isham, 1993). As a consequence, unification of as quantum mechanics and general relativity requires reconciling 2 1 d qac0 2 d − Hac0 − Hac0 + ⋅ ⋅ ⋅ (24) their absolute and relative notions of time. 1 + zac c 2 c −1 with ln ( 0) normalised to 1 and where d2 4.2 Wheeler-DeWitt Equation ln−1[ (t)] ln−1[ (t)] dt2 q = − , (25) An illuminating example is the formulation and interpretation ac0 d 2 ln−1[ (t)] dt of the Wheeler-DeWitt (WdW) equation (DeWitt, 1967). The Wheeler-DeWitt formulation for quantum gravity consists in

4 Caution should be taken here on the mapping a (t) → ln( (t)). This mapping constraining a wave function which applies to the universe as is not a simple direct parametrization of the scale factor based on the real FLRW a whole, — the so called wave function of the universe —, in single-pole metric. Or the result of a direct generalization of Friedmann’s equations. Due to the non-linearity of Einstein’s equations, such a direct generalisation would accordance with the Dirac recipe: not be formally consistent. The present formulation is the outcome of complexify- ̂ ing the FLRW metric and results in a sum of equations associated to infinitely many Ψ = 0 , (27) poles (in tune with Hawking’s assumption of infinite number of primordial universes that occurred simultaneously) arranged along a line in the complex plane with i.e., a stationary, timeless equation, instead of a time- infinitesimal residues (for the details see Vasconcellos et al. (2020)). The multiverse dependent quantum mechanics wave equation as for instance conception corresponds in our formulation to a theoretical mathematical device for implementation of the proposal. César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 5

where R represents the analytically continued scalar cur- ) [gac] i Ĥ Ψ . (28) vature (for the details see Vasconcellos et al. (2021)): )t 2 Here, Ĥ denotes the Hamiltonian operator of a quantum sub- LH d ln−1[ (t)] I dt2 system, while ̂ in the previous equation represents a quantum R[gac] = 6  2c2N2(t) ln−1[ (t)] operator which describes a general relativity constraining, H d ln−1[ (t)] I2 M resulting in a second order hyperbolic equation of gravity dt k 5 + + , (34) variables , a Klein-Gordon-type equation, having therefore a cN(t) ln−1[ (t)] 2 ln−2[ (t)] ‘natural’ conserved associated current ( ),  and represents the energy density of the multiverse. i = Ψ† ∇ ⋅ Ψ − Ψ ∇ ⋅ Ψ† ; with ∇ ⋅ = 0 . (29) Combining (32) and (34) and using the approximation  2  √−g ≈ N(t) ln−3[ (t)], we obtain Primordial technical deficiencies have historically led to a H tendency to underestimate the WdW equation as a consis- 62N(t)c4 ln−2[ (t)] d2 S = ln−1[ (t)] (35) tent formulation of quantum gravity, despite supporting several Ê 16G N2(t)c2 dt2 approaches, from quantum geometrodynamics to loop quan- L d −1 M I ln [ (t)]2 8G tum gravity. More recently, however, an opposite trend has + ln−1[ (t)] dt + k − ln−3[ (t)] d4x. emerged, related to the understanding of the fundamental rea- N(t)c 3c4 sons for the intriguing explicit absence of the time variable in We integrate this equation by parts to remove the second the WdW equation, i.e., that the time variable has no phys- derivative in ln−1[ (t)], resulting in ical significance in general relativity (Rovelli, 2015; Rovelli H d −1 Nc4 ln [ (t)]2 & Smerlak, 2011; Shestakova, 2018). Based on this under- S = − ln−1[ (t)] dt (36) standing, in the following we sketch a quantum formulation of 2G Ê Nc the present approach based on the WdW equation analytically I 8G continued to the complex plane. + k ln−1[ (t)] − ln−3[ (t)] dt . 3c4 Making the choice N = G = c = 1 (see footnote 6), from this 4.3 Analytically continued WdW equation equation we obtain the Lagrangian density of the multiverse: H 4.3.1 Einstein-Hilbert action in the new metric 1 d 2 = k ln−1[ (t)] − ln−1[ (t)] ln−1[ (t)] The define the analytically continued FLRW metric (Vascon-  2 dt cellos et al., 2021): I 8 −3 ds2 2N2 t c2dt2 2 −2 t d 2 r, , , − ln [ (t)] . (37) [ac] = − ( ) + ln [ ( )] Ω ( ) (30) 3 with In the following, on basis of this Lagrangian formulation, L M 2 2 dr 2 2 2 2 we proceed with the quantisation of the system. dΩ (r, , ) ≡ + r (t) d + sin d . 1 − kr2(t) (31) In expression (30), N(t) is an arbitrary lapse function6, and 4.4 Topological Quantisation 2 = 2∕3 is just a normalisation factor. The conjugate momentum pln of the dynamical variable We assume as a starting point a homogeneous and isotropic ln−1[ (t)] is multiverse described by a mini-superspace model (see for ) ̇(t) p =  = − ln−1[ (t)] . (38) instance Kim (1997)) with only one dynamical variable, the ln d ) ln−1[ (t)] (t) scale factor ln−1[ (t)], and a very simple scenario for the dt Therefore the corresponding Hamiltonian becomes Einstein-Hilbert action SEH (He & Cai, 2020) 1 d −1 S = dtd3x (32)  = pln ln [ (t)] −  , (39) EH 16G  dt Ê H 2 I G 1 p 8 1 √ 3 16 3 ln k −1 t −3 t . = −g R[gac]c − dtd x , (33) = − + ln [ ( )] + ln [ ( )] 16G Ê 2c 2 ln−1[ (t)] 3 The quantisation of the Lagrangian density is achieved by 5More precisely, the scale factor a(t), the density (t), the pressure p(t), and raising the dynamical variable ln−1[ (t)] and the conjugate the gravitation constant Λ. 6The lapse function N(t) is not dynamical, but a pure gauge variable. Gauge invariance of the action in general relativity yields a Hamiltonian constraint which requires a gauge fixing condition on the lapse (see Feinberg & Peleg (1995). 6 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer momentum pln to the category of operators in the form contributions implemented by using the projectable Hořava- −1 Lifshitz gravity model78: ln−1[ (t)] → ln̂ [ (t)] (40) )2 ) − + (u) Ψ(u) = 0 , (44) and pln → pln = −i` ; 2 WdWHL ) ln−1[ (t)] )u with (for simplicity, in the following we skip using the hat symbol in (u) = 2 (u) + (u) , (45) the operators p̂ and ln̂ ). Ambiguities in ordering of the opera- WdWHL [ac]WdW HL tors may be overcome by the introduction of an ordering-factor where HL(u) represents an adaptation of the Hořava-Lifshitz in the form gravity potential: H I g k2 g k 2 1 ) − ) (u) = g k − g u − r − s ; (46) p = − − ln [ (t)] , (41) HL c Λ 2 ln ) ln−1[ (t)] ) ln−1[ (t)] u u (for comparison see Bertolami & Zarro (2011); Cordero, with usually chosen in general as = [0, 1]; = 0 corre- Garcia-Compean, & Turrubiates (2019); He & Cai (2020)). sponds to the semiclassical value; intermediate values have no Here, g > 0, stands for the curvature coupling constant meaning. C with the sign of g following the sign of the cosmological Combining (39) and (41), using the prescription = 0, constant (Bertolami & Zarro, 2011); g corresponds to the recovering the original values of the physical constants G and r coupling constant for the radiation contribution and g stands c, and changing variable (u ln−1[ (t)] and du d ln−1[ (t)]) s ≡ ≡ for the “stiff” matter contribution (which corresponds to the we obtain the following expression for the WdW equation: = p equation of state); g and g can be either positive or H I r s `2 )2 E k 4 negative since their signal does not alter the stability of the − + P u2 − u4 Ψ(u) = 0 , (42) 2m )u2 2 3l Hořava-Lifshitz gravity (Bertolami & Zarro, 2011). P 2lP P Combining (42), (44), (45), and (46), the following equation where m , E , and l are the Planck mass, energy and P P P results9: length, respectively (for comparison see for instance He & H I )2 8 g k2 g k Cai (2020)). This expression represents a Schrödinger-type − +g k−g u+ku2− u4− r − s Ψ(u)=0. (47) )u2 c Λ 3 u u2 equation of a particle with the Planck mass mP under the action of the WdW quantum potential This equation, when expressed in terms of non-composed functions, as in the conventional quantum FLRW approach, is EPk 4 ln−1[ (t)] = ln−2[ (t)] − ln−4[ (t)]) . highly non-linear and has no exact solution. [ac]WdW 2 3l 2lP P (43) 4.5.1 Solutions Assuming the first two terms of the potential given by (47), g k 4.4.1 Topological quantisation of spacetime c and −gΛu, are dominant, the substitution The scale factor ln−1[ (t)] as the only dynamical variable of −2∕3 the model may be raised, in a quantum approach, at the level ⇒ (gΛ) gck − gΛu , (48) of a quantum operator. This new status gives the scale factor leads to an Airy equation: ln−1[ (t)] an additional role in describing the evolutionary pro- H I )2 cess of the branch cut universe, that of representing the formal − Ψ() = 0, (49) 2 confluence between the classic description and its quantum ) version through a process that we call spacetime topological whose solution is quantisation, as far as we know, a new nomenclature which Ψ() = C Ai() + C B() , (50) characterises a new perspective in the quantisation of space- 1 2 time. Our formulation describes in short the relative evolution where Ai() and B() are the Airy functions of the first and sec- −1 of the variable ln [ (t)] over worldlines ln associated with ond kind, respectively. The system of equations also supports hypersurfaces ln analytically continued to the complex plane.

7The Hořava-Lifshitz formulation of gravity is an alternative theory to general 4.5 Analytically continued WdW equation relativity which employs higher spatial-derivative terms of the curvature which are added to the Einstein- Hilbert action with the aim of obtaining a renormalisable In the following we consider an extended version of the analy- theory. 8For simplicity, in the following we use natural units. tically continued WdW equation (42) with extrinsic curvature 9 The parameters, gc , gΛ, gr, and gs, although apparently simple, represent trick terms containing diﬀerent orders of spacetime curvatures (Bertolami & Zarro, 2011; Cordero et al., 2019; He & Cai, 2020). César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 7 complex conjugated solutions: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ψ ( ) = C1 Ai ( ) + C2 B ( ) . (51)

4.5.2 Boundary Conditions The evolution phases of the universe can lead to different com- binations of pressure and density characterised by different values of the dimensionless thermodynamical connection (t) (12). In the following, however, for simplicity we chose a sym- metric evolution description of the universe for positive and negative cosmological time. It is expected that the most appropriate solutions of the WdW equation give rise, in the late universe, to a classic FIGURE 1 Left figure: Characteristic plot of the Riemann spacetime, and provide an initial condition for the inflationary surface R associated to the real part of the ln[ (t)] function, period, necessary for the resolution of the flatness and horizon represented by Re[ln[ (t)]], limited to one Riemann sheet in problems of classical cosmology. To meet these expectations, the region surrounding the Planck scale, the transition region it is crucial to impose appropriate boundary conditions to the corresponding to the domain where general relativity and WdW equation. Here, as a first sketch of our quantum pro- quantum mechanics reconcile. Right figure: Plot of the real posal, we do not enter in those specific aspects and impose the part of the Re[ln[1∕ (t)]], assuming that the (t) function is following boundary conditions: orthomodular. The domain of the quantum leap is indicated in the figure. lim Ψ(−) → 0 ; lim Ψ() → 0 . (52) →−∞ →∞

5 RESULTS

In Fig. (1 ) we show characteristic plots of the Riemann surface associated to the real parts of ln[ (t)] and ln[1∕ (t)], assuming that (t) is a orthomodular function. Fig. (2 )shows the corresponding plots of ln−1[ (t)] and ln−1[1∕ (t)].

5.1 Scenarios for the branch cut universe

Our previous results delineate two scenarios for the evolution FIGURE 2 Left figure: Characteristic plot of the Riemann of the branch cut universe which are sketched in an artistic rep- surface R associated to the real part of the ln−1[ (t)] func- resentation (see Fig. (3 )), with a branch point and a branch tion, represented by Re[ln−1[ (t)]] limited to one Riemann cut on the left figure, and no primordial singularity on the right sheet in the region surrounding the Planck scale. Right figure: 10 one . The figures indicate the cosmic contraction and expan- Plot of the real part of the inverse of the previous figure, sion phases of the branch cut universe. In the representation Re[ln−1[1∕ (t)]]. sketched on the left figure, the branch cut universe evolves from negative to positive values of the complex cosmological time tC , — or the thermal time T , as a result of a Wick rotation 6 DISCUSSION —, circumventing continuously a a branch cut and no primordial singularity occurs, only branch points. The right figure The first scenario of the branch cut universe is characterised by on (3 ) sketches an alternative artistic representation of two its continuous expansion and by a systematic decrease of the mirrored evolving universes, originated both from primordial temperature in its positive complex cosmological time sector. singularities. As an example, in Fig. (4 ) characteristic plots In the second scenario, the branch cut and branch point disap- of the unnormalised solutions of equation (50), for the first pear after the realisation of imaginary time by means of a Wick scenario, are shown. rotation, which is replaced here by the real and continuous thermal time (temperature). In this second scenario, a mirrored 10Figures based on an artistic impression originally developed by ESO / M. parallel evolutionary universe, adjacent to ours, is nested in the Kornmesser (Kornmesser, 2020). 8 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer

first scenario the entropy decreases systematically and continuously in the negative thermal time sector until the absolute zero of entropy was reached. And then follows the increase of the entropy systematically in the positive thermal time sector. In the second scenario, entropy increases systematically in the evolution process of our universe but in the parallel mirror- universe, the arrow of time points down the entropy gradient, so the entropy is negative (negentropy sector). Our sketched quantised formulation, based on the Wheeler De Witt equation, brings an alternative ingredient to overcom- FIGURE 3 Left figure shows an artistic representation of ing the primordial singularity of the universe. In particular, the branch cut universe evolution with two scenarios. The the solutions of the quantised version (50) are in line with the figures were based on the artistic impressions by ESO / M. description of the first scenario. A more detailed and in-depth Kornmesser (Kornmesser, 2020). discussion of these aspects involving both scenarios requires a more detailed future analysis.

6.0.1 Observational signatures An expressive challenge is the observational realisation of the proposal presented. Speculations associated with the birth of two universes during the Big Bang, above 13.5 billion years ago, - our universe and another one, which from our perspective is functioning in reverse with time running backward —, as well as the multiverse conception are known and recurring. Fictional literature is lavish in this type of narrative, and from the scientiﬁc point of view, there are renowned scientists who are skeptical of the conception, and others who are proponents of multiverse theories, as S. Hawking for instance. Observations that may give some shelter to such conceptions are very rare or nonexistent. Interpretations of observational data based on such hypotheses were quickly demystiﬁed. More recently, the Antarctic Impulsive Transient Antenna ANITA/- NASA project has detected for the second time (Gorham et al., 2018) a fountain of high-energy particles that resembles an upside-down cosmic-ray shower, generating a pleiades of speculations11 about the meaning of these observations and the possible realisation of a universe specular to ours. Although not supported by the authors of the article, speculations still persist12. From our perspective, the formalism presented here represents a mathematical resource with a view to overcoming singularities in general relativity.

7 FINAL REMARKS FIGURE 4 Characteristic solution of equation (50). As stressed before (Vasconcellos et al., 2021), the present for- structure of space and time, with its evolutionary process going malism presents similarity with quantum bouncing models backwards in the cosmological thermal time negative sector. In this case, the connection between the previous solutions is 11The upward going cosmic rays, generated speculations running from sterile neutrinos and atypical dark matter distributions inside the Earth (Letzter, 2018) to broken as a result of the Wick rotation. A similar result may a topsy-turvy universe created during the Big Bang and existing in parallel with be obtained if we adopt an approach based on the path inte- ours (Cartwright, 2020). 12A subsequent article sought for a consistent explanation for the observed gral formalism with no singularity in the first scenario. In the anomalies with no conclusive results up to now (Smith et al., 2020) . César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer 9 which assume in general a mechanism (or trigger) to keep the APPENDIX A: COSMOGRAPHY bouncing phase stable which could be associated for example PARAMETERS with quantum fluctuations. In this kind of quantum bouncing models, the contraction phase amplifies quantum fluctuations In the following we show our results for the analytically con- and could serve as a trigger for the expansion phase (see for tinued cosmography parameters for the radiation-, matter-, and instance Novello & Bergliaffa (2008)). In turn, the quantum dark-matter dominated eras. formulation sketched in this contribution represents a kind of quantum tunnelling between the contraction and the expansion phases, an effect quite similar to the corresponding tunnelling A.1 Radiation-dominated era effect in ordinary quantum mechanics. For the radiation-dominated era, we obtain Using the causal structure of the McVittie space- (t) ≃ 0 . (A1) time (McVittie, 1933) for a classical bouncing cosmological u t 2 −2 1 2G0 model, Pérez, Bergliaffa, & Romero (2021) recently found ln [ (tP)] + 2 t − tP ln ( 0) 3 out that when the universe reaches a certain minimum scale, Similarly, the trapping horizons disappear and the black hole ceases to ∗ exist, suggesting that neither a contracting nor an expanding ∗ ∗ 0 (t ) ≃ u . (A2) universe can accommodate a black hole at all times. In our t 2 −2 ∗ 1 2G0 ∗ ∗ ln [ (tP)] + 2 t − tP view, the formalism presented here could represent a solution ln ( 0) 3 for the survival of black holes from the contraction sector Additionally, we have to the expanding one. The model presented by Pérez et al. v √ 2G ln2 ln−2[ (t )]+ 1 0 t−t (2021) contains a fundamental ingredient for such description, 4 P 2 P k ln ( 0) 3 Ω (t) = − 3 ln ( 0) , the scale factor a(t) that can be analytically continued to the ccf G 0 −2 1 √ 2G0 ln [ (tP)]+ 2 t−tP complex space, thus enabling a transition from the contraction ln ( 0) 3 phase, in which the universe reaches a certain minimum scale (A3) that causes the trapping horizons to disappear and the black and hole ceases to exist to the expansion phase. An investigation v √ 2G ln2 ln−2[ (t )]+ 1 0 t−t into this topi c is ongoing. 2 4 P ln 2( ) 3 P 3 ln ( 0) 0 Ωcaf (t) = 4 . G0 In physics, the prevailing tendency among scientists is to −2 1 √ 2G0 ln [ (tP)]+ 2 t−tP think of space and time as constituting the central structure of ln ( 0) 3 the universe. Conceptions as physics without time and about (A4) the flow of time being an illusion have enriched the debate Similarly, about its meaning. A question then arises: how to reconcile v u 2G∗ ln2 ln−2[ (t∗ )]+ 1 0 t∗−t∗ these visions with the remarkable predictions of general rel- 3k ln 4( ∗) P ln 2( ∗) 3 P ∗ ∗ 0 0 Ω (t ) = − ∗ u , ativity that implies a materialisation of spacetime, such as in ccf G 2G∗ 0 ln−2[ (t∗ )]+ 1 0 t∗−t∗ P ln 2( ∗) 3 P the detection of gravitational waves, conceived as ‘ripples’ in 0 spacetime? We obviously do not intend to have a definitive (A5) answer to this question. As a final word, as we see, speculations and on this still open questions find a fertile sea in another Ein- v u 2G∗ ln2 ln−2[ (t∗ )]+ 1 0 t∗−t∗ stein’s quotation (Hedman, 2017): ‘time and space are modes 32 ln 4( ∗) P ln 2( ∗) 3 P ∗ ∗ 0 0 Ω (t ) = 4 . caf G∗ u ∗ by which we think and not conditions in which we live’, a state- 0 2G ln−2[ (t∗ )]+ 1 0 t∗−t∗ P ln 2( ∗) 3 P ment so powerful and profound that it will certainly continue 0 to enlighten our creativity and imagination. (A6)

8 ACKNOWLEDGEMENTS A.2 Matter-dominated era For the matter-dominated era, the following result holds (13) P:O:H: acknowledges ﬁnancial support from PAPIIT-DGAPA give (IN100421). The authors wish to thank the referees for valu- ∗ 0 able comments. (t) ≃ . (A7) u 4∕3 −3∕2 1 √ ln [ (tP)] + 3∕2 6G0 t − tP ln ( 0) 10 César A. Zen Vasconcellos, Peter O. Hess, Dimiter Hadjimichef, Benno Bodmann, Moisés Razeira, and Guilherme L. Volkmer

Similarly He, D., & Cai, Q. 2020, Physics Letters B, 809, 135747. Hedman, A. 2017, Consciousness from a Broad Perspective. Berlin, 0 ∗(t∗) ≃ . Germany: Springer. u 4∕3 −3∕2 ∗ 1 √ ∗ ∗ Hess, P. O. 2017, Centennial of General Relativity: A Celebration ln [ (tP)] + 3∕2 6G0 t − tP ln ( 0) (C. A. Z. Vasconcellos, Ed.). Singapore: World Scientific Pub. (A8) Co. Additionally we obtain Hess, P. O., & Boller, T. 2020, Topics on Strong Gravity: A Modern L M View on Theories and Experiments (C. A. Z. Vasconcellos, Ed.). v Singapore: World Scientific Pub. Co. 2 2∕3 −3∕2 1 √ ln ln [ (tP)]+ 3∕2 6G0 t−tP ln ( 0) Hess, P. O., Schäfer, M., & Greiner, W. 2015, Pseudo-Complex k 3 ln ( 0) General Relativity. Heidelberg, Berlin, Germany: Springer. Ωccf (t) = − v , 4G0 4∕3 −3∕2 1 √ Isham, C. J. 1993, Canonical Quantum Gravity and the Problem ln [ (tP)]+ 3∕2 6G0 t−tP ln ( 0) of Time (L. A. Ibort & M. A. Rodríguez, Eds.). Netherlands: (A9) Dordrecht: Springer. Kim, S. P. 1997, Phys. Lett. A, 236, 11. and Kornmesser, M. 2020, History of the Universe (ESO)., Available at L M v https://supernova.eso.org/exhibition/1101/. 2 2∕3 −3∕2 1 √ Letzter, R. 2018, Bizarre particles keep flying out of Antarc- ln ln [ (tP)]+ 3∕2 6G0 t−tP ln ( 0) 2 3 tica’s ice, and they might shatter modern physics., Available ln ( 0) Ωcaf (t) = v , 4G0 at https://www.scientificamerican.com/article/ 8∕3 −3∕2 1 √ ln [ (tP)]+ 3∕2 6G0 t−tP bizarre-particles-keep-flying-out-of-antarcticas ln ( 0) -ice-and-they-might-shatter-modern-physics/ (A10) (Scientific American). Similarly we get McVittie, G. 1933, Mon. Not. R. Astron. Soc., 93. L M Minkowski, H. 1915, Annalen der Physic, 47, 927. v Novello, M., & Bergliaffa, S. 2008, Physics Reports, 463 (4), 127. 2∕3 1 √ ln2 ln−3∕2[ ∗(t∗ )]+ 6G∗ t∗−t∗ P ln 3∕2( ∗) 0 P Pérez, D., Bergliaffa, S. E. P., & Romero, G. E. 2021, Phys. Rev. D, k ln3( ∗) 0 Ω∗ (t∗)=− 0 , 103, 064019. ccf 4G∗ v 0 4∕3 1 √ Rovelli, C. 2004, Quantum Gravity. Cambridge, UK: Cambridge ln−3∕2[ ∗(t∗ )]+ 6G∗ t∗−t∗ P ln 3∕2( ∗) 0 P 0 University Press. (A11) Rovelli, C. 2015, Classical and Quantum Gravity, 32, 124005. Rovelli, C. 2019, The Order of Time. New York, USA: Riverhead and Books. L M v Rovelli, C., & Smerlak, M. 2011, Classical and Quantum Gravity, 2∕3 1 √ ln2 ln−3∕2[ ∗(t∗ )]+ 6G∗ t∗−t∗ 28, 075007. P ln 3∕2( ∗) 0 P 2 ln3( ∗) 0 Schönenbach, T., Caspar, G., Hess, P. O., Boller, T., Müller, A., Ω∗ (t∗) = 0 , caf 4G∗ v Schäfer, M., & Greiner, W. 2014, Month. Not. of the Roy. Astron. 0 8∕3 1 √ ln−3∕2[ ∗(t∗ )]+ 6G∗ t∗−t∗ Society, 442. P ln 3∕2[ ∗) 0 P 0 Shestakova, T. P. 2018, Int. J. Mod. Phys. D, 27, 1841004. (A12) Smith, D., Besson, D. Z., Deaconu, C. et al. 2020, 9, Experimental tests of sub-surface reflectors as an explanation for the ANITA anomalous events. (eprint 2009.13010, arXiv astro-ph.HE) REFERENCES Vasconcellos, C. Z., Hadjimichef, D., Razeira, M., Volkmer, G., & Bodmann, B. 2020, Astronomische Nachrichten, 340 (9,10), 857. Vasconcellos, C. Z., Hess, P., Hadjimichef, D., Bodmann, B., Razeira, Bertolami, O., & Zarro, C. A. D. 2011, Phys. Rev. D, 84, 044042. M., & Volkmer, G. L. 2021, Astronomische Nachrichten. Calmet, X., Graesser, M., & Hsu, S. D. H. (2004, Phys. Rev. Lett., (Accepted for publication) 93, 211101. Cartwright, J. 2020, We may have spotted a parallel universe going backwards in time., Available at https://www.newscientist .com/article/mg24532770-400-we-may-have-spotted-a -parallel-universe-going-backwards-in-time/ (New Scientist). Cordero, R., Garcia-Compean, H., & Turrubiates, F. J. 2019, General Relativity and Gravitation, 51, 138. DeWitt, B. S. 1967, Phys. Rev., 160, 1113. Einstein, A. 2020, Time’s arrow: Albert Einstein’s letters to Michele Besso. Los Angeles, USA: Christie’s. Feinberg, J., & Peleg, Y. 1995, Phys.Rev. D, 52, 1988. Feynman, R., & Hibbs, A. 1965, Quantum Mechanics and Path Integrals. New York, USA: McGraw-Hill. Garay, L. J. 1999, Int. J. Mod. Phys. A, 14, 4079. Gorham, P. W., Rotter, B., Allison, P. et al. 2018, Phys. Rev. Lett., 121, 161102.