Bouncing cosmologies from condensates of quantum geometry

Bouncing cosmologies from condensates of quantum geometry Edward Wilson-Ewing Albert Einstein Institute Max Planck Institute for Gravitational Physics Work with Daniele Oriti and Lorenzo Sindoni Helsinki Workshop on Quantum Gravity E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 1 / 19

Quantum Gravity and the Early Universe Quantum gravity effects are typically expected to become important at either very high energy scales, or in regions of large space-time curvature like near the center of black holes [Simone Speziale s talk], or in the very early universe. In particular, the big-bang singularity of classical general relativity signals that general relativity cannot be trusted in the very early universe a theory of quantum gravity is needed. What does quantum gravity tell us about the early universe? What type of quantum gravity effects are there? What replaces the big-bang singularity? Are there any observable and realistically testable effects? I will consider these questions from the perspective of loop quantum gravity. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 2 / 19

Loop Quantum Gravity: Basic Ingredients Loop quantum gravity is a background independent approach to quantum gravity based on connection and triad variables. [Ashtekar; Immirzi; Barbero; Wolfgang Wieland s talk] A convenient basis for states in the canonical framework are spin networks: graphs coloured by spins on the edges and intertwiners on the nodes. [Penrose] [Rovelli] An important result is that geometrical observables like the volume and the area have a discrete spectrum. Furthermore, each node can be thought of as a fuzzy polyhedron with some volume, and surface areas transverse to the links. In this sense, the spin network is composed of atoms of geometry. [Rovelli, Smolin; Ashtekar, Lewandowski; Freidel, Speziale; Hal Haggard s talk] E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 3 / 19

Loop Quantum Cosmology In loop quantum cosmology, we impose the symmetries of the space-time of interest here the spatially flat Friedmann-Lemaître- Robertson-Walker (FLRW) cosmology directly in the Hamiltonian constraint (expressed in terms of connection A and triad E variables) H = ɛij kei a Ej b 16πGγ 2 dete F ab k + H matter, F k = da k + ɛ k ij A i A j, and promote the various terms to operators. Once we have the Hamiltonian constraint operator, we can calculate the evolution of wave functions of interest. [Bojowald; Ashtekar, Paw lowski, Singh;... ] E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 4 / 19

Loop Quantum Cosmology In loop quantum cosmology, we impose the symmetries of the space-time of interest here the spatially flat Friedmann-Lemaître- Robertson-Walker (FLRW) cosmology directly in the Hamiltonian constraint (expressed in terms of connection A and triad E variables) H = ɛij kei a Ej b 16πGγ 2 dete F ab k + H matter, F k = da k + ɛ k ij A i A j, and promote the various terms to operators. Once we have the Hamiltonian constraint operator, we can calculate the evolution of wave functions of interest. [Bojowald; Ashtekar, Paw lowski, Singh;... ] It is possible to choose a representation such that the triad operators act by multiplication [Ashtekar, Bojowald, Lewandowski], but the field strength is more difficult to handle. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 4 / 19

Field Strength Operator In loop quantum gravity, there is no connection operator. Rather, there only exist operators corresponding to the parallel transport of the connection along a path of finite length. Therefore, we will express the field strength F in terms of the parallel transport of the connection around a finite loop, which we shall choose to be the smallest one possible in LQG: the smallest non-zero eigenvalue of the area operator in LQG [Ashtekar, Paw lowski, Singh]. This essentially corresponds to making the ansatz that in an LQG state corresponding to a cosmological space-time, the dominant contribution to a surface comes from many small areas rather than a few big ones. [Ashtekar, Paw lowski, Singh; Bojowald, Cartin, Khanna; Ashtekar, WE; Paw lowski] E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 5 / 19

Loop Quantum Cosmology Dynamics Skipping technical details, this procedure gives a Hamiltonian constraint operator that generates the evolution of wave functions of interest. The result is that the big-bang singularity is generically resolved [Bojowald] and replaced by a bounce for all states. [Ashtekar, Corichi, Singh] Furthermore, for semi-classical states there exist effective Friedmann equations that provide an excellent approximation to their full quantum dynamics, [Ashtekar, Paw lowski, Singh; Taveras; Willis; Rovelli, WE] H 2 = ( 1 dv 3V dt ) 2 = 8πG 3 ρ (1 ρρc ). E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 6 / 19

Loop Quantum Cosmology Dynamics Skipping technical details, this procedure gives a Hamiltonian constraint operator that generates the evolution of wave functions of interest. The result is that the big-bang singularity is generically resolved [Bojowald] and replaced by a bounce for all states. [Ashtekar, Corichi, Singh] Furthermore, for semi-classical states there exist effective Friedmann equations that provide an excellent approximation to their full quantum dynamics, [Ashtekar, Paw lowski, Singh; Taveras; Willis; Rovelli, WE] H 2 = ( 1 dv 3V dt ) 2 = 8πG 3 ρ (1 ρρc ). Note that if the matter content is a massless scalar field φ, ( ) 2 1 dv = 4πG ( ) 1 π2 φ, 3V dφ 3 2ρ c V 2 and φ can be used as a relational clock. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 6 / 19

Beyond Loop Quantum Cosmology The results of loop quantum cosmology show the form that quantum gravity effects could take in the early universe, and a lot of current work is ongoing in order to derive predictions (which depend on the matter content of the early universe) that could be compared to the high precision observations of the cosmic microwave background. [Barrau, Grain, Cailleteau, Mielczarek; Bojowald, Calcagni, Tsujikawa; WE; Agulló, Ashtekar, Nelson; Bonga, Gupt;... ] Today I will focus on another question. Loop quantum cosmology is obtained through symmetry reducing the classical theory and quantizing the result. Is it possible to directly study cosmology in full LQG? There is no reason to do this blindly. Rather, we can use the intuition developed from LQC over the years in order to guide our efforts in this direction. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 7 / 19

Guidance from LQC LQC suggests that surfaces and volumes are composed of many small quanta (not a few large quanta), and that the universe expands by adding quanta rather than by changing the volume of individual quanta. Therefore, the number of nodes must be free to evolve. Also, the connectivity information (what node is linked to what node) does not play an important role in LQC, so we will ignore that. Finally, in LQC it is typically assumed that all of the nodes and links carry the same labels: all nodes and links are in the same state. This suggests that condensate states are appropriate to study the cosmological sector of LQG. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 8 / 19

Guidance from LQC LQC suggests that surfaces and volumes are composed of many small quanta (not a few large quanta), and that the universe expands by adding quanta rather than by changing the volume of individual quanta. Therefore, the number of nodes must be free to evolve. Also, the connectivity information (what node is linked to what node) does not play an important role in LQC, so we will ignore that. Finally, in LQC it is typically assumed that all of the nodes and links carry the same labels: all nodes and links are in the same state. This suggests that condensate states are appropriate to study the cosmological sector of LQG. This in turn suggests using group field theory (GFT), a field theory reformulation of LQG where the existence of creation and annihilation operators makes it much easier to handle condensate states. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 8 / 19

Group Field Theory with a Scalar Field Group field theory (GFT) can be seen as a second-quantized language for loop quantum gravity, where the field operators ˆϕ(g v1, g v2, g v3, g v4, φ), ˆϕ (g v1, g v2, g v3, g v4, φ), create and annihilate quanta of geometry: spin network nodes [Oriti]. The g vi denote the parallel transport along each of the links leaving the (four-valent) spin network node and the scalar field φ lives on the spin network nodes. The classical GFT action S(ϕ, ϕ) is typically chosen so that the perturbative expansion of the GFT partition function matches the sum over geometries of a spin foam model [Oriti; Sylvain Carrozza s talk]: DϕD ϕ Ψ f [ ϕ]e S[ϕ, ϕ] Ψ i [ϕ] = A e A f A v. two complexes with boundary Ψ i,ψ f j,ι E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 9 / 19 e f v

GFT Condensate A simple family of condensate states are the Gross-Pitaevskii condensate states, i.e., coherent states of the GFT field operator which are, up to a numerical prefactor, [Gielen, Oriti, Sindoni] ( σ exp dφ σ j (φ) ˆφ j ) 0, (φ) j where σ j (φ) is the condensate wave function. Note that σ j (φ) is not normalized; rather, its norm gives the number of fundamental GFT quanta. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 10 / 19

GFT Condensate A simple family of condensate states are the Gross-Pitaevskii condensate states, i.e., coherent states of the GFT field operator which are, up to a numerical prefactor, [Gielen, Oriti, Sindoni] ( σ exp dφ σ j (φ) ˆφ j ) 0, (φ) j where σ j (φ) is the condensate wave function. Note that σ j (φ) is not normalized; rather, its norm gives the number of fundamental GFT quanta. The form of the condensate state has been simplified by choosing it to be isotropic (motivated by FLRW and LQC): all links are identical and the volume is maximized on the created spin network nodes. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 10 / 19

GFT Condensate Also, the scalar field acts as a relational clock: σ j (φ o ) can be interpreted as the condensate wave function evaluated at the time φ o. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 10 / 19 A simple family of condensate states are the Gross-Pitaevskii condensate states, i.e., coherent states of the GFT field operator which are, up to a numerical prefactor, [Gielen, Oriti, Sindoni] ( σ exp dφ σ j (φ) ˆφ j ) 0, (φ) j where σ j (φ) is the condensate wave function. Note that σ j (φ) is not normalized; rather, its norm gives the number of fundamental GFT quanta. The form of the condensate state has been simplified by choosing it to be isotropic (motivated by FLRW and LQC): all links are identical and the volume is maximized on the created spin network nodes.

GFT Action For spin foam models which are based on simplicial interactions, the only interaction terms are five-valent in which case the GFT action contains a kinetic term and ϕ 5 interaction terms, e.g., schematically [ ] S = ϕϕ K 2 + ϕ 5 V 5 + ϕ 5 V 5. g i,φ i g i,φ i Also, from a spin foam perspective it is natural to discretize the scalar field at the vertices of the two-complex. Then, the interaction terms in the GFT action must include delta functions so all φ have the same value at the vertex, while the gradients of φ will be encoded in the propagator of the GFT. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 11 / 19

GFT Action For spin foam models which are based on simplicial interactions, the only interaction terms are five-valent in which case the GFT action contains a kinetic term and ϕ 5 interaction terms, e.g., schematically [ ] S = ϕϕ K 2 + ϕ 5 V 5 + ϕ 5 V 5. g i,φ i g i,φ i Also, from a spin foam perspective it is natural to discretize the scalar field at the vertices of the two-complex. Then, the interaction terms in the GFT action must include delta functions so all φ have the same value at the vertex, while the gradients of φ will be encoded in the propagator of the GFT. Furthermore, if φ is massless and minimally coupled to gravity, the symmetries φ φ + const and φ φ require K 2 ( g 1, g 2, φ 1, φ 2 ) = K 2 ( g 1, g 2, (φ 1 φ 2 ) 2 ). E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 11 / 19

The Kinetic Term Assuming the field operator is analytic in φ, it is possible to perform a derivative expansion of the kinetic term of the GFT action: K = = = d 8 g i dφ 1 dφ 2 ϕ( g 1, φ 1 ) K 2 ( g 1, g 2 ; (φ 1 φ 2 ) 2 ) ϕ( g 2, φ 2 ) d 8 g i dφdδφ ϕ( g 1, φ) K 2 ( g 1, g 2 ; δφ 2 ) ϕ( g 2, φ + δφ) n=0 d 8 g i dφ ϕ( g 1, φ)k (2n) 2 ( g 1, g 2 ) 2n φ 2n ϕ( g 2, φ). E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 12 / 19

The Kinetic Term Assuming the field operator is analytic in φ, it is possible to perform a derivative expansion of the kinetic term of the GFT action: K = = = d 8 g i dφ 1 dφ 2 ϕ( g 1, φ 1 ) K 2 ( g 1, g 2 ; (φ 1 φ 2 ) 2 ) ϕ( g 2, φ 2 ) d 8 g i dφdδφ ϕ( g 1, φ) K 2 ( g 1, g 2 ; δφ 2 ) ϕ( g 2, φ + δφ) n=0 d 8 g i dφ ϕ( g 1, φ)k (2n) 2 ( g 1, g 2 ) 2n φ 2n ϕ( g 2, φ). In the case where δφ is small compared to the Planck scale, a truncation of the sum will provide a good approximation to the full kinetic term. We will only keep the first two terms. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 12 / 19

Quantum Equations of Motion We expect the condensate state to only be an approximate solution to the quantum equations of motion. So, we will only impose the first Schwinger-Dyson equation σ δs σ = 0. [Gielen, Oriti, Sindoni] δ ϕ This gives the non-linear equation 2 φσ j (φ) m 2 j σ j (φ) + w j σ j (φ) 4 = 0, where the numerical values of the mj 2 K (0) 2 /K (2) 2 and w j V 5 /K (2) 2 depend on the details of the GFT action. The Gross-Pitaevskii condensate approximation assumes that interactions are small. (This is also the regime where the spin foam expansion will be useful for perturbative calculations.) Thus, in this regime the interaction term is subdominant. To consider cases when the interaction term becomes important, it will be necessary to go beyond the Gross-Pitaevskii approximation. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 13 / 19

Relational Dynamics Rewriting σ j = ρ j e iθ j and denoting φ f by f, after dropping the subdominant interaction terms the condensate equations of motion are ρ j (mj 2 + θ j 2 )ρ j 0, ρ j θ j + 2ρ jθ j 0. It is easily checked that in this regime, for each j, E j = ρ j 2 + ρ 2 j θ j 2 m 2 j ρ 2 j, Q j = ρ 2 j θ j, are conserved quantities (with respect to the relational time φ). E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 14 / 19

Relational Dynamics Rewriting σ j = ρ j e iθ j and denoting φ f by f, after dropping the subdominant interaction terms the condensate equations of motion are ρ j (mj 2 + θ j 2 )ρ j 0, ρ j θ j + 2ρ jθ j 0. It is easily checked that in this regime, for each j, E j = ρ j 2 + ρ 2 j θ j 2 m 2 j ρ 2 j, Q j = ρ 2 j θ j, are conserved quantities (with respect to the relational time φ). The remaining non-trivial equation can now be rewritten as ρ j Q2 j ρ 3 j m 2 j ρ j 0. Note that ρ j (φ) can never become zero due to the divergent repulsive potential at ρ j = 0. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 14 / 19

Emergent Continuity Equation By calculating the relevant cosmological observables like the total spatial volume and the momentum of the massless scalar field, we can extract the coarse-grained large-scale (cosmological) dynamics from the quantum gravity equations of motion for the condensate state. These cosmological observables are V (φ) = j V j σ j (φ)σ j (φ) = j V j ρ j (φ) 2, π φ (φ) = i 2 [ ] σ j (φ) φ σ j (φ) σ j (φ) φ σ j (φ) = j Q j. It immediately follows that π φ (φ) = 0. This is the continuity equation for an FLRW space-time with a massless scalar field. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 15 / 19

Emergent Continuity Equation By calculating the relevant cosmological observables like the total spatial volume and the momentum of the massless scalar field, we can extract the coarse-grained large-scale (cosmological) dynamics from the quantum gravity equations of motion for the condensate state. These cosmological observables are V (φ) = j V j σ j (φ)σ j (φ) = j V j ρ j (φ) 2, π φ (φ) = i 2 [ ] σ j (φ) φ σ j (φ) σ j (φ) φ σ j (φ) = j Q j. It immediately follows that π φ (φ) = 0. This is the continuity equation for an FLRW space-time with a massless scalar field. Also, since ρ j (φ) is always different from zero, so is V (φ): the big-bang singularity is resolved! E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 15 / 19

The Condensate Friedmann Equation The equation of motion for V, using V = j V j ρ 2 j, V = 2 j V j ρ j ρ j, and the equation of motion for σ j (φ) given earlier, is the emergent Friedmann equation ( ) V 2 2 j V j ρ j E j Q2 j ρ = 2 j 3V 3 j V jρ 2 j + m 2 j ρ2 j 2. The classical Friedmann equation (V /3V ) 2 = 4πG/3 is recovered in the low curvature semi-classical limit (which here corresponds to large ρ j ) for mj 2 = 3πG. This is a condition on the GFT action. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 16 / 19

A Simple Ansatz to (Almost) Recover LQC Recall that LQC suggests that the appropriate condensate state is one where all the quanta are equilateral spin networks with j = 1/2. This motivates choosing a σ j (φ) that only has support on j = j o. (These states dominate in the classical limit for some GFTs. [Gielen]) Then, using ρ = πφ 2/2V 2, the condensate Friedmann equation becomes ( ) V 2 = 4πG ) (1 ρρc + 4V j o E jo 3V 3 9V, with ρ c = 3πG 2 /2V 2 j o (3π/2j 3 o )ρ Pl. This is (almost) exactly the LQC effective Friedmann equation, up to the extra term that depends on E jo (which is determined by the initial conditions and can be chosen to be zero). E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 17 / 19

A Simple Ansatz to (Almost) Recover LQC Recall that LQC suggests that the appropriate condensate state is one where all the quanta are equilateral spin networks with j = 1/2. This motivates choosing a σ j (φ) that only has support on j = j o. (These states dominate in the classical limit for some GFTs. [Gielen]) Then, using ρ = πφ 2/2V 2, the condensate Friedmann equation becomes ( ) V 2 = 4πG ) (1 ρρc + 4V j o E jo 3V 3 9V, with ρ c = 3πG 2 /2V 2 j o (3π/2j 3 o )ρ Pl. This is (almost) exactly the LQC effective Friedmann equation, up to the extra term that depends on E jo (which is determined by the initial conditions and can be chosen to be zero). While E j plays an important role in the dynamics of the GFT condensate, its geometric interpretation remains unclear. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 17 / 19

Conclusions Motivated by LQC, we made a specific ansatz on the type of state in (the GFT reformulation of) LQG that corresponds to cosmological space-times: GFT condensate states. The equations of motion for the condensate states are determined by the GFT action, and from these equations of motion we can extract the continuity and Friedmann equations. The classical Friedmann equations are recovered in an appropriate semi-classical limit for some choices of parameters in the GFT action. The classical singularity is resolved and is generically replaced by a bounce. Also, the LQC effective Friedmann equations are (almost) recovered for a natural choice of the condensate wave function. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 18 / 19

Outlook There are many open questions: Further investigate the condensate Friedmann equation for a variety of condensate wave functions, Consider other types of matter fields, including scalar fields with non-trivial potentials and Maxwell fields, Develop a framework to study cosmological perturbation theory, Include anisotropies, Understand how to handle large interactions, [Andreas Pithis s talk] Include connectivity information in the analysis, Understand the physical interpretation of the energy E j. E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 19 / 19

Outlook There are many open questions: Further investigate the condensate Friedmann equation for a variety of condensate wave functions, Consider other types of matter fields, including scalar fields with non-trivial potentials and Maxwell fields, Develop a framework to study cosmological perturbation theory, Include anisotropies, Understand how to handle large interactions, [Andreas Pithis s talk] Include connectivity information in the analysis, Understand the physical interpretation of the energy E j. Thank you for your attention! E. Wilson-Ewing (AEI) Cosmology from QG Condensates June 1, 2016 19 / 19