Astrophysical and cosmological signatures of Loop Quantum Gravity
Aurélien Barrau (2017), Scholarpedia, 12(10):33321. | doi:10.4249/scholarpedia.33321 | revision #184853 [link to/cite this article] |
The loop quantum gravity theory is a background-independent and non-perturbative quantization of general relativity. It has been developed both in the canonical and covariant formalisms. As for all tentative theories of quantum gravity, one of the key question is to produce clear experimental or observational predictions. As the discrete nature of spacetime at the Planck scale is extremely difficult to probe directly, phenomenological attempts do focus mostly on the cosmological sector, on black holes and on possible high energy astrophysics effects.
The main prediction of loop quantum gravity in the cosmological sector is that the Big Bang is replaced by a Big Bounce: a contracting phase should have taken place before the expanding phase we are now living in. In the black hole sector, the Hawking evaporation spectrum should receive substantial corrections and the very existence of an event horizon can be questioned, potentially leading to black holes bouncing into white holes. Finally, on the astroparticle physics side, the quantum discreteness of spacetime could lead to interesting effects for high energy cosmic rays.
Contents |
Brief introduction to loop quantum gravity
This article uses Planck units except in some formulae where the constants where reinserted to help the understanding.
Canonical approach
In the canonical approach, classical general relativity can be expressed in terms of a spatial metric \(q_{ab}\), describing a 3-slice \(\Sigma\), and the extrinsic curvature \(K_{ab}\) which can be seen as the derivative of this metric with respect to a change of slice. The Hamiltonian and diffeomorphism constraints do then ensure that the fields are indeed solutions to the classical gravitational equations, by generating deformations of the spatial slice and by restoring general covariance. In this framework, it is quite natural to use - instead of the spatial metric - the triad field \(e^a_i\) (square root of the metric) defined by \(q^{ab}=\delta^{ij}e^a_ie^b_i\). This brings an SO(3) gauge freedom which is in this case accounted for by the Gauss constraint.
To quantize such a constrained system, a quite specific situation in physics, one can follow the Dirac prescription. This task and the calculation of observables requires a Hilbert space of functionals \(\psi[q]\) of spatial metrics. The space of metrics - or extrinsic curvature tensors - is however mathematically poorly understood and defining a suitable inner products (the key ingredient of the Hilbert space) is quite difficult. This is one of the reasons why new variables, introduced by Ashtekar, play an important role (an introduction can be found in Bojowald 2006) in loop quantum gravity. Their actual definition is based on the densitized triad \(E^a_i\) which is related to the triad by \(E^a_i=\left|\det e^b_j\right|^{-1} e^a_i\) and its conjugate, the so-called extrinsic curvature coefficients \(K_a^i:=K_{ab}e^b_i\) with \( \{K_a^i(x),E^b_j(y)\}= 8\pi G\delta^b_a\delta^i_j\delta(x-y) \). The Ashtekar connection is finally introduced through \(A_a^i=\Gamma_a^i+\gamma K_a^i\), where \(\gamma\) s the Barbero-Immirzi parameter which has no classical relevance but can play an important role for the quantized theory, and \[ \Gamma_a^i = -\epsilon^{ijk}e^b_j (\partial_{[a}e_{b]}^k+ {\textstyle\frac{1}{2}} e_k^ce_a^l\partial_{[c}e_{b]}^l), \tag{1}\] is the spin connection of the triad field. The Ashtekar connection is conjugate to the densitized triad with the Poisson bracket: \[ \{A_a^i(x),E^b_j(y)\}=8\pi\gamma G\delta_a^b\delta_j^i\delta(x,y). \tag{2}\] The dynamics is given by the Hamiltonian constraint ($N$ being the laps and $F_{ab}^i$ being the curvature of the Ashtekar connection): \[ H[N]=\frac{1}{16\pi\gamma G} \int_{\Sigma} \mathrm{d}^3x N\left|\det E\right|^{-1/2} \left(\epsilon_{ijk}F_{ab}^iE^a_jE^b_k -2(1+\gamma^2) K^i_{[a}K^j_{b]}E^a_iE^b_j\right)\approx 0, \tag{3} \] where \(F_{ab}^i\) is the curvature of the Ashtekar connection.
The Gauss constraint reads \[ G[\Lambda]=\frac{1}{8\pi\gamma G}\int_{\Sigma}\mathrm{d}^3x \Lambda^iD_aE^a_i= \frac{1}{8\pi\gamma G} \int_{\Sigma}\mathrm{d}^3x\Lambda^i(\partial_aE^a_i+ \epsilon_{ijk}A^j_aE^a_k)\approx 0, \tag{4} \] and the diffeomorphism constraint is given by \[ D[N^a]= \frac{1}{8\pi\gamma G}\int_{\Sigma}\mathrm{d}^3x N^aF_{ab}^iE_i^b \approx 0. \tag{5} \]
The Algebra of constraints is said to be "closed" and constitues a first class Dirac Algebra.
When trying to quantize a field theory, it is required to smear fields: they need to be integrated in order to get rid of the \(\delta\)-functions. In this specific framework, this should be performed using Ashtekar variables, taking care not to spoil the background invariance, i.e. without relying on a given a priori metric. To this aim, the connection is integrated along a one-dimensional curve $e$ and then exponentiated in a path-ordered manner (a covariant object is then constructed from the non-Abelian connection). This defines the holonomy, which plays an important role in the theory:
\[ h_e(A)={\cal P}\exp\int_e\tau_i A_a^i\dot{e}^a\mathrm{d}t\tag{6}, \] with \(\dot{e}^a\) a vector tangent to the curve \(e\) and \(\tau_j=-\frac{1}{2}i\sigma_j\) where \(\sigma_j\)'s are the Pauli matrices. Following the same logics, the densitized triads are integrated over a 2-surface denoted \(S\), resulting in the flux \[ F_S(E)=\int_S \tau^i E^a_in_a\mathrm{d}^2y\tag{7}, \] where $n_a$ is the co-normal to \(S\).
The Poisson algebra of holonomies and fluxes is well-defined. It can be represented on a Hilbert space. A unique representation which defines the kinematical Hilbert space is then obtained when diffeomorphism invariance is imposed. The Hilbert space is basically the one of an \(SU(2)\) lattice Yang-Mills theory. Thanks to Penrose theorem, it can be interpreted as a description of quantized geometries. The dynamics is given by the quantum Hamiltonian constraint (see for example Ashtekar 2012). These are the key ingredients of loop quantum gravity (LQG).
Covariant approach
There is also a path-integral formulation of LQG, explicitly covariant, which relies on spin foams (see Rovelli 2011). This approach is based on the formal "sum over 4-geometries" \[ Z \sim \int Dg \ \ e^{\frac{i}{\hbar}\! \int \! \!R\sqrt{g}\,d^4\!x}. \tag{8} \] This famous expression is however hard to use from the practical viewpoint.
In 4-dimensional Lorentzian LQG, the partition function is given by \[ Z_{\cal C}=\sum_{j_f,\mathtt{v}_e} \ \prod_f (2j_f+1)\ \prod_v A_v(j_f,\mathtt{v}_e), \tag{9} \] where \(\cal C\) is a 2-complex with faces \(f\), edges \(e\) and vertices \(v\), whereas \(\mathtt{v}_e\) are the intertwiners of the vertices, and \[ A_v(j_f,\mathtt{v}_e)=Tr\left[\otimes_{e\in v} (f_\gamma \mathtt{v}_{e})\right], \tag{10} \] where \(f_\gamma\) somehow sends \(SU(2)\) spin networks into \(SL(2,\C)\) spin networks. This basically determines the covariant dynamics of LQG. It is worth noticing that the theory is in a way determined by the imbedding of \(SU(2)\) functions into \(SL(2,\C)\) functions.
A key LQG result for phenomenology - that will be used in the following - is that areas are now quantized with a spectrum \[ A_j=8\pi \gamma l_{Pl}^2\sqrt{j(j+1)}, \tag{11} \] where \(j\) is a half integer.
Loop quantum cosmology
Basic ideas
Loop quantum cosmology (LQC) is the application of loop quantum gravity ideas and methods to the Universe itself (see Ashtekar & Singh 2011). Let us begin by considering isotropic and homogeneous loop quantum cosmology. The classical FLRW (Friedmann-Lemaitre-Robertson-Walker) universe is entirely described by the scale factor \(a(t)\), which is a solution to the Einstein's equations written as Friedmann's equations. Using Ashtekar variables, the scale factor and its time derivative can be expressed by a densitized triad component \(\tilde{p}\) with \(|\tilde{p}|=a^2\), determined by the isotropic densitized triad \(E^a_i=\tilde{p}\delta^a_i\), and a connection variable \(\tilde{c}=\gamma\dot{a}\), determined by the isotropic connection \(A_a^i=\tilde{c}\delta_a^i\). The volume of a given region is \(V=\int_Rd^3x\sqrt{|\det E|}\). For an isotropic triad, it reduces to \(V=V_0|\tilde{p}|^{3/2}=|p|^{3/2}\) where the coordinate volume \(V_0\) is absorbed in \(p\). The re-scaled variables \(p\) and \(c=V_0^{1/3}\tilde{c}\) are then independent of the fiducial cell. The Poisson bracket is given by: \[ \{c,p\}=\frac{8\pi\gamma G}{3}\,. \tag{12} \]
Following LQG ideas and prescriptions, these fundamental objects are represented on a Hilbert space with a (orthonormal) basis \(\{|\mu\rangle\}_{\mu\in\R}\) of states made of functions of the connection component by \(\langle c|\mu\rangle = e^{i\mu c/2}\). Terms like \(e^{i\mu'c/2}\) are analogous to holonomies of the full LQG theory and define a shift operator: \[ \widehat{e^{i\mu'c/2}}|\mu\rangle = |\mu+\mu’\rangle, \tag{13} \] while the triad component $p$ acts by multiplication: \[ \hat{p}|\mu\rangle = \frac{1}{6}\gamma l_{Pl}^2\mu|\mu\rangle\,. \tag{14} \] An important property of the "mother theory" is preserved: only exponentials of $c$ can be represented as it is not possible to obtain directly an operator for the component $c$ and the triad operator $\hat{p}$ has a fundmentally discrete spectrum since the eigenstates are normalizable. From \(\hat{p}\) one obtains the operator associated with the volume \(\hat{V}=|\hat{p}|^{3/2}\).
In LQC, the basic equation is a difference equation and not a differential equation as in the usual Wheeler-deWitt theory (see Bojowald 2012). Written in terms of the wave function of the Universe (filled with a scalar field \(\phi\)) it can be shown to be (Bojowald 2003,Bojowald 2004) in the isotropic and flat case:
\[
(V_{\mu+5}-V_{\mu+3})e^{ik}\psi_{\mu+4}(\phi)- 2(V_{\mu+1}-V_{\mu-1})\psi_{\mu}(\phi)
+ (V_{\mu-3}-V_{\mu-5})e^{-ik}\psi_{\mu-4}(\phi)
= -\frac{4}{3}\pi \gamma^3G\ell_{\rm P}^2\hat{H}_{\rm matter}(\mu)\psi_{\mu}(\phi),
\tag{15}
\]
where \(V_{\mu}\) are the volume operator eigenvalues and \(\hat{H}_{\rm matter}(\mu)\) is the matter Hamiltonian.
This equation does not have a singular behavior (Bojowald, 2001). Starting from a large volume, it is possible to evolve the wave function of the Universe backward through the classical singularity without any pathology. The evolution does not stop and one obtains a collapsing branch preceding the classical Big Bang singularity (see details in Bojowald 2006). This is the main ingredient of the LQC Big Bounce scenario. The existence of this bounce, replacing the Big Bang, has been demonstrated in LQC to be true for exactly solvable models (a flat, isotropic universe filled with a massless scalar field), in the presence of spatial curvature, for anisotropic Bianchi models, with both a positive and a negative cosmological constant, for \(\phi^2\) inflationary potentials, for barotropic matter with different equations of state, and for wide states corresponding to highly quantum universes. In addition, numerical simulations (see Singh 2012, Diener et al. 2014) have been performed to show that the quantum evolution is quite generically well described by effective semi-classical equations.
The main basic LQC result is that the effective Friedmann equation now reads:
\[
H^2=\frac{8\pi G\rho}{3}\left(1-\frac{\rho}{\rho_c}\right), \tag{16}
\]
where \(\rho\) is the density and \(\rho_c\) is the critical density (expected to be of the order of the Planck density and more precisely equal to \(0.4\rho_{Pl}\) under quite natural hypotheses). The usual Friedmann equation receives a quadratic correction with the correct sign so that the Universe now bounces.
At this stage, loop quantum cosmology is a framework that imports techniques and ideas from LQG, but it is not yet using the full quantum theory. There could be higher order effects - or other non-perturbative features - missing in the effective cosmological equations. However, the effective equations seem to capture the most relevant and leading quantum effects, and the conclusions are supported by results obtained using techniques in the full theory. This is obviously the case of the singularity resolution in fully covariant LQG (see Bianchi et al. 2010).
In addition, approaches based of gauge fixing (referred to as "reduced LQG", see Alesci et al. 2014) where the quantization is somehow performed before quantization indeed give an Hamiltonian yielding nearly the same modification of the Friedman equations as in standard LQC.
Predictions for the background
The clear description of the background dynamics is one of the key successes of LQC (see Ashtekar et al. 2006). Interestingly, this result has also been recovered in the framework of group field theory (see Oriti et al. 2016). The large scale cosmological dynamics is then described by the hydrodynamics of condensate states in the Gross-Pitaevskii approximation and the same modified equations is derived (under some specific assumptions).
However the model becomes also interesting because it does more than predicting a bounce (see Fig. 1 for the dynamics of the field). Under some hypotheses, it light also predict the duration of inflation. This is something quite specific (see Ashtekar & Sloan 2011). to this approach. The fact that inflation occurs generically in LQC once the appropriate content - e.g. a massive scalar field - is assumed should not come as a surprise: inflation is a strong attractor and starting with an energy density close to the Planck density yields nearly inevitably to inflation. The interest of LQC is here more subtle.
One can define the fractions of potential and kinetic energy, normalized to the maximum energy density, \[ x := \frac{m\phi}{\sqrt{2\rho_{\text{c}}}} \quad \text{and} \quad y :=\frac{\dot{\phi}}{\sqrt{2\rho_{\text{c}}}}, \tag{17} \] so that \[ \rho=\rho_c\left(x^2+y^2\right). \tag{18} \] Interestingly, in the contracting phase of the Universe, far away (before) from the Big Bounce, the density can be shown to be well approximated by \[ \rho=\rho_0\left(1-\frac{1}{2}\sqrt{3\kappa\rho_0}\left( t+\frac{1}{2m}\sin(2mt+2\delta)\right)\right)^{-2}.\tag{19} \] The \(\delta\) parameter is just a phase and can be naturally assigned a flat a priori probability distribution function (PDF). In addition of being the expected distribution for any random oscillatory process of this kind, a flat PDF for \(\delta\) will be preserved over time within the prebounce oscillation phase, making it a quite obvious choice for initial conditions. The important point is that starting from a flat PDF for \(\delta\) and a small enough initial density \(\rho_0\), the distribution of the fraction of potential energy at the bounce \(x_B\) can be explicitly calculated and is far from being flat between 0 and 1. It is sharply peaked around \(3.5\times 10^{-6}\) (this value scales with \(m\), the mass of the scalar field, as \(m\log\left(\frac{1}{m}\right)\) and is here given for the usual value around \(m\approx 10^{-6}\) in Planck units).
Remarkably, this distribution for \(x_B\) can be precisely translated into a PDF for the number \(N\) of e-folds of inflation (defined as the logarithm of the ratio of the scale factor evaluated just after and just before inflation). The number of e-folds could a priori be any number up to a huge value bounded from above by the total energy available. If one assumes a flat PDF for \(x_B\), as sometimes done in cosmology, the most probable value of \(N\) is around \(10^{12}\). But if the causal evolution is taken seriously and if the PDF for \(x_B\) expected from LQC (that is coming from a flat PDF for \(\delta\)) is taken into account, the prediction changes drastically and leads to a value of \(N\) peaked around 140 (see Linsefors & Barrau 2013a).
This result assumes isotropy. However, in bouncing cosmologies, the issue of anisotropies is crucial for a simple reason: the shear term scales as \(1/a^6\) where \(a\) is the mean scale factor of the Universe. When the Universe is in its contraction phase, it is therefore expected that the shear term eventually dominates and drives the dynamics at some point. The LQC background dynamics has also been investigaed in several anisotropic settings (see Ashtekar & Wilson-Ewing 2009, Linsefors & Barrau 2013b). The key result is that the bounce prediction remains correct even if the shear drives the dynamics (see Linsefors & Barrau 2014). In a Bianchi-I universe, the LQC-modified effective Friedmann equation reads as \[ H^2=\sigma_Q+\frac{\kappa}{3}\rho-\lambda^2\gamma^2\left(\frac{3}{2}\sigma_Q+\frac{\kappa}{3}\rho\right)^2, \tag{20} \] where \(\sigma_Q\) defines the "quantum shear" as: \[ \sigma_Q:=\frac{1}{3\lambda^2\gamma^2}\left(1-\frac{1}{3}\Big[\cos(\bar{\mu}_1c_1-\bar{\mu}_2c_2)+\cos(\bar{\mu}_2c_2-\bar{\mu}_3c_3)+\cos(\bar{\mu}_3c_3-\bar{\mu}_1c_1) \Big]\right), \tag{21} \] with \[ \bar{\mu}_1 = \lambda\sqrt{\frac{p_1}{p_2p_3}} \qquad \text{and cyclic expressions,} \tag{22} \] where \(\lambda\) is given by the square root of the minimum area eigenvalue of the LQG area operator (\(\lambda=\sqrt{\Delta}\)) and the \(p_i\)’s are conjugate to the \(c_i\)’s (both are diagonal elements of the Ashtekar variables) so that: \[ \{c_i,p_j\}=\kappa\gamma\delta_{ij}\quad ,\quad \{\phi_n,\pi_m\}=\delta_{mn}. \tag{23} \] In this framework, it is possible to evaluate the PDF for the number of e-folds of inflation. As expected it is lowered when compared to the isotropic setting. This is promising as this might make the actual number of e-folds close to the experimental lower bound which corresponds to cases with a rich phenomenology. (If the number of e-folds is much greater than 70, the interesting quantum gravity effects are beyond the horizon.)
Predictions for the perturbations
The situation for perturbations is less clear than for the background (see Barrau et al. 2013). Different approaches to treat cosmological perturbations have been developed in LQC.
The dressed metric approach (see Agullo et al. 2012, Agullo et al. 2013) relies on a minisuperspace strategy in which the homogeneous and isotropic degrees of freedom, together with the inhomogoneous ones, are all quantized. The former is obtained by the loop quantization and the latter is derived through a "Fock-like" quantization on a fully quantum background (see Ashtekar et al. 2009). The inhomogeneous degrees of freedom considered as small perturbations are given by the usual gauge-invariant Mukhanov-Sasaki variables derived from the linearized classical constraints. This field is essential in the theory of cosmological perturbations. The second order Hamiltonian is promoted to be an operator and the quantization is carried out relying on specific techniques suitable for the quantization of a "test field" evolving on top of a quantum background. The Hilbert space is the tensor product \(\Psi(\nu,v_\mathrm{S(T)},\varphi)=\Psi_\mathrm{FLRW}(\nu,\bar\varphi)\otimes\Psi_\mathrm{pert}(v_\mathrm{S},v_\mathrm{T},\bar\varphi)\) with \(\nu\) the homogeneous and isotropic degrees of freedom and \(v_\mathrm{S(T)}\) the degrees of freedom associated with small perturbations. As long as the backreaction of the perturbations on \(\Psi_{\rm FLRW}\)is negligible, the Schrödinger equation for the perturbations was shown to be identical to the Schrödinger equation for the quantized perturbations evolving in a classical background, but using a specific dressed metric. It reads for tensor modes: \[ i\hbar\partial_{\bar\varphi}\Psi_\mathrm{pert}=\frac{1}{2}\displaystyle\int \frac{d^3k}{(2\pi)^3}\left\{\frac{32\pi G}{\tilde{p}_{\varphi}}\left|\hat\pi_{\mathrm{T},\vec{k}}\right|^2\Psi_\mathrm{pert}+\frac{k^2}{32\pi G}\frac{\tilde{a}^4({\bar\varphi})}{\tilde{p}_{\varphi}}\left|\hat{v}_{\mathrm{T},\vec{k}}\right|^2\Psi_\mathrm{pert}\right\}, \tag{24} \] with \[ (\tilde{p}_{\varphi})^{-1}=\left<\hat{H}^{-1}_\mathrm{FLRW}\right> ~\mathrm{and}~\tilde{a}^4=\frac{\left<\hat{H}^{-1/2}_\mathrm{FLRW}\hat{a}^4({\bar\varphi})\hat{H}^{-1/2}_\mathrm{FLRW}\right>}{\left<\hat{H}^{-1}_\mathrm{FLRW}\right>}, \tag{25} \] where \((\hat{v}_{\mathrm{T},\vec{k}},\hat\pi_{\mathrm{T},\vec{k}})\) are the configuration and momentum operators of the perturbations while \(\hat{H}_\mathrm{FLRW}\) is the Hamiltonian operator for the background. A Fock quantization is finally performed and the modes are solutions of \[ Q''_k+2\left(\frac{\tilde{a}'}{\tilde{a}}\right)Q'_k+\left(k^2+\tilde{U}\right)Q_k=0, \\ h''_k+2\left(\frac{\tilde{a}'}{\tilde{a}}\right)h'_k+k^2h_k=0. \tag{26} \] The gauge-invariant variable \(Q_k\) is related with the Mukhanov-Sasaki variables for scalar modes via \(Q_k=v_{\mathrm{S},k}/a\), and \(\tilde{U}\) is a dressed potential-like term given by \[ \tilde{U}({\bar\varphi})=\frac{\left<\hat{H}^{-1/2}_\mathrm{FLRW}\hat{a}^2({\bar\varphi})\hat{U}({\bar\varphi})\hat{a}^2({\bar\varphi}) \hat{H}^{-1/2}_\mathrm{FLRW}\right>}{\left<\hat{H}^{-1/2}_\mathrm{FLRW}\hat{a}^4({\bar\varphi})\hat{H}^{-1/2}_\mathrm{FLRW}\right>}, \tag{27} \] the quantum equivalent of \[ U({\bar\varphi})=a^2\left(fV({\bar\varphi})-2\sqrt{f}\partial_{\bar\varphi} V+\partial^2_{\bar\varphi} V\right), \tag{28} \] with \(f=24\pi G (\dot{\bar\varphi}^2/\rho)\) the fraction of kinetic energy in the scalar field filling the universe.
This approach leads to spectra that are compatible with the current observations (see Fig. 2). In addition, the large scale modes (appearing in the IR part of the power spectrum) can be excited and not be anymore in the Bunch-Davies vacuum at the bounce time. This opens the exciting possibility to allow for a better fit of the data in the low-l (large scale) part of the CMB spectrum. Furthermore, there are specific predictions for future missions (correlations) that distinguish this LQG based mechanism from non-primordial mechanisms for power suppression.
A second approach (see Bojowald 2006 and Barrau et al. 2014), usually referred to as the "deformed algebra" one, has also been considered in LQC. It is in a way ``less quantum" by construction and probably less deeply rooted in the profound principles of the loop quantum gravity theory. But it is more focused on the important issue of the consistency of the resulting effective theory.
When quantum corrections are indeed taken into account at the effective level, it is difficult to {\it a priori} know whether the consistency conditions summarized in the first-class nature of the classical algebra of constraints remain unaltered.
Especially, in background-independent frameworks like LQG, it is obviously questionnable to rely on standard covariance arguments because the very notion of space-time should in itself emerge from solutions to the fundamental equations. The consistency of the equations must then in principle be ensured before they can be consistently solved. The preserved symmetries imply coherent equations based on first-class constraints (see Bojowald et al. 2008).
The ``deformed algebra" approach is based on considering seriously such gauge issues. It is true that gauge fixing before quantization was shown to be often harmless but the case of gravity is quite different from particule physics theories (dynamics is part of the gauge system).
In the "deformed algebra" approach, one actually considers LQG-inspired corrections to the constraints. Holonomy corrections (see, e.g., Wilson-Ewing 2011) are basically implemented in the relevant equations through the replacement
\[
\bar{k} \rightarrow \mathbb{K}[n] := \frac{\sin(n\bar{\mu} \gamma \bar{k})}{n\bar{\mu}\gamma},
\tag{29}
\]
where \(n\) is some unknown integer, \(\bar{k}\) is still the mean Ashtekar connection, and \(\bar{\mu}\) can bee understood as the coordinate size of a loop.
The quantum-corrected constraints are noted in the following \(\mathcal{C}^Q_I\). If the previous replacement is performed naively in the constraints, the new algebra reads
\[
\{ \mathcal{C}^Q_I, \mathcal{C}^Q_J \} = {f^K}_{IJ}(A^j_b,E^a_i) \mathcal{C}^Q_K+
\mathcal{A}_{IJ},
\tag{30}
\]
where \(\mathcal{A}_{IJ}\) stand for anomaly terms. The consistency condition (that is the algebra closure) imposes \(\mathcal{A}_{IJ}=0\).In turn, this condition leads to severe restrictions on the shape of the quantum corrections. The strategy is the following.
1) The quantum corrected constraints are explicitly written for the perturbations. 2) Then the Poisson brackets are calculated and the associated anomalies are evaluated. 3) Counterterms, which are required to vanish in the classical limit, are added to ensure the anomaly freedom. Importantly, the resulting theory is not only consistent but is also quite uniquely defined under reasonable assumptions. The resulting final algebra is simple and depends on a structure function \(\Omega=1-2\rho/\rho_c\) which encodes all the quantum corrections. Finally, the unknown integers \(n_i\) entering the correction functions \(\mathbb{K}[n_i]\) can all be determined when the matter contain is taken into account. The solution closes the algebra in a non-perturbative ways. This strategy has been followed for tensor, vector and scalar perturbations. It has been shown (see Cailleteau et al. 2011) that a single algebra structure can be consistently written for all perturbations. It basically reads as:
\[
\left\{D[M^a],D [N^a]\right\} = D[M^b\partial_b N^a-N^b\partial_b M^a], \\
\left\{D[M^a],S^Q[N]\right\} = S^Q[M^a\partial_b N-N\partial_a M^a], \\
\left\{S^Q[M],S^Q[N]\right\} = \Omega
D\left[q^{ab}(M\partial_bN-N\partial_bM)\right], \tag{31}
\]
where \(\Omega\) is the "deformation factor" (equal to one in the classical theory with Lorentzian signature). A similar structure was found in lattice LQC (see Wilson-Ewing 2012).
This algebra leads, in particular, to an effective signature change at high density, reminiscent of what happens in the Hartle-Hawking proposal. Perturbations in this framework have been analyzed according to two different philosophical views.
In any case, the equation of motion is now more complex than in the ``dressed metric" setting. It reads for tensor modes, in conformal time
\[
v''_k(\eta)+\left(\Omega k^2-\frac{z_T''}{z_T}\right)v_k(\eta)=0, \tag{32}
\]
where
\(z_T\equiv \frac{a}{\sqrt{\Omega}}\). The mode functions appearing here are related to the amplitude of the tensor modes of the metric perturbation, \(h_k\), via \(v_k=z_T h_k/\sqrt{32\pi G}\). The dynamics of the modes if not anymore driven only by the hierarchy between \(k\) and \(a'’/a\) (and, hence, by the ratio between the considered length scale and the curvature radius) as in the usual case and quite a lot of new phenomena can in principe happen.
The first approach considered is to use the ``silent surface" (that is the surface corresponding to \(\Omega=0\) or \(\rho=\rho_c/2\)) as the natural place (or time) where to set initial conditions. Space points are here decoupled (as anticipated by the BKL conjecture which is here nicely recovered by the quantum corrections) and fluctuations are expected to be described by a white noise spectrum. The resulting cosmological power spectrum can accommodate with the data under some assumptions. It is also possible to take advantage of the mathematical knowledge of the so-called Tricomi problem (a change of regime from hyperbolic to elliptic in the differential equation, see Bojowald & Mielczarek 2015) which leads to an interesting balance between singular Big Bang models and deterministic cyclic models.
The second approach to solving equations in this deformed algebra framework consists in setting initial conditions in the remote past of the contracting branch and to evolve the Fourier transform of the modes through the bounce. The Bunch Davies vacuum initial conditions can be set in the usual manner in the remote pasy. Although this is more tricky, the problem has also been solved for scalar modes. With \(v_S:=\sqrt{\bar{p}}\,\left(\delta\phi + \frac{{\bar{\phi}}'}{\mathscr{H}}\Phi\right)\) and \(v_S :=\sqrt{\bar{p}}\, \frac{{\bar{\phi}}'}{\mathscr{H}}\), \(\mathscr{H}\) being the conformal Hubble parameter, the Mukhanov equation of motion reads \[ \ddot{v}_k + H \dot{v}_k + f_{k}^{\scriptscriptstyle{(v)}}(t) v_k = 0, \tag{33} \] with \(z= a \frac{\dot{\bar{\phi}}}{H}\) and \[ f_{k}^{\scriptscriptstyle{(v)}}(t) := {\bf \Omega}\frac{k^2}{a^2} - \frac{\dot{z}}{z} H - \frac{\ddot{z}}{z} \tag{34} \] being the effective frequency term. In terms of the scalar curvature perturbation \(\mathcal{R} := v/z\), this leads to (Schander et al. 2016) \[ \ddot{\mathcal{R}}_k - \left( 3H + 2 m^2 \frac{\bar{\phi}}{\dot{\bar{\phi}}} + 2 \frac{\dot{H}}{H} \right) \dot{\mathcal{R}}_k + {\bf \Omega}\frac{k^2}{a^2} \mathcal{R}_k = 0. \tag{35} \] The solutions to this equation are indeed regular and were shown to be quite insensitive to the precise choice of initial state ({\it e.g.} instantaneous or WKB vacua). The resulting power spectrum is in strong disagreement with CMB data. This does not disprove LQG or LQC but underlines that a naive propagation through the euclidean phase (where there is strictly speaking no time anymore) is just incorrect. It could also be that the well known transplanckian problem (for a typical duration of inflation all the relevant scales for cosmology were smaller than the Planck length before inflation) needs to be seriously addressed in this framework, together with anisotropies. Preliminary results do suggest that modified dispersion associated with Planck length effects allow to recover the correct spectrum (Martineau et al. 2017).
Important and complementary results have also been recently derived in hybrid LQC (Gomar et al. 2015, Navascues et al. 2014).
Finally, new ideas do suggest that it might be possible te receive gravitational waves from the pre-bounce contracting phase using the specific behavior of the luminosity distance which might decrease with an increasing time interval between events (Barrau et al. 2017).
Black holes in loop quantum gravity
In LQG a black hole is basically described as an isolated horizon (that is a specific surface defined using local spacetime structures only, see Ashtekar 2004) punctured by the edges of a spin network - that is a graph with edges labeled with \(SU(2)\) representations and nodes characterized by intertwiners. A surface punctured by \(N\) edges has an area spectrum given by \[ A_j=8 \pi \gamma l_{Pl}^2\sum_{n=1}^N\sqrt{j_n(j_n+1)}, \tag{36} \] where the sum is carried out over all intersections of the edges with the considered surface. Each state with spin \(j\) has a degeneracy \((2j+1)\). Many works have been devoted to derivations of the Bekenstein-Hawking entropy \(S=A/4\) from the LQG framework (see Rovelli 1996). Recent results from holographic models are very encouraging (see Ghosh et al. 2014).
It is in principle difficult to find LQG astrophysical effects for large macroscopic black holes (although it has recently be claimed, using quite general arguments, that quantum gravity signatures might be found outside the horizon, at a distance of the order of \(\frac{7}{6}R_S\), where \(R_S\) is the Schwarzschild radius, see Haggard & Rovelli 2016). Most of the phenomenology therefore focuses on small black holes.
Evaporating black holes
One approach consists in studying how the usual Hawking evaporation spectrum will be modified by the LQG structure of the area discrete spectrum (see Ashtekar & Bojowald 2005). Monte-Carlo simulations were performed, including the time-consuming projection constraint \[ \tag{37} \sum_{p=1}^N{m_p=0}, \] where the \(m_p\)’s carry information about the curvature of a surface whose area is given by the spin \(j\). The simulation was started around 200 \(A_{Pl}\) and the greybody factors encoding the backscattering probability within the gravitation potential were taken into account (in the general relativistic approximation). A Kolmogorov-Smirnov test was carried out to see if it was possible to distinguish, from the measured spectrum, between the LQG predictions and the standard Hawking spectrum. The conclusion of the study is that with an experimental resolution of 5%, around \(2\times10^5\) black holes are required for a 5\(\sigma\) detection whereas \(1.2\times10^6\) are required with a 20% resolution.
A more recent study was devoted to the discrimination between different approaches to compute the BH entropy within the LQG theory. The first class of models are those where the entropy is calculated using only quantum geometry excitations, leading to \[ \tag{38} S=\frac{\gamma_0}{\gamma} \frac{A}{4} + {o}(A), \] where \(\gamma_0\) is a numerical factor of order one, depending on the details of the state counting methodology. The quantum corrections \(o(A)\) are logarithmic at the leading order. The compatibility with the first law of thermodynamics leads to \(\gamma=\gamma_0\). The second class of models uses the qualitative behavior of matter degeneracy suggested by standard quantum field theory (QFT) with a sharp cut-off at the vicinity of the horizon (i.e. an exponential growth of vacuum entanglement in terms of the BH area). The entropy can then be estimated to be \[ \tag{39} S=\frac{A}{4}+ \sqrt{\frac{\pi A}{6\gamma}} + o(\sqrt{A}). \] As expected, Immirzi parameter \(\gamma\) affects only quantum corrections. The numerical simulations show that the spectra exhibits two distinct parts: a continuous background which corresponds to the semi-classical stages of the evaporation and series of discrete isolated peaks which constitute a specific signature of the deep quantum structure of the black hole (see Barrau et al.2015). The Immirzi parameter has a non-trivial effect on both parts. The number of black holes and the instrumental resolution required to experimentally distinguish between the considered models were also studied and it was shown that the different models can be discriminated one from the other.
Finally, a new idea has also emerged, reviving the hope for possible low energy quantum gravity effects in the evaporating black hole spectrum. The density of energy levels in LQG reads as \[ \rho(M)\sim \rm{exp}(M\sqrt{4\pi G/3}), \tag{40} \] which means that the spectral lines are virtually dense in frequency. This leads to a quasi-continuum of energies for large-mass black holes. However it might be that the evolution of black holes at each emission of a quanta is actually associated with the change of quantum state of only one puncture in the spin network (see Barrau 2016). In that case, somehow in agreement with the old Bekenstein-Mukhanov hypothesis, one should expect clear spectral lines, even for black hole masses arbitrarily far away from the Planck mass. This effect is neither washed out by the dynamics of the process, nor by the existence of a mass spectrum of primordial black holes up to a given width, nor by the secondary component induced by the decay of neutral pions emitted during the time-integrated evaporation.
Bouncing black holes
When matter or radiation reaches the Planck density, quantum geometry effects might generate a sufficient pressure to counterbalance the classically attractive gravitational force. In a black hole, matter's collapse could therefore stop before the central singularity is formed (there are several general singularity resolution theorems in LQG, see Rovelli & Vidotto 2013). The standard event horizon of the black hole would then be replaced by an apparent horizon which is locally equivalent to an event horizon, but from which matter can bounce out. In this scenario, the contracting classical black hole solution is glued to the classical expanding white hole solution. The important point is that such a simple minimal evolution was shown to be possible and consistent: the white hole horizon is in the future of the black hole horizon, bounding the same external Schwarzschild geometry with nothing dramatic happening in the surrounding universe. This interesting possibility is obtained by carving out the relevant solution from a double covering of the Kruskal metric (see Haggard & Rovelli 2014).
One can use coordinates \((u,v,\theta,\phi)\) with \(u\) and \(v\) null coordinates in the \(r\)-\(t\) plane and the metric is then determined by two functions: \[ ds^2=-F(u,v) du dv + r^2(u,v)(d\theta^2+\sin^2\theta d\phi^2). \tag{41} \] In these Kruskal-Szekeres coordinates, the metric can be obtained by taking \[ F(u,v)=\frac{32M^3}{r}e^{\frac{r}{2m}}, \tag{42} \] with \(r\) the function of \((u,v)\) defined by \[ \left(1-\frac{r}{2m}\right)e^{\frac{r}{2M}}=uv. \tag{43} \] The region of interest is bounded by a constant \(v={v}_o\) null line. This constant is a fundamental parameter of the geometry under consideration. By gluing the different parts of the effective metric, and calculating the minimal time for quantum gravitational effects to pile up in the region outside the horizon, one obtains a reasonable estimate for the bounce duration: \[ \tau=-8m \ln v_o> \tau_{q} = 4p\, M^2, \tag{44} \] where \(p\) was computed to be of the order of 0.05. In this model, the bounce time is therefore proportional to \(M^2\), whereas the duration of the Hawking evaporation is proportional to \(M^3\): black holes would bounce faster than they do evaporate. Let us write the actual bounce time as \(x\tau_{q}\) with \(x>1\). This leads to \[ \tau=4k M^2% \frac{M^2}{l_{Pl}}, \tag{45} \] with \(k\equiv xp>0.05\). At this stage \(k\), can be considered as a free parameter whose upper bound is such that the bounce time remains smaller than the Page time (which is smaller than the Hawking time).
Two different components are a priori expected from bouncing black holes. The first one, referred to as the low-energy component, is just determined by a basic dimensional analysis. The Schwarzschild radius of the black hole being the only length scale in the problem, the emitted radiation can be expected to have a mean wavelength of this order. The second one, referred to as the high-energy component, is using the fact that the model is by construction time-symmetric: what goes out should be exactly what went in (the gravitational redshift is compensated by - nearly - exactly the same amount of gravitational blueshift). The energy of the emitted radiation is therefore the same as the one of the null shell assumed to have formed the black holes in the early Universe (as only primordial black holes are phenomenologically interesting in this specific scenario). It has been suggested that the low energy component could explain the mysterious fast radio bursts observed in the Universe (see Barrau at al. 2014).
The detailed electromagnetic component from both those signals was studied taking into account the emission of hadrons - due to quark fragmentation - eventually decaying into neutral pions that will subsequently produce gamma-rays (see Barrau et al. 2015). The maximum distance at which a single bouncing black hole can be detected was evaluated accounting for the absorption and the instrumental sensitivity. The result does depend heavily on the value of \(k\). For the lowest authorized value, the maximum distance at which a bouncing black hole can be detected is larger than the Hubble radius for the low energy channel and of the order of one tenth on the Hubble scale for the high energy channel. It crosses the galactic scale for the low energy channel around \(k\sim 10^{14}\), and for the high energy channel around \(k\sim 10^{8}\). The integrated signal from a possible distribution of bouncing black holes contributing to the cold dark matter was also evaluated. The result of this study is that the shape of the measured signal should be quite close to the one of an individual signal emitted by a single bouncing black hole, but with a slight low-energy distorsion due to redshift effects (see Barrau et al. 2016).
Finally, it has been suggested that the gamma-ray excess from the galactic center, measured by the Fermi satellite, could be accounted for by bouncing black holes. This however requires a specific tuning of the free parameter of the theory and of the width of the emitted signal. This prediction is interesting because a specific redshift dependance for this signal can be estimated. The measure wavelength reads as \[ \lambda_{obs}^{BH}\sim \frac{2GM}{c^2} (1+z) \times \sqrt{\frac{H_0^{-1}}{6\,k\Omega_\Lambda^{\,1/2}}Argsinh\!\!\left[ \left(\frac{\Omega_\Lambda}{\Omega_M}\right)^{\!1/2} (z + 1)^{-3/2}\right]}, \tag{46} \] where we have reinserted the Newton constant \(G\) and the speed of light \(c\) to make things easier to read. The important consequence of this formula is that the redshift dependance of this specific model is much weaker that for other origins (either from astrophysical sources or due to decaying dark matter): the curve becomes quite flat for z>2.
Recently, more rigorous calculations for the transition amplitude \(W(m,T)\), defined such that its modulus squared determines the actual probability density for the bounce to happen in a given time \(T\) for a given mass \(m\), were carried out in loop quantum gravity (see Christodoulou et al. 2016). They are based on the fact that, in LQG, the amplitude associated to a two complex \(C\) is basically given by \[ W_{\mathcal C}(h_\ell)=N_{\mathcal C}\int_{{}_{SU(2)}} dh_{fv}\ \prod_f \delta\Big(\!\prod_{v\in f} h_{fv}\!\Big) \ \prod_v A_v(h_{fv}), \tag{47} \] where \(f\) and \(v\) denote the faces and the vertices, \(h_\ell\in SU(2)\) for every link \(\ell\), and (with \(h_\ell=h_{vf}\)), \[ A_v (h_{\ell})= \int_{{}_{SL(2,\mathbb{C})}}\!\!\!\!\! d{g_{e}}' \, \prod_{\ell} \sum_{j}\ d_{j} \ D_{jn\, jm}^{(\gamma j, j)} (g_{e}g_{e' }^{-1}) \ D_{mn}^{(j)}(h_{\ell}), \tag{48} \] the integration being carried out over one \(g_e\) for each node (edge of \(v\)) - except one. The product is over 10 faces \(f\) per each vertex, and \(D^{(j)}\) and \(D^{(p\, k)}\) are matrix elements of the \(SU(2)\) and \(SL(2,\mathbb{C})\) group representations. Calculating those amplitudes for the considered is difficult and there are no clear evidences that the bouncing time is indeed of the order of \(M^2\). On should notice that the previously given equations are generic for a spin-foam amplitude. The calculation of this phenomenon cannot be done over a background because it is a quantum tunnelling and it is therefore doable in covariant LQG.
Lorentz invariance violation
It has been argued that the discreetness of space, understood as a genuine property of quantum geometry or quantum gravity, could be probed by searching for a Lorentz invariance violation in vacuum. Counter arguments were also given, saving Lorentz invariance, in the sense that a boosted frame might lead to a different mean expectation value for the length operator but not to a different spectrum. The situation is therefore quite unclear at this stage. Several studies (see Girelli et al. 2012) were however devoted to possible Lorentz violations in the specific LQG framework. The associated phenomenological studies are based on an energy-momentum relation typically modified by \(E\simeq p+m^2/2p\pm\xi(E^2/M_{QG})^n\) where \(M_{QG}\) is the energy scale at which quantum gravity begins to play an important role, \(\xi>0\), and \(n\) is an integer expected not to be much higher than one.
This idea can be implemented in several frameworks, from non-commutative space-times to effective field theories and non-linear Poincaré symmetries. In LQG, there is a specific additional helicity dependance than can be added to the previously given relation. It is then possible to carry out calculations by considering the Hamitonian of the electromagnetic field in a semi-classical state which is a discrete approximation to the flat geometry. As the densitized triad operator enters the Hamiltonian expression, its expectation value on this state will receive LQG corrections. The associated modified dispersion relation for photons becomes \(\omega^2_{\pm}=k^2\mp4\chi k^3/M_{Pl}\) with \(\chi\sim1\). Photons would hence be submitted to birefringence in vacuum.
It might be possible to detect chiral gravity with the pure pseudospectrum reconstruction of the cosmic microwave background (CMB) polarized anisotropies (see Ferté & Grain 2014).
Other studies are based on a different approach. Classically, the action \(S[A]=\int_\Sigma\mathcal{S}[A]\) can be used to define a slicing. Quantum mechanically, this slicing does fluctuate. The calculation was performed using a so-called Born-Oppenheimer state \(\Psi_0[A]\chi[A,\phi]\) where \(\Psi_0\) is a semiclassical state. The densitized triad on this state, evaluated close to the classical trajectory, is deformed according to \(E^{(0)~a}_{~~~i}(x,t,\omega)=E^{(0)~a}_{~~~i}(x,t)(1-\alpha L_{Pl}\omega)\). As the triad now explicitly depends on \(\omega\), this induces an \(\omega\)-dependent metric and thus an associated modified dispersion relation: \(m^2=\omega^2-k^2/(1-\alpha L_{Pl}\omega)\). Quantum gravity fluctuations might then lead to a preferred frame. Quantum fluctuations would pick-up a frame \(\tilde{e}_a^\mu\) which is non-linearly related to \(e_a^\mu\) through a \(\pi_\mu\) dependance, \(\tilde{e}_a^\mu=F(e_a^\mu,\pi_\mu)\). As the momenta are measured by \(\tilde{p}_a=\tilde{e}_a^\mu\pi_\mu\), the transformation law would not be given anymore by usual the Lorentz group. This would lead to a deformation of the Poincaré algebra.
There are also indications that different choices of the Immirzi parameter can, in some cases, lead to different outcomes for the modifications of the dispersion relations, depending on the chosen quantization scheme (see Brahma et al. 2016). This might allow one to differentiate between the quantization schemes via testable phenomenological predictions.
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