### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Wheeler–DeWitt cosmology and observations
- 3 Loop quantum cosmology and observations
- 4 Non-Gaussianity
- 5 Outlook
- Acknowledgments
- References
- Biography

The status of quantum cosmologies as testable models of the early universe is assessed in the context of inflation. While traditional Wheeler–DeWitt quantization is unable to produce sizable effects in the cosmic microwave background, the more recent loop quantum cosmology can generate potentially detectable departures from the standard cosmic spectrum. Thus, present observations constrain the parameter space of the model, which could be made falsifiable by near-future experiments.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Wheeler–DeWitt cosmology and observations
- 3 Loop quantum cosmology and observations
- 4 Non-Gaussianity
- 5 Outlook
- Acknowledgments
- References
- Biography

During the last years, quantum gravity has been receiving a great amount of attention from the community of theoreticians. The driving motivation, familiar to anyone who has tried his or her fortune at least once in this broad subject, is to realize a consistent, ultraviolet finite merging of general relativity with quantum mechanics. The programme can be carried out in various forms, from ambitious theories of everything (such as string theory) where all forces are unified to more minimalistic approaches aiming to quantize gravity alone. In the latter category there fall loop quantum gravity (LQG), asymptotic safety, spin-foams, causal dynamical triangulations and many others [1].

A problem endemic to most of these scenarios is their difficulty in making contact with observations. This stems from the highly technical nature of the theoretical frameworks, where the notions of conventional geometry and matter, continuum spacetime, general covariance and physical observables are typically deformed, modified, or disappear altogether. The lack of experimental feedback makes it quite difficult to discriminate among different models and, chiefly, to characterize them as *falsifiable*.

It is natural to turn to cosmology in an attempt to bridge this gap and advance our knowledge [2, 3]. The early Universe is an ideal laboratory where extreme regimes of high energy and high curvature are realized. Under such conditions, it is expected that quantum gravitational effects become sizable. Also, the symmetry reduction entailed in cosmological settings decimates the degrees of freedom of background-independent theories and allows one to simplify the latter to a technically manageable level. The resulting models retain some (or most) of the main features of the full theory and can be better manipulated to extract observables.

Canonical quantum gravity is a popular example of this mechanism. The present review focusses on two of its incarnations, namely, the traditional Wheeler–DeWitt (WDW) model (e.g., [4, 5]) and the more recent loop quantization [6]. The most ancient phase about which we have gathered experimental data is inflation, a period of accelerated expansion of the universe which left a relic in the cosmic microwave background (CMB) radiation. A study of the inflationary perturbations and the associated spectra allows us to track down quantum corrections and confront them with the observed CMB power spectrum. Although the outcome of this procedure is a constraint on the free parameters of the models rather than an actual prediction, time seems ripe for the very next generation of experiments to exclude notable portions of parameter space. As a minimal present-day achievement, we can at least state that quantum cosmology models are compatible with observations.

The stark contrast between the type of quantum corrections arising in these scenarios highlights how sensitive the physics is of the quantization scheme and variables. The typical energy scale during inflation is estimated to be about the grand-unification scale, , corresponding to an energy density . Here is the Hubble parameter, *a* is the scale factor of the universe and a dot denotes differentiation with respect to synchronous time. In contrast, classical gravity is believed to break down at distances shorter than the Planck length , i.e., at energies above 10^{19} GeV. The ratio between the inflationary and Planck energy density is very small,

- (1)

and quantum corrections are expected to be of the same order of magnitude or lower. Thus, quantum-gravity effects would be, in fact, well below any reasonable experimental sensitivity threshold, at least as far as inflation is concerned. WDW quantum cosmology realizes precisely this type of corrections and endorses the above naive argument.

On the other hand, the polymeric quantization of loop quantum cosmology (LQC) [5, 7, 8] generates corrections which are not of the form (1). To get a rough idea of how these corrections arise, one begins by observing that geometry operators representing areas and volumes acquire a discrete spectrum in this context. This is because states of loop quantum gravity, spin networks, are graphs whose edges *e* are labeled by quantum numbers . An area intersected by some of these edges is determined by these quantum numbers, giving the spectrum , where is the Barbero–Immirzi parameter. One single edge defines an “elementary plaquette” of area ; the latter features the Planck area but its actual value depends on the spin quantum number. Since calculations on realistic graphs are very hard in the full theory, it is convenient to focus one's attention on a simplified phenomenological setting. In particular, a homogeneous quantum inflationary universe with small inhomogeneous perturbations may be represented by a quantum semi-classical state Ψ characterized by a length scale *L*. This scale is thought of as encoding the discreteness of the geometry. Any region of volume (arbitrary, if spatial slices are non-compact) can be decomposes into discrete patches of size . The inflationary scale is thus replaced by an effective quantum-gravity scale

- (2)

In general, inverse powers of *L* cannot be quantized to a densely defined operator because the spectrum of the volume contains 0. Inverse volumes appear in the Hamiltonian constraint (of both gravity and matter, as in kinetic matter terms) and hence in the dynamics, and are an unavoidable consequence of spatial discreteness in loop quantum gravity. This requires to reexpress their classical expressions via Poisson brackets, which in turn feature derivatives by *L*. Quantum discreteness then replaces classical continuous derivatives by finite-difference quotients. For example, the expression would become , strongly differing from when *L* is as small as the Planck length, . For larger *L*, corrections are perturbative and of the order , so in general the type of inverse-volume quantum corrections are expressed by the ratio

- (3)

In practice, the actual size of LQC effects will lie well below the over-optimistic upper bound (3), but above the naive estimate (1). It is known that the non-local nature of loop quantum gravity effects prevents the formation of singularities one would typically find classically [6, 9, 10]. This can be shown both at the kinematical level (via the spectra of inverse area and volume operators) and at the exact and effective dynamical level (by looking, respectively, at the state-space spanned by the Hamiltonian constraint acting on volume eigenstates and at the effective dynamics on semi-classical states). The physical interpretation of inverse-volume corrections stems exactly from the same mechanism: classically divergent quantities such as inverse powers of volumes remain finite due to intrinsically quantum effects. Loosely speaking, quanta of geometry cannot be compressed too densely and they determine the onset of a repulsive force at Planck scale [10], which then determine the various corrections to the dynamics.

Before starting, we stress once again the scope of the present review. Although there are many “minimalistic” theories of quantum gravity on the market, at present it is still difficult to do some cosmology with them. Among the scenarios allowing for some phenomenology are asymptotic safety [11] and causal dynamical triangulations [12]. These models do admit a cosmological limit, but either inflationary observables have not been computed yet or there is no unique determination of an effective inflationary gravitational action. Here, on the other hand, we are interested in pitching models based upon canonical quantization (which conventionally go under the umbrella term “quantum cosmology”) against observations. We will leave out string cosmology from the discussion [13, 14], which is based on altogether different techniques. The reader can find the details of various and often interconnected settings in the dedicated literature, such as KKLT and moduli inflation [15, 16], cosmic strings networks [17, 18], brane and DBI inflation [13, 14], string gas cosmology [19, 20], braneworld cosmology [21], ekpyrotic universe [22, 23], non-local cosmology, and others.

### 4 Non-Gaussianity

- Top of page
- Abstract
- 1 Introduction
- 2 Wheeler–DeWitt cosmology and observations
- 3 Loop quantum cosmology and observations
- 4 Non-Gaussianity
- 5 Outlook
- Acknowledgments
- References
- Biography

The effect of quantum corrections goes beyond linear perturbation theory and higher-order observables can be calculated. As the perturbative level increases, the statistics of inhomogeneous fluctuations deviates from the Gaussian one and odd-order correlation functions acquire non-vanishing values. In particular, the bispectrum (three-point correlation function of the curvature perturbation) can be constrained by observations.

To the best of our knowledge, there is only one work on inflationary non-Gaussianity in loop quantum cosmology with inverse-volume corrections [82], and none in the WDW case. A detailed calculation of the momentum-dependent bispectrum shows that no appreciable LQC signal can be detected. We can in fact reach the same conclusion here by a model-independent shortcut, valid only in the so-called squeezed limit (constant non-linear parameter) but beyond perturbation theory and both for LQC and WDW quantum cosmology.

Let ζ be the curvature perturbation on uniform density hypersurfaces. The latter is a gauge-invariant quantity proportional to the comoving curvature perturbation in standard inflation; their relation in the presence of inverse-volume corrections has not been studied yet, but what follows is fairly independent on this detail. In momentum space, the three-point correlation function of ζ is

- (66)

where , called bispectrum, is defined by

- (67)

where , *f*_{NL} is called non-linear parameter and is momentum dependent in general, and is the spectrum of ζ. The form of the non-linear parameter depends on the model of primordial perturbations. In the simplest case [83-85], one decomposes the non-linear curvature perturbation into a Gaussian linear part ζ and a non-linear part:

- (68)

where the non-linear parameter is constant. By definition, . Then, a direct calculation of the bispectrum shows that

- (69)

In fact, the Fourier transform of the non-linear part is

- (70)

The first term stems from the fact that the auto-correlat-ion function is **x** independent. Since all momenta must not vanish at the same time, this piece can be thrown away. The second term enters into the three-point function, which at lowest order is (e.g., [5])

- (71)

which yields Eq. (69) after comparing Eqs. (66) and (67). The decomposition (68) is pointwise in configuration space and for this reason it is called *local model*. For a power-law scalar spectrum , the local bispectrum reads

- (72)

where is a constant amplitude. This expression can be converted into one with spherical multipoles.

### 5 Outlook

- Top of page
- Abstract
- 1 Introduction
- 2 Wheeler–DeWitt cosmology and observations
- 3 Loop quantum cosmology and observations
- 4 Non-Gaussianity
- 5 Outlook
- Acknowledgments
- References
- Biography

Quantum gravitational effects modify the spectra of cosmological perturbations and their imprint in the cosmic microwave background. In this paper, we compared two canonical approaches, the one based on the usual Wheeler–DeWitt quantization and loop quantum cosmology. Wheeler–DeWitt quantum corrections are too small to be detected, even in the most optimistic upper bound, Eq. (34). The model therefore is not falsifiable, at least under the assumptions made in the derivation of the results, but at least it is compatible with what we observe.

In contrast, LQC inverse-volume corrections can be of much greater size and produce an enhancement, rather than suppression, of the large-scale spectra. While in the WDW case quantum corrections change the inhomogeneous dynamics but leave homogeneous background equations unmodified, in LQC the latter are deformed, too. However, this is not the reason why LQC effects are potentially several orders of magnitude larger than the WDW quantization. Rather, the key ingredient is the scale compared with the Planck energy density ρ_{Pl} in the ratio defining the quantum correction: for WDW it is the inflationary scale ρ_{infl}, for LQC it is determined by the characteristic discreteness scale of the semi-classical state describing the quantum universe. This effective energy density can be as large as the Planck density, .

This also highlights the different origin of the observational bounds presented above. While the WDW quantum correction (34) is constrained somewhat indirectly via the usual bounds on the inflationary energy scale, in LQC we have some free parameters on which we have little control theoretically, due to the formidable (and yet unsurmounted) difficulties in explicit constructions of cosmological semi-classical states in the full theory. LQC inverse-volume corrections depend on a phenomenological quantum-gravity scale as well as on partly heuristic, partly quantitative arguments indicating how to implement discrete quantum geometry in a quasi-homogeneous cosmological setting. A multi-variate likelihood analysis involving all the cosmological parameters, including LQC ones, is thus more adequate to the task.

Observations constrain LQC inverse-volume quantum corrections below their theoretical upper bound, but in some instances the signal is above the threshold of cosmic variance. Experiments such as PLANCK or of the next generations should then be able to reach the sensitivity to detect a quantum gravity signal or, in its absence, place yet more stringent constraints. In turn, pressure from actual data will stimulate the quest for a better understanding of the fundamental properties of the states of the full theory, and a greater control over parameters which, as the discreteness scale *L*, are presently phenomenological.