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Primordial Fluctuations from Quantum Gravity

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Primordial Fluctuations from Quantum Gravity Primordial fluctuations from quantum gravity Francesco Gozzini a;b and Francesca Vidotto c;d a CPT, Aix-Marseille Universit´e,Universit´ede Toulon, CNRS, 13288 Marseille, France b Dipartimento di Fisica, Universit´adi Trento, and TIFPA-INFN, 38123 Povo TN, Italy c Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada d University of the Basque Country UPV/EHU, Departamento de F´ısica Te´orica, 48940 Leioa, Spain We study fluctuations and correlations between spacial regions, generated by the primordial quantum gravitational phase of the universe. We do so by a numerical evaluation of Lorentzian amplitudes in Loop Quantum Gravity, in a non-semiclassical regime. We find that the expectation value of the quantum state of the geometry emerging from the early quantum phase of the universe is a homogeneous space but fluctuations are very large and correlations are strong, although not maximal. In particular, this suggests that early quantum gravitational effects could be sufficient to solve the cosmological horizon problem. I. INTRODUCTION The second result is that (contrary to our initial ex- pectation), the variance of these variables is very large. This variance cannot be too small, because the variables The universe emerged from its early quantum gravi- do not commute, but one might have expected a coher- tational phase in a state that included fluctuations with ent state minimizing the spread. Instead, the spread is correlations between distinct regions of space. These pri- almost maximal. This means that (in spite of what the mordial fluctuations play a key role in cosmology | with first result mentioned above might misleadingly suggest), or without inflation | in particular as seeds for struc- the quantum state that emerges from the big bang in this ture formation. Here we explore the physical genesis of approximation is not peaked on a homogeneous geome- these fluctuations from the primordial quantum gravita- try: fluctuations are ample. tional cosmological state. We use Loop Quantum Gravity Finally, we study the correlation between distinct spa- (LQG) and a simple model of the early universe. tial regions. We look directly at the correlation between We consider the quantum transition from an empty two variables in distinct regions, as well as, more gen- state to a 3-geometry. As argued in [1{3], this transi- erally, at the entanglement entropy between one region tion may be relevant in a bounce cosmology, because the and the rest of the geometry. We find that the correla- empty state can be interpreted as the dominant interme- tion is strong and does not vanish with the growth of the diate state in the transition through the bounce. Such scale factor. Moreover, the entanglement entropy has a transition may also describe the very early phase of a Big highly non-trivial behaviour, apparently reaching asymp- Bang cosmology, as the bridge from `nothing' to an initial totically a stable value. quantum geometry. We treat the dynamics of the gravi- All this indicates that the universe emerging from an tational degrees of freedom non-perturbatively, using the early quantum era has a rich structure of fluctuations, covariant formulation of the dynamics of LQG. This is a homogeneity properties, and large scale correlations, due fully Lorentzian calculation. This dynamics can describe to the common quantum origin of spatially separated re- the quantum tunneling from nothing to a 3-geometry. gions, which can be studied theoretically and appear to The quantum state generated by this transition deter- be compatible with the observed universe. Inflation or a mines a probability distribution over 3-geometries, which bounce, in particular, might not be strictly necessary to includes correlations between spatial regions. We trun- circumvent the horizon problem. That is, the result of arXiv:1906.02211v1 [gr-qc] 5 Jun 2019 cate the degrees of freedom of the gravitational field to classical general relativity stating that distant regions of a small finite number in addition to the scale factor. We the universe were causally disconnected in the past in a view this as a finer truncation than the common symme- (non-inflationary) expansion is not anymore true if the try reduction that treats only the scale factor non per- initial quantum phase is taken into account. turbatively [2,3]. Even for this simple model, exact ana- The numerical calculation we develop uses the am- lytical results are hard, hence we use numerical methods. plitude of covariant Loop Quantum Gravity in a non- We report here the implementation of the calculations semiclassical regime, which can be loosely understood and three results. as a quantum `tunneling' transition. We find that the The first regards the expectation value of the variables relevant contribution from the loop transition amplitude describing the geometry at a given value of the scale fac- comes from the so-called vector geometries [4,5]. This tor. The result is that this turns out to be rather precisely clarifies their physical meaning as a non-suppressed com- given by (the truncation of) a metric 3-sphere. At first ponent of the amplitude, that appeared quite mysterious this result may seem pretty obvious, for reasons of sym- so far. metry, but combined with what follows it is much less The model we analyse is crude and insufficient to de- so. velop a finer quantitative analysis. In particular, the 2 truncation we use implies that any two regions are in where Vj is the spin-j representation space of SU(2). We contact, and this prevents us from distinguishing short shall focus on the subspaces Hj of H defined by jl = j. from long distance correlation. We leave open an in- These have the tensorial structure triguing question, which might be addressed with these O Hj = In (4) methods: could the quantum epoch directly yield almost n scale-invariant cosmological fluctuations? where each In is isomorphic to (Vj⊗Vj⊗Vj⊗Vj)=SU(2). The basis states are tensor states, which we denote as II. STRUCTURE OF THE MODEL O jj; ini = jini (5) n A. Truncation (by this we mean jj; i1; : : : ; i5i = ji1i ⊗ · · · ⊗ ji5i). We choose a basis in In as usual, that is: we fix a pairing We discretize a closed cosmological 3-geometry into of the links at each node and we choose the basis that five tetrahedra glued to one another, giving an S3 topol- diagonalises the modulus square of the sum of the SU(2) ogy. This is the simplest regular triangulation of a topo- generators in the pair. logical 3-sphere, and it corresponds to the boundary of In covariant LQG, the transition amplitude from an a geometrical 4-simplex. This geometry has twenty de- empty state to a state jj; ini in Hj is given by the spin- grees of freedom, which capture the gravitational field in foam amplitude of the boundary state jj; ini alone. The this truncation. To first order in the spinfoam expansion, reason is that this transition corresponds to the ampli- the result of the transition from nothing to a 3-geometry tude of a boundary state that has only one connected is described in this truncation directly by the covariant component, here interpreted as the future one. To first LQG Lorentzian vertex amplitude. order in the spinfoam expansion, the amplitude of a We take the further simplification consisting in assum- boundary state is given by a single vertex. Hence the ing all areas of all the faces of the tetrahedra to be equal nothing-to-jj; ini amplitude is just the vertex amplitude and we use this common value as a proxy for the scale for the boundary state factor. The remaining degrees of freedom are those char- acterizing the shapes of the five tetrahedra. We are in- hj; inj?i = W (j; in) ≡ hj; inj oi (6) terested in the fluctuations of these variables and the where W (j; i ) is the Lorentzian EPRL vertex amplitude correlations between (variables in) distinct tetrahedra, n [7]. at different values of the scale factor. Notice that this implies that we can view the ket j oi that has components W (j; in) on the spin network ba- sis, as the quantum state emerging from the big bang. B. Quantum theory This is the analogue, in (Lorentzian) LQG, of the Hartle- Hawking `no-boundary' initial state in (Euclidean) path In Loop Quantum Gravity, the state space of a trun- integral quantum gravity [8]. This is the state we are cation of the degrees of freedom of the gravitational field interested in. We want to study the mean geometry it is given by the Hilbert space defines and the quantum fluctuations and correlations it 2 L N incorporates. To this aim, we study the expectation value H = L [SU(2) =SU(2) ]Γ (1) hAi = h ojAj oi; (7) where Γ is an oriented graph with L links and N nodes that determines the way SU(2)N acts on SU(2)L [6]. Here the spread we consider the truncation of general relativity defined by p 2 2 the complete graph Γ with five nodes, hence N = 5 and ∆A = h ojA j oi − hAi (8) L = 10. We label the nodes as n = 1;:::; 5. We label the and the (normalized) correlations (oriented) links as l = 1;:::; 10, or alternatively in term 0 of the two nodes they link: l = nn . Geometrically, this h ojA1A2j oi − hA1ihA2i graph corresponds to the discretization of a three-sphere C(A1;A2) = (9) (∆A1) (∆A2) with five tetrahedra, joined via their 10 shared faces. The spin network basis in H is given by the states of local geometry operators A; A1;A2; ::: defined on H. When relevant, the dependence of the states and results jjl; ini (2) from the scale parameter j is denoted as j oij and hAij.
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