The first stage of quantum gravity began with a series of papers by Bergmann [128] pursuing a program concerned with the quantization of field theories which are covariant with respect to general coordinate transformations, like the general theory of relativity. All these theories share the property that the existence and form of the equations of motion is a direct consequence of the covariant character of the equations. Dirac's work [129,130] on the quantization of constrained systems is a cornerstone in this pursuit. A more sophisticated and systematic treatment can be found in the book by Henneaux and Teitelboim [131].

#### **ADM Quantization**.

As most researchers would agree, the canon of canonical quantization of general relativity is the ADM formalism for quantizing 3-geometries in a 3+1 (space-time) decomposition, developed by Arnowitt, Deser and Misner in a series of papers 1959-1961. See their summary in [132]. Read also Kuchaˇr [133], Ashtekar [134], the monographs of Rovelli [135] and Thiemann [136].

#### **Wheeler-DeWitt equation**.

The Wheeler-DeWitt equation [137,138] is a Hamilton-Jacobi equation formulation of the ADM quantization on the superspace, the space of all 3-geometries. (For a short introduction with some historical tidbits, read [139].) Minisuperspace refers to the truncation of the infinite dimensional superspace to a few dimensions, such as the scale factors of the Bianchi universes. Quantum cosmology was studied by Misner [50,52] by applying the ADM quantization to the mini-superspaces of Bianchi Type I and IX universes. Read also Ryan [74].

The second wave of quantum cosmology arrived in the mid-80s with the considerations of the initial conditions for the wave functions of the universe, notably, Hartle and

Hawking's no boundary condition [140] and Vilenkin's tunneling wave function [141–143]. That was also the time when the two major programs of quantum gravity took off, namely, superstring theory [144,145] and loop QG [146,147] (and its modern version of spin foam network [148–150] which has a more obvious microscopic appeal). Supersymmetric quantum cosmology is well represented by the monographs of D'Eath [151] and of Moniz [152]. Other major approaches to quantum gravity are nicely represented in Oriti's edited book [153]. A state-of-the-art overview of multiple approaches from the words of 37 prominent practitioners is recorded in this most recent book [154]. Please refer to the original papers, reviews and monographs on this subject.

#### *3.1. Quantum Cosmology: Bounce in Gowdy T*3 *Universe*

Here we summarize Berger's work [96,97]. The Gowdy *T* 3 universe [94,95] can be interpreted as consisting of a single polarization of gravitational waves whose amplitudes satisfy a linear wave equation propagating in a non-linear background spacetime. One can then view these wave amplitudes acting quadratically as a source on the first-order equations for the non-linear background part of the gravitational field as a backreaction problem. Classically the model has an initial "big bang" singularity near which it is velocity-dominated behaving as a different Kasner solution at each value of the spatial coordinate orthogonal to the symmetry plane. Far from the singularity, the method of Isaacson [155,156] may be used to identify the gravitational waves. Their effective stress-energy tensor is that for a *p* = *ρ* fluid in two dimensions propagating in a spatially-homogeneous background spacetime. At the quantum level, Berger performed a canonical quantization of the dynamical degrees of freedom. An adiabatic vacuum state was introduced and adiabatic regularization used to obtain non-divergent stress-energy tensor vacuum expectation values. The regularized expectation value was used as a source for the classical background spacetime in the spirit of semi-classical gravity. Berger found that "The effect of the regularization of the stress-energy tensor expectation value is to replace the classical singularity with a symmetric bounce. The semi-classical spatially-averaged Gowdy *T* 3 cosmological model thus collapses from a state of large volume to a minimum volume near *r* = 0 and then re-expands. The classical singularity is replaced by a symmetric bounce".

#### *3.2. Quantum BKL-Mixmaster Scenarios*

Earlier we mentioned applications of dynamical system methods and chaos theories to cosmology, in particular, to the analysis of the BKL-mixmaster behavior [157–159]. For quantum chaos applied to the BKL-M models, we mention the numerical solutions of the Wheeler-DeWitt equation by Furusawa [160,161] and Monte Carlo simulations of the vacuum Bianchi IX universe by Berger [162]. For more recent work we mention two groups of authors on this subject. The first group's authors are Bergeron, Czuchry, Gazeau, Małkiewicz and Piechocki. Of particular interest to our theme here is their results on singularity avoidance [163,164]. How to get the universe to enter an inflationary stage without soliciting the aid of an inflaton field (cf. Starobinsky inflation [123] invokes a scalar field, at least the trace anomaly of it, as was also studied in [15]) is certainly of interest [165]. (See [166,167] for an earlier work on mixmaster inflation via an interacting quantum field). These authors constructed the quantum mixmaster solutions using affine and Weyl–Heisenberg covariant integral quantizations. According to these authors these quantization methods can regularize classical singularities which, they claim, the commonly implemented canonical quantization cannot. From this they show some promising physical features such as the singularity resolution, smooth bouncing, the excitation of anisotropic oscillations and a substantial amount of post-bounce inflation as the backreaction to the latter. For a recent summary of these authors' work on this subject, see [168].

The other group's authors are Gó´zd´z, Kiefer, Kwidzinski, Piechocki, Piontek and G. Plewa. Kiefer et al. [169,170] formulated the criteria of singularity avoidance for general Bianchi class A models in the framework of quantum geometrodynamics based on the Wheeler–DeWitt equation and give explicit and detailed results for Bianchi I models with

and without matter. The singularities in these cases are big bang and big rip. They find that the classical singularities can generally be avoided in these models.

Gó´zd´z et al. [171] use affine coherent states quantization in the physical phase space and show that during quantum evolution the expectation values of all considered observables are finite. The classical singularity of the BKL scenario is replaced by the quantum bounce that presents a unitary evolution of the considered gravitational system. In a later paper [172] they asked whether their claims may change because the affine coherent states quantization method depends on the choice of the group parametrization. Using the two simplest parameterizations of the affine group, they show that qualitative features of their quantum system do not depend on the choice. They maintain that the quantum bounce replacing a singular classical scenario is expected to be a generic feature of the considered system.

#### *3.3. Loop Quantum Cosmology (LQC): 'Big Bounce'?*

We devote this space to LQC because singularity avoidance is supposed to be a major accomplishment of LQG6 . This is referred to as the 'Big Bounce', in capital letters. There are many reviews of LQC, e.g., Ashtekar et al. [178–182], Bojowald [183,184], and a recent book [185]. (Note the loop QG community does not sing in unison in regard to many major claims in LQC. Read Ashtekhar and Singh's 2011 status report [182] and the critiques of Bojowald [186,187]).

Without getting into the detailed claims and critiques we ask two key physical questions: (1) What role would genuine quantum degrees of freedom play in the cosmological singularity proven to exist in classical general relativity? (2) Does quantization of gravity alter the Big Bang? If it does, will there be a minimum finite scale in the quantum geometry—the quantum bounce? (Here, the canonical formulation of cosmic 'time' is often provided by a relational variable such as the scale factor in the '3-volume' or a scalar field.)

There seems to be some consensus in the LQC community on the existence of a quantum bounce. On the first issue Bojowald asked the question [188]: How Quantum is the Big Bang? Studying isotropic models with an interacting scalar field he concluded that quantum fluctuations do not affect the bounce much. Quantum correlations, however, do play an important role and could even eliminate the bounce.

In [189] a new model is studied which describes the quantum behavior of transitions through an isotropic quantum cosmological bounce in loop quantum cosmology sourced by a free and massless scalar field. Using dynamical coherent states Bojowald provided a demonstration that in general quantum fluctuations before and after the bounce are unrelated. "Thus, even within this solvable model the condition of classicality at late times does not imply classicality at early times before the bounce without further assumptions. Nevertheless, the quantum state does evolve deterministically through the bounce. These analytical bouncing solutions corroborate results from numerical simulations attesting to robustly smooth bounces under the assumption of semiclassicality".

There seems to be no controversy in this last statement within the LQC community. However, how much of the state of the universe before the bounce is retained, is a subject of debates.

#### **Cosmic memory over bounces**

For theorists favoring the bounce scenario, especially the cyclic cosmologies, an important question to ask is, does the universe retain, after the bounce, its memory about the previous phase.

Bojowald is of the opinion of 'cosmic forgetfulness' [190], meaning, not all the fluctuations (and higher moments) of a state before the bounce can be recovered after the bounce, and values depend very sensitively on the late time state. In [191] he showed that quantum fluctuations before the big bang are generically unrelated to those after the big bang. A reliable determination of pre-big bang quantum fluctuations of geometry would thus require exceedingly precise observations.

Countering this claim, Corichi and Singh [192,193], in one exactly solvable model, found that a semiclassical state at late times on one side of the bounce, peaked on a pair of canonically conjugate variables, strongly bounds the fluctuations on the other side, implying semiclassicality. From this result these authors assert that cosmic recall is almost perfect. See Bojowald's Comments [194] and Corichi & Singh's Reply [195].

#### **Bianchi I model as a prototype for a cyclical Universe**

For more recent work on this topic, we mention two. Montani, Marchi and Moriconi [196] investigated the classical and quantum behavior of a Bianchi I model with stiff matter, an ultra-relativistic component, and a small negative cosmological constant (at the turnover point). They apply the Vilenkin wave function, using the volume of the universe, a quasiclassical variable, as the dynamical clock for the pure quantum degrees of freedom. They found that in a model where the isotropic variable is considered to be on a lattice, the big-bang singularity is removed at a semiclassical level in favor of a big bounce. The mean value of the Universe anisotropy variables remains finite during the whole evolution across the big bounce, assuming a value that depends on the initial conditions fixed far from the turning point.

An even simpler picture is obtained by Wilson-Ewing [197–199]: "In the loop quantum cosmology effective dynamics for the vacuum Bianchi type I and type IX space-times, a non-singular bounce replaces the classical singularity. The bounce can be approximated as an instantaneous transition between two classical vacuum Bianchi I solutions, with simple transition rules relating the solutions before and after the bounce. These transition rules are especially simple when expressed in terms of the Misner variables: the evolution of the mean logarithmic scale factor Ω is reversed, while the shape parameters *β*± are unaffected. As a result, the loop quantum cosmology effective dynamics for the vacuum Bianchi IX space-time can be approximated by a sequence of classical vacuum Bianchi I solutions, following the usual Mixmaster transition maps in the classical regime, and undergoing a bounce with this new transition rule in the Planck regime." At the level of effective dynamics in loop quantum cosmology, for this type of bounce solutions, Gupt and Singh [200] found the selection rules and underlying conditions for all allowed and forbidden transitions.

#### **Open Issues of LQC**

Bojowald raised a few cautionary points about the bounce solutions in LQC [201]: (a) A conceptual gap exists in the form of several unquestioned links between bounded densities, bouncing volume expectation values, and singularity avoidance, made commonly in arguments in favor of a generic bounce in LQC. (b) While bouncing solutions exist and may even be generic within a given quantum representation, they are not generic if quantization ambiguities such as choices of representations are taken into account. Small-volume BKL behavior in a collapsing universe is shown to be crucially different from the large-volume behavior exclusively studied so far in LQC.

#### **4. Quantum Phase Transition and Quantum Backreaction**

In addition to the popularly known string and loop theories, there is a handful of proposals of quantum gravity theories describing sub-Planckian (length scale) physics [153]. Our present aim is to see how the WCH fares in these theories, i.e., whether some of these theories predict that the Weyl curvature diminishes by comparison with other components of the Riemann tensor near the singularity or bounce. Note that this question may not be answerable by all of them because many of these theories have not yet presented a full account of how their basic constituents interact, and at the Planck length and above, give rise to the familiar manifold structure of spacetime described by GR. What makes this nontrivial and non-straightforward is the possibility of the emergence of mesoscopic structures between the micro (QG) and the macro (GR), or the occurrence of phase transitions which, in our opinion, is very likely. Quantum gravity not being discussed here since it is a topic

covered by many books and reviews, we will limit our attention to identifying those microtheories which predict a robust and smooth (near-isotropic and homogeneous) geometry at the Planck scale rising from sub-Planckian scale by phase transitions or other processes or mechanisms (e.g., polymerization, crystalization).

#### *4.1. Quantum Phase Transition: Continuum to Discrete*

#### **Asymptotic Safety**

First proposed by Weinberg [202], the asymptotic safety program in quantum gravity [203–211] asserts that the short-distance behavior of gravity is governed by a nontrivial renormalization group fixed point. As describd by Bonano and Saueressig [212] It provides an elegant mechanism for completing the gravitational force at sub-Planckian scales. At high energies, the fixed point controls the scaling of couplings such that unphysical divergences are absent, while the emergence of classical low-energy physics is linked to a crossover between two renormalization group fixed points. These features make asymptotic safety an attractive framework for the building of a cosmological model.

D'Odorico and Saueressig [213] studied quantum corrections to the classical Bianchi I and Bianchi IX universes. The correction terms induce a phase transition in the dynamics of the model, changing the classical, chaotic Kasner oscillations into a uniform approach to a point singularity. This seems to be consistent with the results obtained from the backreaction of particle creation from quantum fields, namely, the isotropization of anisotropy, which we mentioned in Section 2.3 and which will be described in Section 5.1.

#### **Causal Dynamical Triangulation CDT**

A systematic and vigorous treatment of discrete quantum geometry is the Causal Dynamical Triangulations (CDT) program pursued by Ambjorn, Jurkiewicz, Loll et al. extensively for decades. CDT is a lattice model of gravity that has been used to study non-perturbative aspects of quantum gravity. Read the review of Loll [214], and for the most recent development, Ambjorn et al. [215]: "CDT has a built-in time foliation but is coordinate-independent in the spatial directions. The *higher-order phase transitions* observed in the model may be used to define a continuum limit of the lattice theory. Some aspects of the transitions are better studied when the topology of space is toroidal rather than spherical. In addition, a toroidal spatial topology allows us to understand more easily the nature of typical quantum fluctuations of the geometry. In particular, this topology makes it possible to use massless scalar fields that are solutions to Laplace's equation with special boundary conditions as coordinates that capture the *fractal structure of the quantum geometry*. When such scalar fields are included as dynamical fields in the path integral, they can have a dramatic effect on the geometry." One can view this as a discrete modeling of the Gowdy *T* 3 universe we described in Section 3.1. The quantum phase transition is of course the interestingly new and exciting development.

#### **Causal Set: Continuum to crystalline phase transition**

Causal set theory, first proposed by Sorkin [216] and developed extensively by Dowker [217,218] and their associates, has seen significant developments both in the number of adherents and in scope of its research topics. How the classical continuum spacetime as we know it evolved from discrete causal sets is one of the key issues. Brightwell et al. [219] noted that non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Yet these configurations are entropically dominant and pose an obstacle to obtaining the desired classical limit. These authors examine this 'entropy problem' in a model of causal set quantum gravity corresponding to a discretization of 2D spacetimes. Using results from the theory of partial orders they show that, in the large volume or continuum limit, its partition function is dominated by causal sets which approximate a region of 2D Minkowski space.

Surya [220] continued this investigationand presented evidence for a continuum phase in a theory of 2D causal set quantum gravity, which contains a dimensionless non-locality parameter *e* ∈ (0, 1]. She also found a phase transition between this continuum phase and a new crystalline phase which is characterized by a set of covariant observables. For more recent developments, see her review [221].

#### **Berezenskii-Kosterlitz-Thouless (BKT) transition**

Antoniadis, Mazur, and Mottola [222] presented a simple argument which determines the critical value of the anomaly coefficient in four dimensional conformal factor quantum gravity, at which a phase transition between a smooth and elongated phase should occur. The argument is based on the contribution of singular configurations ("spikes") which dominate the partition function in the infrared. The critical value is the analog of c = 1 in the theory of random surfaces, and the phase transition is similar to the Berezenskii-Kosterlitz-Thouless transition. The critical value they obtain is in agreement with the previous canonical analysis of physical states of the conformal factor and may explain why a smooth phase of quantum gravity has not yet been observed in simplicial simulations.

#### **2D Quantum Gravity and Random Geometry**

This was an active topic of research in the second half of the 80s. As a representative work, Gross and Migdal [223] proposed a nonperturbative definition of two-dimensional quantum gravity, based on a double scaling limit of the random matrix model which they solve to all orders in the genus expansion. They derived an exact differential equation for the partition function of two-dimensional gravity as a function of the string coupling constant that governs the genus expansion of two-dimensional surfaces. The works of the principal contributors to this subject are collected in a book [224].

#### *4.2. Quantum Backreaction of Inhomogeneous Superspace Modes*

We have mentioned the backreaction problem earlier. It may refer to several different effects. We use different symbols to denote them: (C) the backreaction of inhomogeneous modes of classical gravitational perturbations on the homogeneous modes in classical general relativity, as in the works of Buchert, Clifton, Ellis et al. versus Green & Wald. (Q) The backreaction of the inhomogeneous modes of quantum geometry (in the full superspace of 3-geometries) on the mini-superspace quantum cosmology as studied in [225,226]. In between the classical and quantum levels lie the semiclassical and stochastic. (S) The backreaction in semiclassical and stochastic gravity refers to the effects of quantum matter field processes, such as the trace anomaly or particle creation, back-reacting on a classical background geometry: (SC) Semiclassical gravity when the mean value of the stress energy tensor of matter fields act as the source in the semiclassical Einstein equation; (ST) stochastic gravity when the fluctuations of the quantum fields are included as sources of the Einstein-Langevin equation. The 'quantum backreaction' in the title of this paper refers to the middle two levels between the classical and the quantum.

These 4 levels (C-SC-ST-Q) of backreaction are summarized in a recent review with an extensive bibliography [225]. These authors start by assessing the question of backreaction, i.e., whether cosmological inhomogeneities have an effect on the large scale evolution of the Universe, focusing on the purely quantum mechanical backreaction. They present one recent approach based on mathematical tools inspired by the Born–Oppenheimer approximation to include backreaction in quantum cosmology.

We mention three groups of representative work on this issue in the 80s to 90s , then leave this topic for the readers to explore with the help of this excellent review.

(1) Using path-integral quantization, Hawking and Halliwell [227] assume that the Universe is in the quantum state defined by a path integral over compact four-metrics. This can be regarded as a boundary condition for the wave function of the Universe on superspace, the space of all three-metrics and matter field configurations on a three-surface, the same as was proposed by Hartle and Hawking earlier. They treated the homogeneous and

isotropic degrees of freedom of the Friedmann Universe exactly and the inhomogeneous and anisotropic degrees of freedom in superspace to second order in the perturbations. For the same model Kiefer [228] calculated explicitly the wave functions for all multipoles of the matter field and for the tensor modes of the metric. D'Eath and Halliwell [229] have considered the backreaction of the fermionic perturbations on the homogeneous modes. Origin of structure in supersymmetric quantum cosmology was studied by Moniz [230]. By employing the supersymmetry and Lorentz constraint equations a set of quantum states was obtained and a particular quantum state which has properties typical of the conventional no-boundary Hartle-Hawking solution was identified.

(2) In canonical quantum gravity Kuchaˇr and Ryan [231] questioned the validity of physical predictions based on minisuperspace quantization of Einstein's theory of gravitation. They proposed to investigate a hierarchy of models with higher symmetry embedded in models of lesser symmetry, to spell out the criteria under which minisuperspace quantum results can be expected to make meaningful predictions about full quantum gravity. As a concrete example they studied a homogeneous, anisotropic cosmological model of higher symmetry (the Taub model) embedded in one of lesser symmetry (the mixmaster model) and showed that the respective behavior is widely different.

(3) With quantum field theory placed in an open systems setting, Sinha and Hu [232,233] used the example of an interacting *λ*Φ4 scalar quantum field in a closed Robertson-Walker universe, where the scale factor and the homogeneous mode of the scalar field model the minisuperspace degrees of freedom. They explicitly computed the back-reaction of the inhomogeneous modes on the minisuperspace sector using a coarse-grained effective action and show that the minisuperspace approximation is valid only when this backreaction is small.

For backreaction due to quantized matter fields, its decoherence effects, and how the semiclassical limits are reached in quantum cosmology, see, e.g., [67,234–237].

Concerning quantum backeaction, specifically the question of how the inhomogeneous quantum modes backreact on the lower dimensional superspace modes, and whether the results from mini- or midi-superspace can give a fair representation of the full picture, a recent paper by Bojowald [238] also expressed such a concern: "Even though the BKL scenario allows us to use the classical dynamics of homogeneous models to understand space-time near a spacelike singularity, it is a poor justification of minisuperspace models in quantum cosmology. Using a minisuperspace model to evolve from a nearly homogeneous geometry at late times to a BKL-like geometry at early times means that we begin with a well-justified, approximate infrared contribution of the full theory, but then push the infrared scale all the way into the ultraviolet."

Meanwhile, we shall wait to see further research results from the practitioners of loop and spin-foam quantum cosmology to see how the singularity avoidance in LQG might be altered by including the backreaction of inhomgeneous quantum modes.

In the next section we shall discuss the backreaction of quantum field processes and how that affects the dynamics of classical background spacetimes. Before ending, we mention two backreaction effects, one in inflation, the other in classical GR. Finelli et al. [239] studied the backreaction of scalar fluctuations on the space-time dynamics in the long wavelength limit using a set of gauge invariant variables. Below, we add a short interlude in the classical backreaction research programs for completeness, as it bears on gravitational entropy, albeit less so on the WCH at the cosmological singularity or bounce.

#### **Backreaction of inhomogeneous mode in classical GR**

There are two schools of thought on classical backreaction: This research began around 2000, with the work of Buchert [240] and Ellis with collaborators [241]. The central themes are summarized in this letter [242] where these authors make the observation that "A large-scale smoothed-out model of the universe ignores small-scale inhomogeneities, but the averaged effects of those inhomogeneities may alter both observational and dynamical relations at the larger scale". It also comments briefly on the relation to gravitational entropy. A broader representation of their work can be found in these two reviews [243,244] See also [245,246].

Green and Wald, on the other hand, showed in a series of papers from 2011–2014 [247] that the backreaction effects of classical inhomogenous modes are insignificant. These issues are related to how one should treat the inhomogeneous modes of shorter wavelengths, such as using the Brill-Hartle-Isaacson average [155,156,248,249], and how much they impact on the behavior of the theory if only the lowest modes are retained. Buchert, Ellis et al. [250] countered the Green-Wald arguments, reasserting their claims. In turn Green and Wald replied to this [251,252] with a simple derivation of their no backreaction results.

#### **5. Semiclassical Backreaction Supports WCH**

In this last section we examine to what degree is WCH consistent or contradictory with the results of semiclassical backreaction, namely, the backreaction of quantum field processes involving vacuum polarization or fluctuations on the geometrodynamics near the Planck time, like those due to the trace anomaly or cosmological particle creation. These processes have been studied rather thoroughly in the twenty years 1977–1996 as we mentioned before. Earlier, in Sections 1.2, 2.3 and 3.1, we focused on the avoidance of singularity due to the backreaction effects of the trace anomaly and particle creation. Here we complete our narrative by adding the considerations of the damping of anisotropy and inhomogeneity.

#### *5.1. Damping of Irregularities Ensures a Smooth Transition, Critical for Cyclic Cosmologies*

Sparing the reader the details of the derivations of regularized stress energy tensors for particle creation and the calculations of the backreaction effects from the solutions of the semiclassical Einstein equation (which can be found in the papers referred to in earlier sections or in a recent book [28]), we shall just highlight the key features with relevance to WCH in the following. More comments about gravitational entropy in this context can be found in [30].

(1) Vacuum fluctuations of a quantum field are ubiquitous, one need not add them in by hand, nor can one wish them away. Their effects on the background spacetime have to be included in considerations of whether the universe contracts to a singularity or undergoes a bounce.

(2) Many vigorous quantum field- theoretical studies of particle creation in (weakly) anisotropic and inhomogeneous spacetimes show that vacuum particle production is abundant near the Planck time, their backreaction effects are strong and the dissipation of these irregularities happens swiftly. (Note the significant qualitative differences between semiclassical backreaction of quantum field processes and classical backreaction of inhomogeneous modes in GR mentioned earlier).

(3) The damping of irregularities drives down the Weyl curvature with their strong and swift actions. This seems to suggest that quantum field processes act as an effective protector, maybe even a guarantor, of WCH.

(4) These quantum field processes act in such a way as to draw the spacetime into a *fixed point* in superspace, a three-geometry of high isotropy and homogeneity, the special status bestowed by the WCH.

(5) The overall backreaction effects of quantum field processes can be summarized by something resembling a *Lenz law*: the production of particles acts in such a way as to diminish their production. This is because, when the universe is isotropized and homogenized, it becomes conformally-flat, whence no more conformal particles are produced. (There is ground to believe that in the very early universe massless particles are overwhelmingly more abundant.)

(6) For any aficionado of cyclic cosmology, in addition to showing that a bounce is possible, the 'reining-in' of the irregularities near the bounce–in fact, near every bounce– is particularly important [118]: If the isotropic expansion were unstable as the scale of the universe approaches zero, then huge irregularities and anisotropies could accumulate and the successive cycles would be very different and increasingly anisotropic. Some powerful and ever-ready damping mechanisms need be there to prevent this from happening.

In summary, if we ponder upon the ubiquitous nature of these quantum field processes, their strongly dissipative effects on the spacetime irregularities, and the isotropized and homogenized outcome 'locked-in' by the Lenz law, we can see that WCH is more likely to be validated when these factors are included in our considerations. Add to this the fact that these processes apply both in the expansion and contraction phases of the universe, rendering both the beginning and ending of a cycle to be smooth, we can see that quantum field processes are good facilitators of cyclic cosmology.

#### *5.2. Quantum Gravity: Macro Spacetime Manifold from the Interaction of Micro Constituents*

Going from the Planck scale down toward the singularity or bounce, even from our rather skimpy description earlier, it may appear to the reader that some quantum gravity theories, e.g, those based on the WdW equation, Hamiltonian quantum gravity, and loop quantum cosmology are more capable of providing a detailed description of quantum spacetime than others, e.g., strings, spin foams, causal sets. This is not the case. One needs to first scrutinize whether some tacit, commonplace, yet unproven assumption has crept in, that *the macro variables are the same as the micro variables*. One should ask, "Are they?". This is a pivotal question, because by assuming there is no change of variables from micro to macro, one only needs to deal with the quantum to classical transition issue. In my opinion, a more important and challenging question is that of micro to macro transition.

Theoretically, this cuts into the key issue of whether the dynamical variables in GR are 'fundamental' all the way from the scale of our universe now, captured amazingly well by Einstein's theory, to the sub-Planckian scales, and, for the former group of theories, even all the way down toward the singularity. What if the variables are collective variables, that the macroscopic spacetime is emergent, and that general relativity is an effective theory valid only at large scales and low energies? We know Nature has many, possibly an infinite number of, levels of structure, separated by characteristic scales, and each level can be described by a suitably constructed effective theory. If so, the theories we now know of the large scale structures of spacetime will very likely become inapplicable at scales below the Planck length.

Are we to believe that once a macro-variable is quantized, it becomes a micro-variable? This is a big leap of faith. In Nature, a quantized macro-variable can be a very different physical entity from a micro-variable, which can be of a quantum nature. E.g., sound is a collective excitation of atoms. Quantizing them yields phonons. But quantization of sound will never lead to the microvariables we want–atoms. In fact, these collective variables lose their meanings once the atomic scale is reached. If the metric or connection forms are collective variables, quantizing them is the wrong way to discover the sub-Planckian microscopic constituents. In contrast, those theories which propose certain entities as the microscopic constituents of spacetime, such as strings or causets, need to show that their interactions can indeed produce the macroscopic spacetime we know, thus is the difference between quantum and emergent gravity [83].

#### *5.3. Quantum Cosmology: Singularity or Bounce Ruled by the Interaction of Micro-Constituents*

Varied as the many schools of thought and practice are, there is agreement that quantum gravity refers to *theories of the microscopic structures of spacetime*. Quantum cosmology is supposed to be the application of quantum gravity to the description of the universe from the Planck scale down to the singularity or the bounce. Thus, some knowledge of the underlying theory of quantum gravity is required. Many theories of quantum cosmology assume the existence of a metric or connection, and many papers, such as in string cosmology, start with a statement like, "Let us consider strings in the FLRW or de Sitter universe". Right there, I must say, an ontological issue already shows up. Where does your spacetime manifold come from? If we believe that strings or loops are the fundamental microscopic constituents of spacetime at scales below the Planck length, then, before one talks about

string or loop cosmology in terms of the dynamics of the metric or connection, one needs to explain how the spacetime manifold, this macroscopic structure of our universe, befittingly described by general relativity, comes into being.

It appears that a clear demonstration of how the spacetime manifold emerges from string or loop interactions is still largely lacking. If one takes the dynamical variables of a macroscopic theory of spacetime and quantizes them, thinking that one would get the microscopic degrees of freedom in this way, while disregarding the possibly huge structural and behavioral differences between the micro and the meso- and macro-scopic emergent structures, the claims made pertaining to the nature of a singularity or a bounce in the sub-Planckian length regions may not make much sense or even be outright inconsistent.

Therefore, unless one can provide a proof that the micro-variables describing sub-Planckian physics are the same as the macro variables of the large scale spacetime we are familiar with, in our opinion, all predictions and claims about bounces made in quantum cosmology based on quantizing the macro geometric variables in general relativity need be reassessed.

It is with this broader perspective that we should look at the meaning and applicability of WCH. It may very well be that it is enough to examine the validity and implications of WCH at the Planck scale and not extend it to the sub-Planckian scale, because, after all, the Weyl curvature, like all the geometric objects, is well defined only when the spacetime it lives in has a manifold structure, which could be an emergent entity of the underlying QG theories for the microscopic constituents of spacetime.

**Funding:** This research received no external funding.

**Acknowledgments:** I would like to thank Paul Anderson and Paulo Moniz for a careful proofreading of the completed manuscript with helpful suggestions, and for providing the references for the approach to the Big Crunch and for supersymmetic quantum cosmology, respectively. The assistance of Cici Xia at MDPI Wuhan to put the references in the MDPI format is much appreciated.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Notes**

6 We could include group field theory (GFT) [173–176] in the present considerations (only) to the extent that (partial) equivalence of models employed or results reported can be shown to exist between these two theories [177]. Of course we need to hear the views of the GFT community on this.

#### **References**

## *Review* **Quantum Cosmology with Third Quantisation**

**Salvador J. Robles-Pérez**

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain<EMAIL_ADDRESS>
**Abstract:** We reviewed the canonical quantisation of the geometry of the spacetime in the cases of a simply and a non-simply connected manifold. In the former, we analysed the information contained in the solutions of the Wheeler–DeWitt equation and showed their interpretation in terms of the customary boundary conditions that are typically imposed on the semiclassical wave functions. In particular, we reviewed three different paradigms for the quantum creation of a homogeneous and isotropic universe. For the quantisation of a non-simply connected manifold, the best framework is the third quantisation formalism, in which the wave function of the universe is seen as a field that propagates in the space of Riemannian 3-geometries, which turns out to be isomorphic to a (part of a) 1 + 5 Minkowski spacetime. Thus, the quantisation of the wave function follows the customary formalism of a quantum field theory. A general review of the formalism is given, and the creation of the universes is analysed, including their initial expansion and the appearance of matter after inflation. These features are presented in more detail in the case of a homogeneous and isotropic universe. The main conclusion in both cases is that the most natural way in which the universes should be created is in entangled universe–antiuniverse pairs.

**Keywords:** quantum cosmology; multiverse; superspace; third quantisation; universe–antiuniverse pair

**Citation:** Robles-Pérez, S.J. Quantum Cosmology with Third Quantisation. *Universe* **2021**, *7*, 404. https:// doi.org/10.3390/universe7110404

Academic Editor: Jaime Haro Cases

Received: 25 June 2021 Accepted: 25 October 2021 Published: 27 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **Contents**

#### **1. Introduction**

Quantum cosmology is the application of the quantum theory to the universe as a whole. However, it was clear from the beginning that the customary formalism of the Copenhagen interpretation cannot be applied to the quantisation of the universe because the Schrödinger equation and the measurement process on which the Copenhagen formalism is based cannot be fundamental elements of a quantum theory of the spacetime. Let us notice that if the quantum theory must represent the quantum state of the spacetime, then, as Wheeler showed [1], its quantum state must be at the Planck length with a superposition of spacetime geometries that is impossible to visualise or represent and where one cannot even *use the word "observation" at all* (cf. [1]). However, as we approach the macroscopic scale, the quantum state of the universe must represent the approximately stable spacetime where we live and perform measurements of particles and other matter fields. It means that the description of the universe that we observe must be an emergent feature of the quantum representation of the universe.

In this article, we review the canonical formulation of quantum cosmology. We start from the foliation of the spacetime into space and time that allows us to express the Einstein–Hilbert action as a functional action of the components of the 3-metric of the spatial sections of the spacetime. The evolution of the universe turns out to then be a trajectory in the space of 3-Riemannian metrics, *M*, and its quantum state is represented by a wave function that is the solution of the Wheeler–DeWitt equation, which, in principle, contains all the information about the spacetime and the matter fields that propagate therein. However, as we have already said, the full quantum state is a superposition of solutions that correspond to different paths, i.e., different evolutions, in the space *M*. It is only in the semiclassical regime where a particular kind of solution emerges by a decoherence process1 . This kind of solution includes the semiclassical solutions that represent a fixed classical spacetime background with matter fields propagating therein and in which the Schrödinger equation appears as an approximated equation at order *h*¯ 1 . In particular, we analyse the semiclassical wave function of a homogeneous and isotropic universe with small inhomogeneities that can be treated as perturbations. In that case, explicit solutions can be given for which it is easier to analyse the different boundary conditions that can be imposed on the state of the universe. They give rise to different scenarios for the creation of the universe, which are analysed in detail.

On the other hand, the space of 3-dimensional space-like metrics, *M*, defined at any point in the space, turns out to be isomorphic to a 1 + 5 dimensional Minkowski spacetime2 . This analogy between the space *M* and the spacetime allows us to consider the wave function of the universe as a field that propagates in *M*, and the Wheeler– DeWitt equation as the field equation. In that case, a procedure of quantisation called *third quantisation* can formally be performed in a similar way as it is done in a quantum field theory. For instance, we can define quantum operators representing the creation and annihilation of particular modes of the spacetime, i.e., different universes, and the corresponding Fock space will then allow us to represent the quantum state of a multiply connected spacetime manifold. It turns out to then be the appropriate framework to describe the quantum state of the multiverse. Moreover, as it happens in a quantum field theory, the isotropy of the background space implies that the creation of universes must be in pairs with opposite values of the components of the momentum conjugated to the configuration variables. We shall analyse the charge and parity relation between the matter fields that propagate in one of these pairs to see that the matter content of one of the universes must be CP inversely related with the content of the other universe. They thus

form a universe–antiuniverse pair. We analyse this pair creation in detail in the case of a homogeneous and isotropic universe where the period of reheating after inflation is investigated. The decay of the inflaton field into the particles of the Standard Model is produced in a CP-conjugated way in the two universes, so any excess of matter over antimatter in one of the universes of the entangled pair is balanced with the excess of the antimatter over matter in the partner universe, these two concepts (matter and antimatter) always having a relative meaning, i.e., an internal observer in any of the universe always interprets the content of his/her universe as matter.

This proposal is however still far from being directly testable. The effects of quantum cosmology and, in particular, the effects of the existence of an entangled universe are mainly restricted to the very early stage of the universe. Perhaps with future advances in the detection of gravitational waves or of a cosmic neutrino background we will be able to test the pre-inflationary stage of the universe where the effects of quantum gravity may be significant. Moreover, the third quantisation formalism can also be a proposal for the quantisation of the spacetime, and thus a better understanding of the formalism and its application to other gravitational scenarios can provide us with a new line of research for the search of a quantum theory of gravity.

#### **2. Quantisation of a Simply Connected Spacetime Manifold**

#### *2.1. Quantisation of the Spacetime Geometry*

Following the customary approach3 , the spacetime can be foliated in space and time by assuming a global time function *t* such that each surface *t* = *constant* is a spacelike Cauchy hypersurface, Σ*t* . The proper distance between the point *x*0 in Σ*t* and the point *x*0 + *dx* of Σ*t*+*dt* is given by [7,8] (see, Figure 1)

$$ds^2 = g\_{\mu\nu}dx^\mu dx^\nu = \left(N\_d N^a - N^2\right)dt^2 + 2N\_d dx^a dt + h\_{ab} dx^a dx^b,\tag{1}$$

where *NaNa* = *habNaNb* and *hab* is the three-dimensional metric induced on each hypersurface Σ*t* , with unit normal *nµ*, satisfying, *n µnµ* = −1. The functions *N* and *Na* are called the lapse and the shift functions, respectively. They are the normal and tangential components of the vector field *t µ* , which is the vector field that transport the point *x*0 from Σ*t* to Σ*t*+*dt*.

With the split of the spacetime in space and time, the spacetime can be seen as a spacelike hypersurface evolving in time. The geometry of the hypersurface Σ*t* at a given time *t*0 is determined by the metric tensor *hab*(*t*0), so eventually the evolution of the universe is encoded in the evolution of the metric *hab*(*t*). It is then remarkable that in the end, as Wheeler says [9], *Eintein's geometrodynamics deals with the dynamics of* 3*-geometry, not* 4*-geometry!* (emphasis his). From this 3 + 1 viewpoint of the spacetime, we can cast the Einstein-Hilbert action4 [7]

$$S\_{EH} \int\_{\mathcal{M}} d^4 \mathbf{x} \sqrt{-g} \left( ^4 \mathbf{R} - 2\Lambda \right) - 2 \int\_{\partial \mathcal{M}} d^3 \mathbf{x} \sqrt{h} \mathbf{K},\tag{2}$$

and the action of the matter fields5

$$S\_{\rm matter} = \int\_{\mathcal{M}} d^4 \mathbf{x} \sqrt{-g} \left( \frac{1}{2} g^{\mu \nu} \partial\_{\mu} \varphi \partial\_{\nu} \varphi - V(\varphi) \right), \tag{3}$$

into the standard Lagrangian form,

$$S = \int dt \, L(q^i, q^i, t) ,\tag{4}$$

where *qi* and *q*˙ *i* will be here the spatial metric, *hab*(*x*), the matter field(s), encoded in the variable *ϕ*(*x*), and their corresponding velocities. In (2), 4*R* is the Ricci scalar associated to the 4-dimensional metric *gµν*, Λ is the cosmological constant, *K* = *habK ab*, is the trace of the extrinsic curvature, which can be written as,

$$K\_{ab} = \frac{1}{2N} \left( \dot{h}\_{ab} - D\_a N\_b - D\_b N\_a \right), \tag{5}$$

where, *Da* is the covariant derivative on the spatial section Σ*t* , and *∂*M is the boundary of the manifold M. After some manipulation (see Ref. [7]), one finds that the Einstein–Hilbert action (2) can be written as

$$S\_{EH} = \int\_{\mathcal{M}} dt d^3 \mathbf{x} N \Big( G^{abcd} K\_{ab} K\_{cd} + \sqrt{h} \Big( ^3 \mathcal{R} - 2\Lambda \Big) \Big), \tag{6}$$

where *h* is the determinant of the spatial metric *hab*, and

$$G^{abcd} = \frac{\sqrt{h}}{2} \left( h^{ac} h^{bd} + h^{ad} h^{bc} - 2h^{ab} h^{cd} \right), \tag{7}$$

is the DeWitt's metric [10]. The structure of the action (6) is very interesting. First, it is of the standard form (4),

$$S\_{EH} = \int dt \, L\_{EH} = \int dt d^3 \mathbf{x} \, \mathcal{L}\_{EH\prime} \tag{8}$$

with *LEH* the Lagrangian associated to the Einstein–Hilbert action (6) and L*EH* the *Lagrangian density*. Second, from (5) one can see that the extrinsic curvature *Kab* contains the time derivative of the metric tensor *hab*, and 3*R* only depends on *hab*. Thus, the action (6) presents the customary structure of a kinetic term that is quadratic in the *velocities* plus a potential term. Furthermore, the supermetric (7) defines a metric structure on the space of spacelike metrics, called the superspace6 . Thus, the action (6) looks like the action of a particle that moves in a curved space; the coordinates of the "particle" are the time-dependent values of the components of the metric tensor, *hab*(*t*) and the curved space where this particle moves in the space of symmetric Riemannian 3-metrics. That is, the evolution of the universe can be seen as the *trajectory* in the superspace7 (see Figure 2).

**Figure 1.** Splitting the spacetime into space and time.

The momenta conjugated to the metric tensor components, *hab*, are given by [7]

$$p^{ab} \equiv \frac{\partial \mathcal{L}\_{EH}}{\partial \dot{h}\_{ab}} = G^{abcd} K\_{cd} = \sqrt{h} \left( K^{ab} - K h^{ab} \right). \tag{9}$$

Thus, in terms of the momenta, the total action (the gravitational action plus the action of the matter field) can be written as

$$\mathcal{S} = \mathcal{S}\_{EH} + \mathcal{S}\_{\text{matter}} = \int dt d^3 \mathbf{x} \left( p^{ab} \dot{h}\_{ab} + p\_\varphi \phi - N\mathcal{H} - N^a \mathcal{H}\_a \right), \tag{10}$$

where the lapse and the shift functions act as Lagrange multipliers, with [7]

$$\mathcal{H}\_{\text{a}} = -\mathcal{G}\_{\text{abcd}} p^{\text{ab}} p^{\text{cd}} - \sqrt{h} \left( ^3 \mathcal{R} - 2\Lambda \right) + \mathcal{H}\_{\text{matter}} \tag{11}$$

$$\mathcal{H}\_{a} = -2D\_{b}p\_{a}^{b} + \sqrt{h}f\_{a} \tag{12}$$

where [7], *Ja* ≡ *h µ a Tµνn ν* , and *Gabcd* is the inverse of the DeWitt metric (7),

$$G\_{abcd} = \frac{1}{2\sqrt{h}}(h\_{ac}h\_{bd} + h\_{ad}h\_{bc} - h\_{ab}h\_{cd})\tag{13}$$

with [10],

$$\mathbf{G}^{abcd}\mathbf{G}\_{cdef} = \frac{1}{2} \left(\delta^a\_\varepsilon \delta^b\_f + \delta^a\_f \delta^b\_\varepsilon\right). \tag{14}$$

Therefore, variation of the action (10) with respect to the lapse and the shift functions yields the classical Hamiltonian and momentum constraints, respectively, i.e.,

$$
\mathcal{H} = 0 \quad , \ \mathcal{H}\_{\mathfrak{a}} = 0. \tag{15}
$$

Let us now focus for a moment on the gravitational part of these constraints. The Hamiltonian constraint, *H* = 0, can be seen as the analogue of the momentum constraint of a particle that propagates in the spacetime,

$$
\mathfrak{g}^{\mu\nu} p\_{\mu} p\_{\nu} + \mathfrak{m}^2 = 0,\tag{16}
$$

in which the mass is substituted by a non-constant potential (this analogy will be further exploited in Section 3). On the other hand, the function H*a* generates infinitesimal diffeomorphisms (change of coordinates) in the spatial hypersurfaces, Σ. Thus, the second constraint in (15), H*a* = 0, means that the Einstein–Hilbert action (6) is invariant under such diffeomorphisms, which turns out to be like a gauge freedom [7]. Thus, the real configuration space is the quotient space of all Riemannian 3-metrics, *M* ≡ Riem(Σ), in which all three metrics related by diffeomorphisms correspond to the same class, i.e.

$$\mathcal{S}(\Sigma) = \frac{M}{\text{Diff}(\Sigma)},\tag{17}$$

which is called the *superspace* [7–10].

**Figure 2.** Left: the evolution of the universe can be seen as a path in the abstract space spanned by the component of the spatial metric tensor, *hab*. Right: an example of evolution is depicted for the case that the spatial metric is diagonal, i.e., given by, *hab*(*t*) = diag(*h*11(*t*), *h*22(*t*), *h*33(*t*)).

Following Dirac [11], the canonical procedure of quantisation consists in assuming the quantum version of the classical constraints (15) by promoting the classical variables and their conjugate momenta into quantum operators and applying them to a wave function, *φ*, that is defined in the configuration space,

$$
\hat{\mathcal{H}}\phi = 0 \quad , \quad \hat{\mathcal{H}}\_a \phi = 0. \tag{18}
$$

The quantum version of the momentum constraint in (15), Hˆ *aφ* = 0, assures that the wave function *φ* is invariant under spatial diffeomorphisms in the 3-dimensional slices Σ*t* [7]. For our purposes, much more interesting is the constraint, Hˆ *φ* = 0, which under canonical quantisation becomes the Wheeler–DeWitt equation [7,8,10],

$$\left(-\hbar^2 \nabla \cdot \vec{\nabla} + \sqrt{\hbar} \left(-\,^{(3)}\mathbb{R} + 2\Lambda + \hat{T}^{00}\right)\right) \phi(\boldsymbol{h}\_{ab\prime} \cdot \boldsymbol{\varrho}) = 0,\tag{19}$$

where, ∇· and ∇~ , are the divergence and the gradient, respectively, defined in the space of 3-dimensional Riemannian metrics, *M*, and for a scalar field *T*ˆ 00 reads

$$
\hat{T}^{00} = \frac{-\hbar^2}{2\hbar} \frac{\delta^2}{\delta \varphi^2} + \frac{1}{2} \hbar^{ij} \varphi\_{,i} \varphi\_{,j} + V(\varphi). \tag{20}
$$

The Wheeler–DeWitt Equation (19) is the keystone of the canonical formalism of quantum cosmology. The solution, *φ*(*hab*, *ϕ*), is usually called the *wave function of the universe* [12] because it represents the quantum state of the spacetime and the matter fields that propagate therein. This is usually applied to the case of a single universe. However, as we shall see in Section 3, the whole spacetime manifold can present a more complicated structure and represent something more than what is typically called a universe. In that sense, the name can be misleading. However, for historical reasons we shall retain sometimes the name *wave function of the universe* even in the cases where it may represent the state of many different universes.

The Wheeler–DeWitt Equation (19) can be seen as a Schrödinger-like equation with no time variable, which is a consequence of the invariance of the quantum state of the universe with respect to the time variable. In that sense, there is no preferred time in the quantum description of the universe (additionally, there are no preferred spatial coordinates because the constraint, H*a* = 0). It is then sometimes stated that there is no time evolution of the quantum state of the universe. However, this is not true, or at least it is not accurate. As we have already pointed out, the evolution of the universe can be seen as the trajectory in the superspace. The spacetime coordinates are the parameters that parametrise the trajectory, which is therefore invariant under reparametrisations, but that does not mean that there is no evolution. It is similar to the description of the path followed by a particle in the spacetime, which is independent of the parametrisation of the path, but that does not mean that the particle does not move.

There is also a path integral approach to quantum gravity and quantum cosmology [7,8,12]. It is a generalisation of Feynman's idea that the amplitude for a particle to go from one to another point point is given by a functional integral that weighs all the paths that start from the point **x**0 at time *t*0 and end in the point **x**1 at *t*1 (see Figure 3). Following a parallel reasoning, Hartle and Hawking propose [12] that the amplitude for the universe to change from the hypersurface Σ, in which the spatial geometry and the field configuration are given by *hab* and *ϕ*, respectively, to the hypersurface Σ 0 , where they are given by the values *h* 0 *ab* and *ϕ* 0 , is given by [12]

$$<\langle h'\_{ab\prime} \, \rho' | h\_{ab\prime} \, \rho \rangle = \int \delta \mathbf{g} \, \delta \boldsymbol{\varrho} \, e^{iS[\boldsymbol{\varrho}, \rho]} \,, \tag{21}$$

where the integral must be performed over all 4-geometries and field configurations that match the given values on the two spacelike hypersurfaces [12], Σ and Σ 0 . Following that approach, the wave function of the universe is then given by,

$$\phi(h\_{ab\prime}\,\varphi) = \sum \int\_{\mathcal{C}} \delta \mathbf{g} \,\delta \boldsymbol{\varrho} \, e^{i\mathcal{S}(\boldsymbol{\varrho},\boldsymbol{\varrho})} ,\tag{22}$$

where *C* denotes the class of spacetimes and matter configurations that fulfil the boundary requirements on the hypersurfaces Σ, and the sum is performed over all kind of topologies. In order to make well defined the path integral in (22) one has to rotate to Euclidean time. However, that does not remove all the technical problems, which we are not going to deal with here. In practice, as it happens with the Wheeler–DeWitt equation, the path integral can only be performed for spacetimes and matter field configurations with a high degree of symmetry. Moreover, both formulations become equivalent because the requirement of invariance of the wave function *φ*(*hab*, *ϕ*) under reparametrizations of the time variable implies the constraint [12], *δS δN* = 0, whose quantised version is the Wheeler–DeWitt equation.

Regardless, the path integral formulation has two interesting points. First, the analogy with the Feynman's path integral formulation of the trajectory of a particle in spacetime makes it very intuitive. As we have seen, the classical evolution of the universe can be seen as a path in the superspace. Applying Feynman's idea to gravity means that the quantum state of the universe is given by a quantum superposition of all the paths that go from one to another configuration of the spatial hypersurfaces. As it happens in the spacetime, the *classical path* (i.e., the classical evolution) emerges in some specific limit because of the constructive interference between the paths of the quantum superposition, and in the same limit the non-classical paths suffer from a destructive interference or *decoherence* [3,4,13,14]. The second interesting point of the path integral approach is the following: let us assume that we already know how to construct the quantum amplitude for the universe to go from one to another given configuration of the spacelike section. Then, what is the amplitude for the birth of the universe? What are the boundary conditions that one must impose on the state of the universe to obtain the appropriate probability amplitude for the universe to be created?

**Figure 3.** (**Left**): path integral in spacetime. (**Right**): path integral in quantum gravity.

#### *2.2. Boundary Conditions*

In classical mechanics, as well as in quantum mechanics, we usually work with some given conditions that we know or assume for certain at some initial time. Then, knowing the law of evolution, the initial conditions determine the state of the system at any later time. In the universe the thing is a bit different. What we only know for certain is the state of the observable universe, say from the inflationary period8 to the current stage of accelerated expansion, and we have to make some guess about the initial conditions that give rise to a universe like that. However, this is all classical cosmology. The question in quantum cosmology is: what are the conditions at the quantum level that give rise to a specific initial boundary, Σ0, that is propitious to inflate? The probability for the creation of such universe would be given by the modulus squared of the wave function of the universe, when this is evaluated at the initial hypersurface9 . From that point of view, one can say that the initial state of the (classical) universe is the final state of the amplitude for the universe to be created ... *from what*? That is actually the issue behind the question of the boundary condition of the universe.

Using the path integral approach, Hartle and Hawking proposed [12] that the class *C* over which the integral (22) has to be performed is the class of compact geometries (in principle of all topologies) that have Σ0 as their only boundary [7,12] (see Figure 4) and matter fields that are regular on those geometries. It means that the *boundary of the universe is that the universe has no boundary*, or equivalently, that the boundary Σ0 is created from *nothing*. We shall see later on that for the case of an inflationary spacetime, the quantum state that results from the no-boundary condition is [7,12,15]

$$\phi\_{NB} \propto \exp\left(\frac{1}{3V(\varphi)}\right) \cos\left(\frac{(a^2V(\varphi) - 1)^{\frac{3}{2}}}{3V(\varphi)} - \frac{\pi}{4}\right),\tag{23}$$

where *V*(*ϕ*) is the potential of the inflaton field and *a* is the scale factor, which goes from the initial value, *a*0 = *V*(*ϕ*) −1/2, to infinity. It is important to notice that the wave function (23) can be written as

$$
\phi\_{NB} \propto e^{\frac{1}{3V(\phi)}} \left( e^{iS} + e^{-iS} \right) \tag{24}
$$

where,

$$S = \frac{(a^2V(\varphi) - 1)^{\frac{3}{2}}}{3V(\varphi)} - \frac{\pi}{4}. \tag{25}$$

We shall see in the next section that in terms of the same time variable, one of the two terms in (24), say the *branch* with *e* −*iS*, describes an expanding universe, and the branch with *e iS* describes a contracting universe. It means that the result of imposing the no-boundary condition on the quantum state of the universe is that it is given by the linear combination of two states: one representing an expanding universe and one representing a contracting universe. Typically, one considers the expanding branch of the universe as representing our universe and disregards the contracting branch for being unphysical. However, we shall see that there is another interpretation. In terms of the *physical* time variable of each universe, the two universes can both be seen as expanding universes but with their matter fields being CP-conjugated. They can be interpreted then as an expanding universe–antiuniverse pair (see Sections 2.5.3 and 3). From that point of view, the no-boundary proposal would yield the creation of universes in entangled universe–antiuniverse pairs.

**Figure 4.** Left: the class *C* over which the path integral has to be performed is the class of compact geometries that have Σ0 as their only boundary [12]. Right: the tunnelling proposal states that the only modes that survive the quantum barrier are the "outgoing" modes that represent expanding universes.

Vilenkin's tunneling proposal is quite different10. Perhaps more based on practical grounds, he proposes that the only mode that survives the tunnelling from the Euclidean region (the region located at, *a* < *V*(*ϕ*) −1/2) is the one that represents an expanding universe. Imposing the tunnelling boundary condition, the resulting wave function of the universe is

$$\phi\_T \propto \exp\left(-\frac{1}{3V(\varphi)}\right) \exp\left(\frac{-i}{3V(\varphi)} (a^2 V(\varphi) - 1)^{\frac{3}{2}}\right). \tag{26}$$

The main difference with respect to the Hartle–Hawking wave function (23) is the negative sign in the exponent of the exponential pre-factor, which may have important consequences. Let us notice that the probability for the universe to be created from nothing is *P* ∝ |*φ*| 2 ; so, in the case of the no-boundary condition, we have

$$P\_{HH} \propto e^{\frac{2}{3V(\phi)}},\tag{27}$$

while in the case of the Vilenkin's tunnelling condition, we have

$$P\_T \propto e^{-\frac{2}{3V(\varphi)}}.\tag{28}$$

One immediate consequence is that the Vilenkin's condition seems to favour the creation of a universe with a large value of the potential, which is a necessary condition for the initial hypersurface Σ0 to inflate. That would in principle reject the Hartle–Hawking proposal because, on the contrary, the no-boundary proposal seems to favour the creation of a universe with a small (or zero) value of the potential (*PHH* → ∞, as *V* → ∞). However, this result changes when high-order corrections are taken into account, so the result is not conclusive [7].

From a purely theoretical point of view, it seems that the Hartle–Hawking proposal is more fundamental in the sense that these authors put the focus on the *natural* condition that one should impose on the Euclidean region of the spacetime. As a consequence, they obtain that the universe is represented by two branches, one corresponding to an expanding universe and the other describing a contracting branches. These two branches can be considered independently once they suffer a process of decoherence, so in practice they may represent two different universes. Vilenkin's proposal seems to be more practical (although it is also based on a parallelism with some processes of quantum mechanics).

#### *2.3. Semiclassical Quantum Gravity*

The first thing that the quantum state of the spacetime must provide us with is a consistent explanation of how the classical background can emerge from the full quantum state of the spacetime. Fortunately, not only can this be done in a beautiful manner but it is in fact one of the greatest achievements of quantum cosmology.

Again, the path integral formulation of the spacetime supplies a clear picture of how it can be obtained. At some appropriate limit, which is generally a large length or mass scale compared with their corresponding Planck values, the contribution of most of the paths in the integral vanishes because of their destructive interference. The only paths that survive the interference are those that are *in phase*, i.e., those for which *δS* ≈ 0. These are actually the trajectories of the superspace given by the classical constraints. Therefore, much in a similar way as the classical trajectory of a particle emerges from the constructive interference in the path integral approach of the quantum mechanics of a particle, the classical background spacetime emerges as the constructive interference among the paths in the superspace. In the quantum mechanics of a particle that we can then compute the quantum corrections to the trajectory of the particle in terms of *quantum uncertainties*. We shall see in this section that the quantum corrections to the classical background of the universe are caused by the matter fields.

Following the customary approach [5,7,8,15], let us consider the following semiclassical wave function [5],

$$
\phi(h\_{ab\prime}\varphi) = \Delta(h\_{ab})e^{\pm \frac{i}{\hbar}S\_0(h\_{ab})}\psi(h\_{ab\prime}\varphi),\tag{29}
$$

where ∆ and *ψ* are slowly varying functions of the metric tensor *hab*, and *S*0(*hab*) is the classical action for gravity alone, given by (2). Essentially, the wave function (29) contains two parts: one that only depends on the geometric variables of the spacetime and another part that contains all the matter degrees of freedom in the wave function *ψ*. The basic idea is that we expect that the quantum fluctuations of the spacetime will weaken more rapidly than the quantum fluctuations in the state of the matter fields. In that case, we shall find a regime where the spacetime behaves nearly classically with quantum matter fields propagating therein. That is exactly what we need to describe the universe we observe. Also notice the presence of the Planck constant in (29). It means that the classical behaviour of the spacetime will be present whenever the gravitational action is large with respect to the Planck constant.

Now, insert the semiclassical wave function (29) into the Wheeler–DeWitt Equation (19) and solve it order by order in *h*¯ in the geometrical degrees of freedom. At order *h*¯ 0 , one finds

$$G\_{abcd} \frac{\delta S\_0}{\delta h\_{ab}} \frac{\delta S\_0}{\delta h\_{cd}} - \sqrt{h} \left( ^3 \mathcal{R} - 2\Lambda \right) = 0,\tag{30}$$

which is the gravitational part of the Hamiltonian constraint (11) if we make the identification,

$$p^{ab} = \frac{\delta S\_0}{\delta h\_{ab}}.\tag{31}$$

At first order in *h*¯, neglecting second derivatives of the slowly varying terms with respect to the 3-metric, two equations are obtained. The first equation [5]

$$G\_{abcd} \frac{\delta}{\delta h\_{ab}} \left(\Delta^2 p^{cd}\right) = 0,\tag{32}$$

is actually the condition for the function ∆(*hab*) to be a *slowly varying* function. In other words, whenever (32) is satisfied (or to the extent it is satisfied) the wave function (29) can be a good candidate to describe the observable universe. Equation (32) is also the equation of the conservation of the probability current, ∆ 2 *p cd*. The other equation that is obtained at order ¯*h* is

$$\mp \Im i\mathcal{G}\_{abcd} \frac{\delta \mathcal{S}\_0}{\delta h\_{ab}} \frac{\delta \psi}{\delta h\_{cd}} + \sqrt{h} \mathcal{T}^{00} = 0,\tag{33}$$

where the ∓ signs correspond to the ± signs of the exponent of (29). It suggests the identification of a time variable *t*, given by

$$\frac{\partial}{\partial t} = \pm 2G\_{abcd} \frac{\delta S\_0}{\delta h\_{ab}} \frac{\delta}{\delta h\_{cd}}.\tag{34}$$

In that case, (33) becomes the Schrödinger-like equation of the matter fields

$$i\frac{\partial\psi}{\partial t} = \sqrt{h}\,\hat{T}^{00}(\varphi\_{\prime} - i\partial\slash\partial\varphi)\psi. \tag{35}$$

Therefore, in the semiclassical regime we obtain at order *h*¯ 0 the classical behaviour of the spacetime and, at first order in *h*¯, the quantum evolution of the matter fields. Thus, the semiclassical wave function (29) contains all the physical information of the observable universe. One could say that recovering the classical equations for the spacetime degrees of freedom and the Schrödinger equation for the matter fields does not add anything to what we already knew before the quantisation of the universe. That is true; these two

features are nothing more than a test of consistency for quantum cosmology. After all, the recovering of the classical spacetime must not be surprising. We started the process of quantisation from the classical action of the spacetime and the matter fields, and the quantisation procedure consists basically in promoting the classical variables to noncommuting quantum operators, [ ˆ*hab*, *p*ˆ *ab*] ∝ *h*¯. Then, it should not be surprising that in the limit *h*¯ → 0 we recover the classical behaviour (this is the essence of the correspondence principle) and something similar for the quantum behaviour of the matter fields.

The canonical quantisation of the whole universe that we have seen in this section is interesting for several reasons. First, it suggests that the quantisation processes that leads to the wave function of the universe is consistent and, in that case, one can assume that the wave function of the universe, *φ*(*hab*, *ϕ*), would contain in principle all the physical (classical and quantum) information of all the degrees of freedom of the universe. Any physical process should be describable within the formalism of quantum cosmology. Of course, this reductionist point of view is not practical at all, but from the conceptual point of view it results appealing. A more interesting feature is that it allows us to analyse higher-order corrections to the semiclassical universe, and this should give novel features that cannot be foreseen in the classical scenario. It might help us to go beyond the quantum description of matter fields in a classical spacetime background. In particular, it can help us to find some exclusive features of the quantum regime of the spacetime, i.e., small deviations from the known behaviour caused by the high-order corrections of quantum gravity [17–20].

Another interesting feature of quantum cosmology is the appearance of time. From the point of view of the superspace, time is just the parameter that parametrises the curve that describes a particular trajectory. In the picture given by the path integral, the quantum state of the universe is given by the set of all paths that join together the initial and final states. If the quantum wave-packet is spread, no definite time variable can be chosen mainly because there is no definite curve that describes the evolution of the universe. It is only when the wave-packet is peaked around a particular solution (ideally becoming a delta function) when we have a definite curve, the classical evolution of the universe, that it can therefore be parametrised in terms of a parameter that we can call time11. It therefore appears as the result of a decoherence process between the different *histories* of the universe [3–5,21].

Furthermore, quantum cosmology relates the two concepts of time of contemporary physics: the one of the theory of relativity and the one of quantum mechanics. This is a very subtle point. Both the theory of relativity and the quantum theory work with a time variable, say *tr* and *tq*, respectively. We usually assume that both time variables are the same, *tr* = *tq* = *t*, but this is an assumption that is not guaranteed from the beginning. Of course, they (must) coincide in the Newtonian limit of both theories, but in general, they only coincide if the time variable of theory of relativity would be measured with an actual clock, which is made of matter fields. However, the theory of relativity deals with "ideal clocks," and the consideration of an actual clock may entail some problems12 .

#### *2.4. Minisuperspace Model*

Despite its conceptual importance, it is not hard to see that the Wheeler–DeWitt equation found in the previous chapter is very difficult if not impossible to solve for a general configuration of the spacetime and the matter field(s). In order to make computations, one generally has to assume some symmetries in the underlying spacetime of the universe. This process, called *symmetry reduction* [7], reduces the number of variables of the superspace and the so-reduced superspace is called *minisuperspace*13 .

Furthermore, a minisuperspace model is not necessarily an unrealistic model. On the contrary, the observational data indicates that for most of the history of the universe the spacetime presents a high degree of symmetry. Even more, if the initial hypersurface Σ0 from which the universe starts evolving is large enough compared with the Planck length, a minisuperspace model could describe the whole history of the universe14. In other cases, it can be taken as a *toy model* from which we can obtain relevant information about some (classical and quantum) aspects of the universe like, for instance, the creation of the initial hypersurface Σ0 or the *quantum-to-classical* transition and the appearance of time, among others.

They can also allow us to study the effect of small deviations from the symmetric picture. For instance, we shall consider later on small departures from the homogeneous spacetime and the matter field in the form of gravitons and matter particles, respectively. These are local perturbations of the otherwise homogeneous and isotropic background. The picture then becomes quite realistic and allows us to analyse potentially observable effects like the kind of correlations between the modes of the matter fields in different regions of the spacetime or the quantum gravitational corrections to the Schrödinger equation of the matter fields. In all those cases, the study of the minisuperspace model turns out to be justified.

Therefore, let us consider the minisuperspace that is obtained from the foliation of a 4-dimensional spacetime with closed homogeneous and isotropic spatial sections. The geometry of the spacetime is then characterised by a Freedman–Robertson–Walker (FRW) metric

$$ds^2 = -N^2(t)dt^2 + a^2(t)d\Omega\_{3\prime}^2\tag{36}$$

where *d*Ω2 3 is the line element on the unit three sphere. The foliation of the spacetime into space and time is in that case parametrised by just two functions, the scale factor *a*(*t*) and the lapse function *N*(*t*). The geometry of the spatial sections, *hab*, is then fully characterised by the scale factor *a*(*t*) that parametrises the variation in the distance between two fixed points of the space along the evolution of the universe. It parametrises therefore the expansion or the contraction of the universe. On the other hand, the lapse function *N*(*t*) determines the time parametrisation of the foliation. Different values of *N*(*t*) entail different time variables (i.e., different time parametrisations), with some special cases. For instance, if *N* = 1, the time variable *t* is called cosmic time, and if *N* = *a*(*t*), *t* is customarily renamed with the Greek letter *η* and is called conformal time because in terms of *η* the metric becomes conformal to the metric of a closed static spacetime. We know that the evolution of the universe is invariant under the choice of time reparametrisation, so, at the end of the process, we can take the preferable time variable15 .

The line element of the spacetime is then fully determined by these two functions, *a*(*t*) and *N*(*t*). The total action, i.e., the Einstein–Hilbert action (2) plus the action of the scalar field (3), can be written as [7]

$$S = S\_{EH} + S\_m = \frac{1}{2} \int dt N \left( -\frac{a\hbar^2}{N^2} + a - \frac{\Lambda a^3}{3} + \frac{a^3 \phi^2}{N^2} - 2a^3 V(\phi) \right), \tag{37}$$

where an integration over the spatial variables has been performed and absorbed with a definition of units in which, 2*G*/3*π* = 1, and the rescalings, *ϕ* → *ϕ*/ √ 2*π* and *V* → *V*/2*π* 2 . The total action has been simplified considerably. The only dynamical degrees of freedom are the scale factor, *a*(*t*), and the scalar field, *ϕ*(*t*), which according to the homogeneity condition must only depend on the time variable. The superspace has then been reduced to a two dimensional space. Now, we can proceed as described in the preceding sections. The momenta conjugated to the configuration variables are,

$$p\_a \equiv \frac{\delta L}{\delta \dot{a}} = -\frac{a\hbar}{N} \; , \; p\_\phi \equiv \frac{\delta L}{\delta \dot{\phi}} = \frac{a^3 \phi}{N} \; , \tag{38}$$

and the Hamiltonian then reads

$$H = N\mathcal{H} = \frac{N}{2} \left( -\frac{1}{a}p\_a^2 + \frac{1}{a^3}p\_\varphi^2 - a + \frac{\Lambda a^3}{3} + 2a^3V(\varphi) \right). \tag{39}$$

The momentum constraint is automatically satisfied by the symmetries of the spacetime, and the Hamiltonian constraint, *δH δN* = 0, then becomes, H = 0. Promoting the momenta into quantum operators and applying the quantum version of the Hamiltonian constraint, H = 0, to the wave function of the universe, *φ*(*a*, *ϕ*), which depends now on the two variables of the minisuperspace, *a* and *ϕ*, we obtain the Wheeler–DeWitt equation, Hˆ *φ*(*a*, *ϕ*) = 0.

#### 2.4.1. Inflationary Universe

Let us first analyse the initial stage of the universe where the scalar field is assumed to be approximately constant on the small time scale in which the universe rapidly undergoes an inflationary expansion [25], *ϕ*˙ ≈ 0 and *ϕ*0 1. In that case, the kinetic term of the scalar field in (37) can be neglected and the potential turns out to be approximately constant, *V*(*ϕ*0). From (37), it can be seen that a constant value of the potential is equivalent to a cosmological constant term, so the universe effectively behaves like a DeSitter spacetime. Then, we can write (37) as

$$S = \frac{1}{2} \int dt N \left( -\frac{a d^2}{N^2} + a - H\_0^2 a^3 \right),\tag{40}$$

where, *H*2 0 = 2*V*(*ϕ*0), in the case of the inflationary universe or, *H*2 0 = Λ/3, in the case of a "pure" DeSitter spacetime, or the sum of both in a general case. The corresponding Hamiltonian constraint turns out to be

$$\mathcal{H} = -\frac{1}{a}p\_a^2 - a + H\_0^2 a^3 = 0,\tag{41}$$

which is nothing more than the Friedmann equation expressed in terms of the momentum conjugated to the scale factor, *pa*. In terms of the time derivative of the scale factor, using *pa* = *aa*˙ (in cosmic time16, with *N* = 1 in (38)), the Hamiltonian constraint (41) can be written as

$$d = \sqrt{H\_0^2 a^2 - 1},\tag{42}$$

whose solutions can easily be obtained,

$$a(t) = \frac{1}{H\_0} \cosh H\_0 t. \tag{43}$$

If *t* ∈ (−∞, ∞), the scale factor (43) describes a universe that starts shrinking from an infinite volume, bounces at the minimum value, *a*0 = *H* −1 0 , and ends up in an eternal expansion (see, Figure 5 Left). However, it does not seem quite plausible that the universe is created with an infinite volume, so the most reasonable possibility consist in restricting ourselves to the domain, *t* ∈ (0, ∞), which describes a "bubble" of spacetime that is created with radius *a*0 = *H* −1 0 at *t* = 0 (the origin of time), and starts expanding exponentially in an inflationary like expansion. If the length scale of the initial "bubble" is some orders of magnitude greater than the Planck scale, i.e., *H* −1 0 >> *lP*, then, the quantum fluctuations of the spacetime would be small, and the homogeneous and isotropic picture described here could reasonably represent the creation of a universe like ours17 .

**Figure 5. Left**: the closed DeSitter spacetime describes a universe that starts shrinking from an infinite volume, bounces at the minimum value, *a*0 = *H* −1 0 , and ends up in an eternal expansion. **Right**: the value *a*0 can be seen as a barrier where the incoming waves are reflected and converted into outgoing waves.

Let us now analyse the solutions of the corresponding Wheeler–DeWitt equation. Making the substitution, *pa* → −*ih*¯ *∂a*, and leaving aside the ambiguity of the factor ordering, the Hamiltonian constraint (41) can conveniently be written as

$$
\hbar^2 \frac{d^2 \phi(a)}{da^2} + \omega^2(a)\phi(a) = 0,\tag{44}
$$

where,

$$
\omega(a) = \sqrt{H\_0^2 a^4 - a^2}.\tag{45}
$$

Written in this way, the Wheeler–DeWitt Equation (44) resembles the equation of a harmonic oscillator with time-dependent frequency (such parallelism will be further exploited in Section 3). Equation (44) is not exactly solvable. However, far from the turning point, *a*0 = *H* −1 0 , we can approximate the solutions by the WKB wave functions

$$\phi^{\pm}(a) = \frac{N^{\pm}}{\sqrt{\omega(a)}} e^{\pm \frac{i}{\hbar}S(a)},\tag{46}$$

where *N*± is a normalisation constant and *S*(*a*) is given by

$$S(a) = \int da \,\omega(a) = \frac{\left(H\_0^2 a^2 - 1\right)^{\frac{3}{2}}}{3H\_0^2}.\tag{47}$$

The two wave functions *φ* ± in (46) represent incoming and outgoing wave functions. Let us see it by inserting them into the Wheeler–DeWitt Equation (44). In that case, at order *h*¯ 0 the Hamilton–Jacobi equation is obtained

$$\left(\frac{\partial S}{\partial a}\right)^2 + a^2 - H\_0^2 a^4 = 0,\tag{48}$$

which corresponds to the Hamiltonian constraint (41) if one makes the identification

$$p\_a = \pm \frac{\partial S}{\partial a}.\tag{49}$$

In fact, let us note that at leading order in ¯*h* it is satisfied

$$-a\hbar \equiv p\_a \approx \langle \phi^{\pm} | \not p\_a | \phi^{\pm} \rangle = \pm \frac{\partial S}{\partial a} + \mathcal{O}(\hbar^1). \tag{50}$$

Therefore, it is obtained

$$d = \mp \frac{\omega(a)}{a} = \mp \sqrt{H\_0^2 a^2 - 1},\tag{51}$$

which is the Friedmann Equation (42), with two signs: the − sign that corresponds to *φ* + and the + sign that corresponds to *φ* −. Thus, *φ* − describes an outgoing wave, i.e., a wave that travels towards greater values of the scale factor—it thus represents an expanding universe—and *φ* + an incoming wave, i.e., a wave that travels towards smaller values of the scale factor, which corresponds therefore to a contracting universe. Let us notice however that the interpretation of *φ* ± in terms of incoming and outgoing waves must be taken carefully [26]. The Friedmann Equation (41) is invariant under the time reversal change, *t* → −*t*, and in terms of −*t* (*a*˙ → −*a*˙), *φ* + would represent the expanding universe (i.e., an outgoing wave) and *φ* − the contracting universe (i.e., an incoming wave). Clearly, it depends on the (time) parametrisation. The important thing is that the general quantum state of the universe (far from the turning point) can be written as

$$
\phi(a) = A\_0 \phi^+(a) + B\_0 \phi^-(a),
\tag{52}
$$

which represents a quantum superposition of incoming and outgoing waves, irrespective of the particular wave function that represents each one. Another important feature that is worth noticing is that the minimum value *a*0 constitutes a classical barrier below which the universe cannot go through. Let us notice that for a value *a* < *a*0, there is no real solution of the Friedmann Equation (51), so an incoming wave function representing a contracting universe is classically reflected (bounced) into an outgoing wave (see Figure 5 Right).

#### 2.4.2. Small Perturbations and Backreaction

Let us now consider a more realistic scenario by introducing two important changes. First, we shall not neglect the kinetic term of the scalar field in the action (3), and we shall consider a general form (i.e., not necessarily a constant) for the potential *V*(*ϕ*). Contrary to what may be thought, that will not introduce qualitative changes. In return, it will allow us to represent other stages of the evolution of our universe as well as many other types of universes. The second change that we are going to make is the introduction of small perturbations around the homogeneous and isotropic background spacetime. This will help us to analyse several phenomena like the behaviour of the matter fields in the semiclassical universe or the appearance of a physical time variable.

Regarding the last question, we can assume that except for the very beginning, the inhomogeneities of the universe are relatively small18. Therefore, to a good order of approximation, the universe can be represented by a homogeneous and isotropic background with relatively small inhomogeneities propagating in the background. In that case, it seems reasonable to expand the variables of the spacetime and the matter fields around the homogeneous and isotropic values and study the inhomogeneities as perturbations of the homogeneous and isotropic background. We can still consider the 3 + 1 splitting of the spacetime given in (1). However, the idea is now to expand the configuration variables (*hab*, *N*, *Na* and *ϕ*) around their homogeneous and isotropic values and retain just the first-order terms. Then, let us consider the following expansions [6,27]

$$h\_{ab}(t, \mathbf{x}) \quad = \quad a^2(t)\Omega\_{ab} + a^2(t)\sum\_{\mathbf{n}} 2d\_{\mathbf{n}}(t)G\_{ab}^{\mathbf{n}}(\mathbf{x}) + \dots \tag{53}$$

$$
\varphi(t, \mathbf{x}) \quad = \frac{1}{\sqrt{2\pi}} \varphi(t) + \sum\_{\mathbf{n}} f\_{\mathbf{n}}(t) Q^{\mathbf{n}}(\mathbf{x}), \tag{54}
$$

where Ω*ab* is the metric on the unit three-sphere, *ϕ*(*t*) is the homogeneous mode of the scalar field, *Q***n** (**x**) are the scalar harmonics on the three-sphere, and *G* **n** *ij*(**x**) the transverse traceless tensor harmonics [27], with, **n** ≡ (*n*, *l*, *m*). More harmonics can be present in (53). However, we shall only focus on the tensor modes, *d***n**, as the representative of the perturbation of the spacetime. Eventually, these modes will represent gravitons propagating in the background spacetime, and, analogously, the perturbation modes *fn* will represent the particles of the scalar field(s). The lapse and shift functions must also be expanded in terms of the spherical harmonics. Then, all these perturbed functions are inserted in the

action (10). The configuration variables are now the scale factor *a*(*t*), the homogeneous mode of the scalar field *ϕ*(*t*), and the infinite number of modes *d***n**(*t*) and *f***n**(*t*), denoted generically by *x***n**(*t*). The minisuperspace is then the infinite dimensional space spanned by the variables (*a*, *ϕ*, *x***n**), and the time evolution of the universe is represented by a parametrised trajectory in that space, (*a*(*t*), *ϕ*(*t*), *x***n**(*t*)). We should have included the modes of the perturbed lapse and shift functions. However, as it happens with their homogeneous counterparts, they are not dynamical variables but Lagrange multipliers that generate a set of constraints that can be used to simplify the equations [27]. After a cumbersome computation, one arrives at the Hamiltonian constraint

$$
\mathcal{H} = \mathcal{H}\_0 + \mathcal{H}\_m = 0,\tag{55}
$$

where the Hamiltonian H0 contains only degrees of freedom of the homogeneous and isotropic background, and H*m* is the Hamiltonian density that contains also the degrees of freedom of the perturbation modes.

Let us first consider the Hamiltonian of the background spacetime, which is given by the Hamiltonian density of (39),

$$\mathcal{H}\_0 = \frac{1}{a} \left( -p\_a^2 - a^2 + \frac{1}{a^2} p\_\varphi^2 + 2a^4 V(\varphi) \right) = 0. \tag{56}$$

After canonical quantisation, the Wheeler–DeWitt equation reads

$$\left(\hbar^2 \frac{\partial^2}{\partial a^2} - \frac{\hbar^2}{a^2} \frac{\partial^2}{\partial \varphi^2} + a^4 H^2(\varphi) - a^2\right) \phi\_0(a, \varphi) = 0,\tag{57}$$

where, *H*2 (*ϕ*) ≡ 2*V*(*ϕ*), is not necessarily constant now. Let us consider the semiclassical solutions,

$$
\phi\_0^{\pm}(a,\varphi) = \Delta(a,\varphi)e^{\pm \frac{i}{\hbar}S(a,\varphi)},\tag{58}
$$

where ∆(*a*, *ϕ*) is a slow varying field of the scale factor. Inserting the wave function (58) into the Wheeler–DeWitt Equation (57), as we did in the previous section, and disregarding second-order derivatives with respect to the background variables, one obtains at order *h*¯ 0 the Hamilton–Jacobi equation [6]

$$-\left(\frac{\partial S}{\partial a}\right)^2 + \frac{1}{a^2}\left(\frac{\partial S}{\partial \varphi}\right)^2 + a^4 H^2(\varphi) - a^2 = 0. \tag{59}$$

Following the semiclassical development of the previous sections, let us define a WKB-time parameter *tw* as [6]

$$\frac{\partial}{\partial t\_w^{\pm}} \equiv \pm \left( -\frac{1}{a} \frac{\partial S}{\partial a} \frac{\partial}{\partial a} + \frac{1}{a^3} \frac{\partial S}{\partial \varphi} \frac{\partial}{\partial \varphi} \right) \,\tag{60}$$

in terms of which,

$$d^2 = \frac{1}{a^2} \left(\frac{\partial S}{\partial a}\right)^2, \ \phi^2 = \frac{1}{a^6} \left(\frac{\partial S}{\partial \varphi}\right)^2. \tag{61}$$

and the Hamilton–Jacobi Equation (59) turns out to be

$$a^2 + 1 - a^2 \left(\phi^2 + 2V(\varphi)\right) = 0,\tag{62}$$

which is the Friedmann equation of the background spacetime. The WKB wave functions *φ* ± 0 describe universes with a background spacetime that evolves according to the Friedmann Equation (62). The wave function *φ*0 may thus represent the quantum state of a large variety of classical models of the universe. The two signs in (60) are irrelevant at the classical level because the Friedman Equation (62) is invariant under the reversal change of the time variable. However, they will play an important role at the semiclassical level.

Let us now consider the complete Hamiltonian constraint (55). The corresponding Wheeler–DeWitt equation is

$$(\hat{\mathcal{H}}\_0 + \hat{\mathcal{H}}\_m)\phi(a, \varphi; \mathbf{x}\_\mathbf{n}) = 0,\tag{63}$$

where Hˆ 0 is the Hamiltonian of the background that we have already seen (56) and (57), and Hˆ *m* is the Hamiltonian of the perturbation modes, which for the moment we do not need to specify. The wave function of the universe can be separated in two factors, the wave function *φ*0(*a*, *ϕ*) of the background (58) and another wave function that contains the matter degrees of freedom19. Therefore, the semiclassical wave function can be written now as

$$
\phi^{\pm}(a,\varphi;\mathfrak{x}\_{\mathbf{n}}) = \Delta(a,\varphi)e^{\pm \frac{i}{\hbar}S(a,\varphi)}\psi\_{\pm}(a,\varphi;\mathfrak{x}\_{\mathbf{n}}),\tag{64}
$$

If we insert the wave function (64) into the Wheeler–DeWitt Equation (63) and we solve it order by order in *h*¯, it at order *h*¯ 0 the Hamilton–Jacobi Equation (59) is obtained. Therefore, the wave function (64) still describes the background spacetime that evolves according to (62). On the other hand, at order ¯*h* 1 in H0, one obtains

$$\pm i\hbar \left( -\frac{1}{a} \frac{\partial S}{\partial a} \frac{\partial}{\partial a} + \frac{1}{a^3} \frac{\partial S}{\partial \varphi} \frac{\partial}{\partial \varphi} \right) \psi\_{\pm} = \hat{\mathcal{H}}\_m \psi\_{\pm}. \tag{65}$$

Here comes a subtle but crucial point of the semiclassical regime. In terms of the initial proper time, *t*, the two branches of the universe represent an expanding and a contracting universe because, from (38),

$$-a\frac{da}{dt} = p\_a \approx \langle \phi\_0^{\pm} | \not p\_a | \phi\_0^{\pm} \rangle \sim \pm \frac{\partial S}{\partial a'} \tag{66}$$

so *φ* − 0 describes a universe whose spacetime background is expanding and *φ* + 0 a universe whose background spacetime is contracting. In that case, in order for the WKB-time (60) to represent the proper time variable *t* in the two branches, we have to choose for the branch *φ* − the WKB-time variable *t*− defined by

$$\frac{\partial}{\partial t\_{-}} \equiv -\left(-\frac{1}{a} \frac{\partial S}{\partial a} \frac{\partial}{\partial a} + \frac{1}{a^3} \frac{\partial S}{\partial \rho} \frac{\partial}{\partial \rho}\right) \tag{67}$$

in which case,

$$\frac{\partial a}{\partial t\_{-}} = \frac{1}{a} \frac{\partial S}{\partial a'} \tag{68}$$

which represents an expanding universe. For the WKB-time variable in the *φ* + branch, we must choose

$$\frac{\partial}{\partial t\_+} \equiv -\frac{1}{a} \frac{\partial S}{\partial a} \frac{\partial}{\partial a} + \frac{1}{a^3} \frac{\partial S}{\partial \varphi} \frac{\partial}{\partial \varphi'} \tag{69}$$

in terms of which,

$$\frac{\partial a}{\partial t\_{+}} = -\frac{1}{a} \frac{\partial S}{\partial a'} \tag{70}$$

describes a contracting universe. It is worth noticing that this assignation is somehow arbitrary [26,28]. If we would have started with a time variable, *t* → −*t*, then, *φ* + 0 would represent the contracting universe and *φ* − 0 the expanding one, and the assignations of the WKB-time variables would have been the other way around. However, in terms of the definitions (69) and (67), the Equation (65) reads,

$$i\hbar\frac{\partial}{\partial t\_{\pm}}\psi\_{\pm}(t\_{\pm\prime},\boldsymbol{\varrho})=\hat{\mathcal{H}}\_{m}\psi\_{\pm}(t\_{\pm\prime},\boldsymbol{\varrho}),\tag{71}$$

where *ψ*±(*t*±, *ϕ*) ≡ *ψ*±[*a*(*t*±), *ϕ*], evaluated in the background solutions of (70) and (68). Therefore, we have ended up with two universes, one contracting and another expanding, both filled with matter.

There is however a different interpretation. One may assume that the physical time variable, i.e., the time variable measured by actual clocks that are made of matter, is the time variable that appears in the Schrödinger equation. In that case, it is worth noticing that the physical time variable of the two universes is reversely related, *t*+ = −*t*−. Let us assume that we fix the time variable by fixing the time that a particular observer measures and consider thus *t*− as the physical time. Then, in terms of the time variable *t*− the evolution of the scale factor is given by (70) so the two wave functions, *φ* + and *φ* −, represent both with a universe with an expanding background spacetime, i.e from the point of view of this hypothetical observer both universes are expanding. The Schrödinger Equation (65) in the observer's universe becomes,

$$i\hbar\frac{\partial}{\partial t\_{-}}\psi\_{-}(t\_{-},\mathbf{x\_{n}})=\hat{\mathcal{H}}\_{m}\psi\_{-}(t\_{-},\mathbf{x\_{n}}).\tag{72}$$

However, the Schrödinger equation for the fields of the partner universe is

$$-i\hbar \frac{\partial}{\partial t\_{-}} \psi\_{+}(t\_{-}, \mathbf{x\_{n}}) = \hat{\mathcal{H}}\_{m} \psi\_{+}(t\_{-}, \mathbf{x\_{n}})\_{\prime} \tag{73}$$

for the wave function *ψ*+. The "wrong sign" in (73) is not problematic. It is only indicating that (73) is the Schrödinger equation of the complex conjugated wave function *ψ* ∗ + with a *CP*-transformed Hamiltonian [26]. Let us notice that (73) can be written as

$$i\hbar\frac{\partial}{\partial t\_{-}}\psi\_{-}(t\_{-},\mathfrak{x}\_{\mathbf{n}})=\hat{\mathcal{H}}\_{\mathcal{W}}(\mathfrak{x}\_{\mathbf{n}})\psi\_{-}(t\_{-},\mathfrak{x}\_{\mathbf{n}}).\tag{74}$$

It is therefore the Schrödinger equation for the antimatter fields of the observer's universe. In this case, we have ended up in the description of two universes, both expanding but one filled with matter and the other filled with antimatter20. The two interpretations raised here for the linear combination of incoming and outgoing wave look very similar to the two interpretations made in QED of an electron–positron pair (see Figure 6). In Section 3.2.4, we shall interpret this combination as the creation of a universe–antiuniverse pair.

**Figure 6.** An electron–positron pair can equivalently be seen as either an electron propagating backwards in time, bouncing, and then propagating forward in time or as a particle–antiparticle pair propagating forward in time.

Let us now specify the Hamiltonian of the perturbation modes. If we restrict to small linear perturbations, the different modes do not interact, and H*m* turns out to be the sum of the Hamiltonian of a set of harmonic oscillators [4,29]

$$\mathcal{H}\_{\mathfrak{m}} = \sum\_{\mathbf{n}} \mathcal{H}\_{\mathfrak{n}\nu} \tag{75}$$

with

$$\mathcal{H}\_{\mathbf{n}} = \frac{1}{2M} p\_{\mathbf{x}\_{\mathbf{n}}}^2 + \frac{M\omega\_{\mathbf{n}}^2}{2} \mathbf{x}\_{\mathbf{n}'}^2 \tag{76}$$

where, *M*(*t*) = *a* 3 (*t*), and [6]

$$
\omega\_n^2 = \frac{n^2 - 1}{a^2},
\tag{77}
$$

for the tensorial modes of the spacetime (*x***n** ≡ *d***n**), and

$$
\omega\_n^2 = \frac{n^2 - 1}{a^2} + m^2,\tag{78}
$$

for the perturbation modes of the scalar field (*x***n** ≡ *f***n**). Thus, for small perturbations, the Hamiltonian of the modes turns out to be the Hamiltonian of a set of uncoupled harmonic oscillators with time-dependent mass, *M*(*t*) = *M*[*a*(*t*)], and frequency given by, *ωn*(*t*) = *ωn*[*a*(*t*)], where *a*(*t*) is the solution of the Friedmann equation of the background spacetime (possibly including the backreaction).

The Schrödinger Equation (71) of the perturbation modes is then the Schrödinger equation of a set of uncoupled harmonic oscillators, whose general solution can be written as [4,6,30]

$$
\psi\_{\pm} = \prod\_{\mathbf{n}} \psi\_{\mathbf{n}}(t\_{\pm \prime} x\_{\mathbf{n}}) \,\tag{79}
$$

where the function *ψ***n**(*t*, *x***n**) is the wave function of a harmonic oscillator with timedependent mass and frequency, which can be written in terms of the wave function of a harmonic oscillator with constant mass and frequency [31–34]. The general solution of *ψ***n**(*t*, *x***n**) can then be expanded in the basis of number eigenstates of the invariant representation, *ψN*,**n**, as

$$
\psi\_{\mathbf{n}} = \sum\_{N} c\_{N} \,\psi\_{N, \mathbf{n}'} \tag{80}
$$

where *cN* are constants coefficients, and the wave function of the invariant number state, *ψN*,**n**, is given by [32,35]

$$\left<\psi\_{\mathbf{N},\mathbf{n}}(a,\phi;\mathbf{x\_{n}})\equiv\langle a,\phi;\mathbf{x\_{n}}|N\_{\mathbf{n}}\right> = \frac{1}{\sigma^{\frac{1}{2}}}\exp\left\{\frac{iM}{2}\frac{\dot{\sigma}}{\sigma}\mathbf{x\_{n}^{2}}\right\}\bar{\psi}\_{N}\left(\frac{\mathbf{x\_{n}}}{\sigma},\mathbf{r}\right) \tag{81}$$

where *ψ*¯*N*(*q*, *τ*) is the customary wave function of the harmonic oscillator,

$$\psi\_N(q,\tau) = \left(\frac{1}{2^N N! \pi^{\frac{1}{2}}}\right)^{\frac{1}{2}} e^{-i(N+\frac{1}{2})\tau} e^{-\frac{q^2}{2}} \mathcal{H}\_N(q) \tag{82}$$

with H*N*(*q*) the Hermite polynomial of order *N*, *q* ≡ *x***n** *σ* ,

$$
\tau(t) = \int^t \frac{1}{M(t')\sigma^2(t')}dt',\tag{83}
$$

and *σ*(*t*) is an auxiliary function that satisfies the non-linear equation [31,32]

$$
\ddot{\sigma} + \frac{\dot{M}}{M}\dot{\sigma} + \omega\_n^2 \sigma = \frac{1}{M^2 \sigma^3} \,\prime \tag{84}
$$

plus some boundary condition [36]. The interesting property of the invariant representation is that once the *field*21 is in a number state of the invariant representation, it remains in the same state along the entire evolution of the field. For instance, let us assume that the perturbation modes are in the vacuum state of the invariant representation, |0i = ∏**n** |0**n**i. In that case, the mean value of the energy of the perturbations reads22

$$
\langle \mathcal{H}\_{\mathfrak{M}} \rangle = \sum\_{\mathbf{n}} \hbar \omega\_{\mathbf{n}} \left( \langle \hat{\mathbf{N}}\_{\mathbf{n}} \rangle + \frac{1}{2} \right) = \sum\_{\mathbf{n}} \frac{\hbar \omega\_{\mathbf{n}}}{2} \to \frac{\hbar}{2} \int^{\eta\_{\text{max}}} dn \ n^2 \,\omega\_{\mathbf{n}\prime} \tag{85}
$$

where *ωn* is given by (77) and (78) for matter particles or spacetime gravitons, respectively. The sum in (85) diverges and some cut-off *n*max must be taken. The energy hH*m*i is the backreaction of the perturbations on the homogeneous and isotropic background that would induce a modification of the Friedmann Equation (62),

$$
\left(\frac{d}{a}\right)^2 = 2\rho\_\varphi - \frac{1}{a^2} + \frac{\langle \mathcal{H}\_m \rangle}{a^3},\tag{86}
$$

where, *ρϕ* = 1 2 *ϕ*˙ 2 + *V*(*ϕ*) is the energy of the homogeneous mode of the matter field and hH*m*i ∝ *a* −1 for the massless tensor modes or hH*m*i ∝ *m* for the perturbations of the scalar field.

#### *2.5. Paradigms for the Creation of the Universe in Quantum Cosmology*

In the preceding section we ended up with a Friedman equation that was corrected by the backreaction of the perturbation modes of the spacetime. We shall see in this section that such a term may induce important consequences in the way in which the universes can be created. However, for historical reasons, instead of considering the backreaction of the perturbation modes, we shall consider the model of a massless scalar field conformally coupled to gravity and a cosmological constant, which eventually raises the same term in the Friedmann equation, so the two models effectively entail similar effects. Later on, we shall briefly comment on the nature of this term. The conformally coupled massless scalar field is the field used by Hartle and Hawking in Ref. [12] to describe the quantum state of the universe and the process of quantum creation. Besides, it will also help us to introduce different paradigms for the creation of the universe.

Therefore, let us consider the following action for the spacetime and the massless, i.e., *V*(*ϕ*) = 0, scalar field,

$$S = S\_{EH} + S\_m = \frac{1}{2} \int dt N \left( -\frac{a d^2}{N^2} + a - \frac{\Lambda a^3}{3} + \frac{a^3 \phi^2}{N^2} - \frac{1}{6} a^3 \,^4 \mathbb{R} \phi^2 \right), \tag{87}$$

where the last term represents the conformal coupling of the scalar field, and 4*R* is the Ricci scalar. Then, with the change, *χ* = *aϕ*, and after an integration by parts, the total action (87) can be written as [12]

$$\mathcal{S} = \mathcal{S}\_{EH} + \mathcal{S}\_m = \frac{1}{2} \int dt \mathbf{N} \left( -\frac{a\hbar^2}{N^2} + a - H\_0^2 a^3 + \frac{a\dot{\chi}^2}{N^2} - \frac{\chi^2}{a} \right),\tag{88}$$

where *H*2 0 can be a pure cosmological constant, *H*2 0 = Λ/3, or the constant value of the potential of an auxiliary inflaton field (different from the scalar field *ϕ* that we are considering in (88)). In any case, it will be assumed that it is a constant. Now, the momenta conjugated to the scale factor and the conformally coupled massless field *χ* can be easily obtained, and the Hamiltonian constraint associated to the action (88) reads

$$H = N\mathcal{H} = \frac{N}{2a} \left( -p\_a^2 - a^2 + H\_0^2 a^4 + p\_\chi^2 + \chi^2 \right) = 0,\tag{89}$$

which, taking into account that *p* 2 *χ* + *χ* 2 is nothing more than (twice) the energy of the scalar field [36], 2*E*, can also be written as

$$\frac{1}{2}p\_a^2 + \mathcal{U}(a) = E,\tag{90}$$

with

$$\mathcal{U}(a) = \frac{1}{2} \left( a^2 - H\_0^2 a^4 \right). \tag{91}$$

which is formally similar to the energy equation of a particle that propagates under the action of the potential (91). The first term of the l.h.s. of (90) would be the kinetic energy, the second term would be the potential energy, and *E* would be the total energy23

Quantum-mechanically, the wave function of the universe, *φ*(*a*, *χ*), is the solution of the Wheeler–DeWitt equation associated to the Hamiltonian constraint (89),

$$\left(\frac{\partial^2}{\partial a^2} + H\_0^2 a^4 - a^2 - \frac{\partial^2}{\partial \chi^2} + \chi^2\right)\phi(a, \chi) = 0,\tag{92}$$

which can be solved by the method of the separation of variables. Making, *φ*(*a*, *χ*) = *ξ*(*χ*)*ψ*(*a*), the Wheeler–DeWitt Equation (92) can be split into the two following equations,

$$\left(-\frac{d^2}{d\chi^2} + \chi^2\right)\xi(\chi) = 2E\xi(\chi),\tag{93}$$

$$\frac{d^2\psi(a)}{da^2} + \left(H\_0^2 a^4 - a^2\right)\psi(a) = -2E\psi(a). \tag{94}$$

The first of these equations is the equation of a quantum harmonic oscillator with unit mass and frequency. It can be solved in terms of Hermite polynomials, H*n*(*x*),

$$\mathfrak{F}(\chi) \equiv \mathfrak{F}\_n(\chi) = \frac{1}{\sqrt{2^n n!}} \left(\frac{1}{\pi}\right)^{\frac{1}{4}} e^{-\frac{\chi^2}{2}} \mathcal{H}\_n(\chi),\tag{95}$$

with a quantised energy given by

$$E \equiv E\_n = (n + \frac{1}{2}).\tag{96}$$

On the other hand, Equation (94) can be written as

$$\frac{1}{2}\frac{d^2\psi(a)}{da^2} + (E - \mathcal{U}(a))\psi(a) = 0,\tag{97}$$

with U(*a*) given by (91). Equation (97) is formally similar to the Schrödinger equation of a particle of energy *E* moving under the action of the potential U(*a*) (see Figure 7). For the value *E* ∈ (0, U*max*), where U*max* is the maximum value of the potential, we can distinguish three regions: two classically allowed regions (regions *I* and *III* in Figure 7) separated by a classically forbidden region (region *II*). We have already analysed in the preceding section the behaviour of the wave function in region *I*. There are incoming and outgoing waves that represent a contracting universe that shrinks to the minimum value of the scale factor *a*+ (see, Figure 7) and bounces (or it is reflected), becoming an expanding universe. In the other classically allowed region, region *III*, we shall see that there are also incident and reflected waves that represent a universe that is confined to oscillate between *a* = 0 and the maximum value *a*−.

In the case of a particle moving under the action of a similar potential, the particle that is placed in the region *III* is classically confined to move in that region like the universes in our example. However, we know that quantum-mechanically there is a nonzero probability for the particle to tunnel though the quantum barrier and appear in the region *I*. Something similar happens with the universe. If *E* > 0, the small oscillating universe of region *III* can tunnel out through the Euclidean barrier appearing in region *I* as a new-born universe. The universe is then said to be created from *something*. In the limiting case, *E* = 0, which was the case analysed by Hartle and Hawking [12,37–39] and by Vilenkin [40–42] for the creation of the universe, there is no region *III* from which to tunnel out to region *I*. The universe appears in region *I* from a pure tunnelling phenomena (like the creation of particles from the quantum vacuum). In that case, the universe is said to be created from *nothing*24. Let us analyse the two cases separately.

**Figure 7.** Potential U(*a*). For *E* > 0, there are three regions: two Lorentzian regions separated by an Euclidean one. Compare it with the case *E* = 0 analysed in Figure 5.

#### 2.5.1. Creation of the Universe from *Nothing*

Let us first analyse the creation of the universe from nothing, i.e., *E* = 0 in (90). In that case, in terms of the time derivative of the scale factor, *pa* = −*aa*˙, the Hamiltonian constraint (90) reduces to the Friedmann Equation (42) already studied in Section 2.4.1,

$$
\dot{a} = \sqrt{H\_0^2 a^2 - 1},
\tag{98}
$$

which yields the well-known solution,

$$a(t) = a\_0 \cosh H\_0 t,\tag{99}$$

with, *a*0 = *H* −1 0 . If we restrict ourselves to the "expanding branch," *t* ≥ 0, the scale factor (99) represents a universe that starts expanding from the initial boundary Σ(*a*0) until infinity. For values *a* < *a*0, there is no real solution, and the value *a* = *a*0 constitutes a classical barrier for the universe (see Figure 5). However, one can perform a Wick rotation into Euclidean time, *t* = −*iτ*, in terms of which the Friedmann Equation (98) becomes

$$\frac{da\_E}{d\tau} = \sqrt{1 - H\_0^2 a\_E^2}.\tag{100}$$

Now, the Euclidean Equation (100) has the solution

$$a\_E(\tau) = a\_0 \cos H\_0 \tau,\tag{101}$$

where *τ* ∈ (− *π* 2*H*0 , 0). Let us notice that, in Euclidean time, −*dt*2 → +*dτ* 2 , the line element turns to be

$$ds^2 = d\tau^2 + a\_E^2(\tau)d\Omega\_{3\prime}^2\tag{102}$$

which is the line element of a 4-sphere of radius 1/*H*0 embedded in a 5-dimensional flat Euclidean spacetime. The "spatial section" starts expanding from a single point of the sphere (at *τ* = −*π*/2*H*0) until it reaches the value *H* −1 0 at the Euclidean time, *τ* = 0 (see, Figure 8). It is called a DeSitter *instanton*, and it gives the maximum contribution to the tunnelling wave function (it is the extremal solution of the Euclidean action [12]). At the boundary hypersurface Σ(*a*0), it appears in the Lorentzian region as a new born DeSitter universe that starts expanding exponentially.

**Figure 8. Left**: a DeSitter instanton can be seen as a 4-dimensional sphere in the 5-dimensional Euclidean flat spacetime. **Right**: the creation of a DeSitter spacetime from a DeSitter instanton of the Euclidean sector of the spacetime.

Quantum-mechanically, the situation is the following. The initial boundary Σ(*a*0) separates the Lorentzian region of the spacetime (see, Figure 8), where classical solutions are allowed, from the Euclidean region, where only tunnelling solutions can exist. The wave function that represents a classical spacetime in the Lorentzian region is given in terms of the oscillatory wave functions *e* +*iS* and *e* −*iS*, where *S* is given by (47)

$$S(a,\rho) = \frac{(H^2 a^2 - 1)^{\frac{3}{2}}}{3H^2}.\tag{103}$$

In the Euclidean sector, *a* < *a*0, the wave function of the universe, can be written in terms of the tunnelling wave functions, *e I* and *e* −*I* , where *I* is given by the integral (47) with *ω*(*a*) being replaced by |*ω*(*a*)|, i.e.,

$$I(a,\varphi) = \frac{(1 - H^2 a^2)^{\frac{3}{2}}}{3H^2}.\tag{104}$$

The picture is then similar to the problem of a wave-particle tunnelling through a quantum barrier. The oscillatory wave functions *e* ±*iS* can be seen as the incoming and the reflected waves that may represent a photon or another quantum mechanical particle. Classically, the boundary Σ(*a*0) acts as a barrier that cannot be crossed (see Figure 9). Quantummechanically, however, there is a non-null probability of penetrating into the barrier, although the amplitude is exponentially suppressed in the Euclidean region. Analogously, one can say in cosmology that there is a non-zero probability for the universe to appear *from nothing*, i.e., from the Euclidean barrier of the spacetime. An essential difference is that here the tunnelling is not from another classically allowed region of the spacetime. It is therefore more similar to the creation of virtual particles from the quantum vacuum in a quantum field theory. In a quantum field theory, the pair of virtual particles can only exist at a small amount of time compatible with Heisenberg's uncertainty relations, otherwise the principle of energy conservation would be violated. In the universe, however, the energy is zero (the negative gravitational energy balances the positive energy of the matter fields) so the creation of the universe from nothing does not violate the conservation of energy.

From the above reasoning it is clear that the name "creation from nothing" does not refer to the absolute meaning of nothing, i.e., to something to which we can ascribe no properties. As we have seen, the Euclidean region of the spacetime has geometrical properties. In the standard literature (see, for instance, Refs. [7,8,16,37,38,40]), it usually refers to two meanings. One, perhaps the most consistent, is that the universe is created from a region of the spacetime where nothing real exists; in particular, there is no actual time (i.e., time measured by clocks). In that sense, there is nothing. Another sense with which the term "nothing" is used in the creation of the universe is that the universe, in the paradigmatic case of the creation of a DeSitter spacetime from the Euclidean instanton (101), begins from a single non-singular point of the Euclidean 4-sphere (which is geometrically equivalent to any other point in the sphere). In that case, the single point is meant to be "nothing".

#### Vilenkin's vs. Hartle–Hawking's Versions

The general quantum state of the universe is therefore given in the region *I* by the linear combination

$$\phi\_I(a,\varphi) = A\_I \frac{1}{\sqrt{\omega(a)}} e^{+\frac{i}{\hbar}S(a)} + B\_I \frac{1}{\sqrt{\omega(a)}} e^{-\frac{i}{\hbar}S(a)},\tag{105}$$

and in the tunnelling region *II* by

$$\phi\_{II}(a,\varphi) = A\_{II}\frac{1}{\sqrt{|\omega(a)|}}e^{+\frac{1}{\hbar}I(a)} + B\_{II}\frac{1}{\sqrt{|\omega(a)|}}e^{-\frac{1}{\hbar}I(a)}.\tag{106}$$

The particular combination of WKB wave functions in the Lorentzian and in the Euclidean regions of the spacetime, i.e., the particular values of the constants *AI*,*II* and *BI*,*II*, depend on the boundary condition that we impose on the state of the universe.

Hartle and Hawking [12,37,39] propose as the boundary condition that the path integral must be performed over compact Euclidean geometries that fit with the "final" values25 *a*0 and *ϕ* in the boundary Σ(*a*0) (see Figure 8). In the homogeneous and isotropic minisuperspace, it is equivalent to the conditions [15]

$$a(\tau\_0) = 0, \,\frac{da}{d\tau}(\tau\_0) = 1, \,\frac{d\varphi}{d\tau}(\tau\_0) = 1. \tag{107}$$

The wave function obtained for the Euclidean sections is *e* −*I* , with

$$I = \int\_0^a da \, a \, (1 - H^2 a^2)^{\frac{1}{2}} = \frac{-1}{3H^2} \left( 1 - (1 - H^2 a^2)^{\frac{3}{2}} \right). \tag{108}$$

It yields [7,12,15]

$$\phi\_{HH}^{\rm II}(a,\rho) = \frac{1}{(1 - H^2 a^2)^{\frac{1}{4}}} \exp\left(\frac{1}{3H^2} \left(1 - (1 - H^2 a^2)^{\frac{3}{2}}\right)\right). \tag{109}$$

From the matching conditions of the WKB method, the Hartle–Hawking wave function (109) implies a linear combination of oscillatory wave functions in the Lorentzian region (*a* > *a*0) [7,12,15],

$$\phi\_{HH}^{I}(a,\varphi) = \frac{1}{\left(H^2 a^2 - 1\right)^{\frac{1}{4}}} \exp\left(\frac{1}{3H^2}\right) \cos\left(\frac{(H^2 a^2 - 1)^{\frac{3}{2}}}{3H^2} - \frac{\pi}{4}\right). \tag{110}$$

**Figure 9.** Hartle–Hawking "no-boundary" boundary proposal vs. Vilenkin's tunnelling boundary proposal. The H–H state implies a linear combination of expanding and contracting wave functions in the Lorentzian region, where the tunnelling wave function corresponds only to expanding universes.

On the other hand, Vilenkin [40–42] proposes as the boundary condition, in an analogy with the tunnelling process of quantum mechanics, that the only modes that survive the quantum barrier are the "outgoing" modes, i.e., those that represent an expanding universe26. It means that in the region *I*, the wave function of the universe is given by [7,15,41,43]

$$\phi\_T^I(a,\varphi) = \frac{A\_I(\varphi)}{(H^2 a^2 - 1)^{\frac{1}{4}}} \exp\left(-i \frac{(H^2 a^2 - 1)^{\frac{3}{2}}}{3H^2}\right) \tag{111}$$

where *AI*(*ϕ*) is a normalisation "constant" that can be found by imposing the regularity conditions [15], *∂φ*/*∂ϕ* → 0 as *a* → 0. Then, *AI*(*ϕ*) = exp −1/3*H*2 , for which *φT* ∼ *e* − 1 2 *a* 2 , that is regular at *a* → 0 for any value of the scale factor. By following the same WKB procedure of matching conditions, we found in the Euclidean sector [15]

$$\phi\_T^{II}(a,\rho) = \frac{1}{(1 - H^2 a^2)^{\frac{1}{4}}} \left( e^I - \frac{i}{2} e^{-I} \right) \approx \frac{1}{(1 - H^2 a^2)^{\frac{1}{4}}} e^I. \tag{112}$$

where *I* is given by (108). Except for the values of the scale factor close to *a*0, the second term in (112) is exponentially smaller than the first, so it is usually neglected.

Besides their conceptual meaning, the main difference between the wave function of the two proposals is the different sign in the exponent of the pre-factor, exp −1/3*H*2 , in the case of the tunnelling wave function, and exp 1/3*H*2 , in the case of the no-boundary wave function. The probability measures are different in both cases. Because the similarity of the Wheeler–DeWitt equation with the Klein–Gordon equation, Vilenkin proposes to use the probability current27 [43,44]

$$J = \frac{i}{2} (\phi^\* \nabla \phi - \phi \nabla \phi^\*)\_\prime \tag{113}$$

which is conserved because in virtue of the Wheeler–DeWitt equation it satisfies ∇ · *J* = 0. The Hartle–Hawking wave function is real, so the probability measure (113) would yield zero. These authors propose instead to use the customary probability measure of quantum mechanics,

$$J = \left|\phi\right|^2. \tag{114}$$

With these two choices, the probability for the creation of the universe reads [15]

$$P = \mathcal{J} \cdot d\Sigma \approx \exp\left(\pm \frac{2}{3H^2(\varrho)}\right) d\varrho,\tag{115}$$

where the + sign is for the no-boundary wave function and the − sign for the tunnelling wave function. Thus, as we have already noticed in Section 2.2, the no-boundary proposal seems to favour small values of the potential (*PHH* → ∞ for *H*(*ϕ*) → 0), and the tunnelling proposal seems to favour the creation of universe with a large value of the potential (*PT* → 0 for *H*(*ϕ*) → 0). Therefore, it is usually stated that Vilenkin's tunnelling condition fits better with the inflationary scenario [25,43].

#### 2.5.2. Creation of the Universe from *Something*

Let us now analyse the case *E* ∈ (0, U*max*) in (90). The corresponding Friedman equation is obtained by substituting the value, *pa* = −*aa*˙, in (90). It yields

$$\left(\frac{d}{a}\right)^2 = H\_0^2 - \frac{1}{a^2} + \frac{2E}{a^4}.\tag{116}$$

We can see that the conformally massless scalar field can reproduce the effect of a radiation content of the universe (*ρ* ∼ *a* −4 , hH*m*i ∝ *a* −1 in (86)). The Friedman Equation (116) can also be written as

$$d = \frac{H\_0}{a} \sqrt{(a^2 - a\_+^2)(a^2 - a\_-^2)},\tag{117}$$

with [26,45]

$$a\_{\pm}^2 = \frac{1}{2H\_0^2} \left( 1 \pm \left( 1 - 8EH\_0^2 \right)^{\frac{1}{2}} \right). \tag{118}$$

For the value, 1 8*H*2 0 = U*max* > *E* > 0, the two Lorentzian regions are located at *a* > *a*+ and *a* < *a*−, respectively. These two sectors represent two separated regions where the universe may exist. In between, there is the Euclidean sector, which is a classically forbidden region. We have therefore two different types of universes separated by a quantum barrier (see Figure 7). In region *III*, the solution of (117) can be written as [36]

$$a(t) = \left(a\_-^2 \cosh^2 H\_0 \Delta t - a\_+^2 \sinh^2 H\_0 \Delta t\right)^{\frac{1}{2}},\tag{119}$$

where, ∆*t* = *t* − *t*0, with

$$
\tilde{t}\_0 = \frac{1}{H\_0} \operatorname{arctanh} \frac{a\_-}{a\_+}.\tag{120}
$$

It represents a small universe that starts in a big-bang-like singularity at *t* = 0, expands to the maximum value *a*−, at *t* = ˜*t*0, and then re-collapses to a big-crunch-like singularity, at *t* = 2˜*t*0. For a value *H*0 1, the evolution of this type of universe is like the evolution of a radiation-dominated universe (see Figure 10). This kind of universes are called *baby* universes [46], which are typically associated with quantum fluctuations of the spacetime.

**Figure 10.** The universes of region *III* are cyclic universes that start from a big-bang singularity and end in a big-crunch one. In region *I*, the universe effectively behaves like a closed DeSitter spacetime. On the other hand, the solution of (116) in region *I* can be written as [36]

$$a(t) = \left(a\_+^2 \cosh^2 H\_0 \Delta t - a\_-^2 \sinh^2 H\_0 \Delta t\right)^{\frac{1}{2}},\tag{121}$$

with ∆*t* ∈ (−∞, ∞). It represents a universe that contracts from infinity to the minimum value *a*+, reached at ∆*t* = 0, and then, it expands again to infinity (see, Figure 10). This solution is essentially very similar to the closed DeSitter universe. In fact, it can continuously be transformed into the customary solution of the closed DeSitter spacetime in the limit *E* → 0, for which *a*− → 0 and *a*+ → 1/*H*0.

In between, there is a tunnelling region, where the solution of the Euclidean version of the Friedman equation is the Euclidean instanton (102) with the scale factor given by

$$a\_E(\tau) = \left(a\_+^2 \sin^2 H\_0 \Delta \tau + a\_-^2 \cos^2 H \Delta \tau\right)^{\frac{1}{2}},\tag{122}$$

where *a*± is given by (118), *a* ∈ (*a*−, *a*+), and

$$
\Delta \tau = \tau - \tau\_0 \in (0, \frac{\pi}{2H\_0}). \tag{123}
$$

The Euclidean instanton with scale factor (122) connects the maximum expansion hypersurface Σ(*a*−) of the baby universes of region *I* with the initial hypersurface Σ(*a*+) of the large parent universe in region *I* (see Figure 11). It therefore connects the two regions, and it also provides the first-order contribution to the probability of crossing the quantum barrier and appearing in region *I* as a new-born universe (see, Ref. [36]). The universe is then said to be created *from something*.

**Figure 11.** The creation of a large parent universe from a baby universe.

Gott and Li: The Universe Is Its Own Mother

The question of whether the universe is created from *nothing* or from *something* seems to be rooted in the value of the energy *E* in the Friedman Equation (90). If *E* = 0, the universe must be created from nothing, and if *E* > 0, the universe must be created from something. However, the value *E* = 0 is controversial because from (96), *E* = *n* + 1/2, so the value *E* = 0 would violate the uncertainty principle of quantum mechanics [45]. In their original study [12], Hartle and Hawking suggested that this term might be cancelled by some renormalisation procedure. However, Gott and Li [45] argue that there is no expectation for such an exact cancellation, and in fact, Barvinsky and Kamenshchik [47–49] computed the renormalisation corrections and not only is the energy term not cancelled but new similar terms appear. Even more, we have seen that the backreaction of the perturbation modes of the spacetime produces a similar term in the Friedmann equation (their energy density is given by hH*m*i/*V* ∝ *a* −4 , where hH*m*i ∝ *a* −1 in (85) for the perturbations of the spacetime). Therefore, a term with *E* > 0 seems to be unavoidable. One may argue that it can be effectively small, and thus it could be neglected. However,

regardless of how small it can be, we have seen that the consequences for the creation of the universe are very important.

A conceptual problem arises however if the universe is created from "something" because then we need the existence of that "something" prior to the creation of the universe, i.e., if the universe is created from the tunnelling of a quantum fluctuation of a pre-existing spacetime, then, one should explain how the *first* spacetime has been created from which the rest of the universes have subsequently been generated. Gott and Li give an apparently exotic although quite interesting explanation. They argue and show [45] that in region *III* there can exist closed temporal curves (CTC's). In that case, the spacetime fluctuations of a large parent spacetime can travel through a CTC and become the baby universe that "tunnelled out" through the Euclidean barrier to give rise to the parent universe in an atemporal process in which terms like "after" or "before" become meaningless. They are only meaningful within the large parent regions of the spacetime where CTCs do not exist. Thus, according to these authors, the universe could be its own mother.

#### 2.5.3. Creation of Universes in Pairs