Late time acceleration of the universe from quantum gravity

: We show that the accelerating expansion phase of the universe can emerge from the 1 group ﬁeld theory formalism, a candidate theory of quantum gravity. The cosmological evolution 2 can be extracted from condensate states using mean ﬁeld approximation, in a form of modiﬁed 3 FLRW equations. By introducing an effective equation of state w , we can reveal the relevant 4 features of the evolution, and show that with proper choice of parameters, w will approach to − 1, 5 corresponds to the behaviour of cosmological constant, results in a late time acceleration and leads 6 to de Sitter spacetime asymptotically. 7

some dark energy fields do exist, their behaviour may subject to the quantum gravity 23 effects, for instance some future singularities could be avoided [11]. 24 In this work we also follow this approach, and show that in GFT condensate 25 cosmology, the late time acceleration of the universe expansion will emerge naturally 26 [10], without introducing any dark energy field which may have mysterious properties. 27 As a candidate theory of quantum gravity, instead of the familiar spacetime degrees 28 of freedom, GFT provides more abstract, non-spatiotemporal entities, from which the 29 continuum spacetime should emerge [12][13][14]. Furthermore, the cosmological evolution For the emergence of 4d spacetime, one usually chooses the GFT field to be a 48 map over four copies of SU(2) group, ϕ : SU(2) 4 → C, ϕ(g v ) = ϕ(g 1 , g 2 , g 3 , g 4 ) [12]. 49 And geometrically to associate the basic quanta of our theory with a tetrahedra, we 50 further require that the field ϕ(g v ) to be right invariant ϕ(g v h) = ϕ(g 1 h, g 2 h, g 3 h, g 4 h) = 51 ϕ(g v ), ∀h ∈ SU(2) [12]. Furthermore, since the spacetime is emergent, especially there 52 is no time to start with, one usually needs a free massless scalar field as a relational clock 53 to track the evolution, this way the field ϕ(g v , φ) becomes time dependent.

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The right invariance of ϕ(g v , φ) under SU(2) enables us to project the field using 55 Peter-Weyl decomposition onto the complete and orthonormal basis of L 2 (SU(2) 4 /SU(2)), 56 the space of square integrable functions over quotient space SU(2) 4 /SU(2). Such basis 57 can be given by the spin network vertex functions κ x (g v ), which associates graphically  More explicitly, in the second quantized form, the field operators can be written aŝ whereĉ x (φ) andĉ † x (φ) are annihilation and creation operator respectively.
Having the basic operators in hand, the next step would be to find a suitable state for the interested spacetime, at least approximately. In our case, the homogeneous universe at a given time can be approximated by the coherent peaked state (CPS), which is constituted by a large number of quantas concentrated in a fixed relational time φ 0 [16,22] with |0 is vacuum state, defined byφ(g v , φ) |0 = 0 for all g v and φ, and N (σ) is a 66 suitable normalization factor. Peaking function η ε (φ − φ 0 , π 0 ) is peaked around φ 0 with 67 a typical width given by ε, and the fluctuations of the operator corresponding to the 68 conjugate momentum of the scalar field φ are controlled by π 0 . We callσ(g v , φ) the 69 reduced condensate wave function and assume that it does not modify the peaking property  The wave functionσ(g v , φ) can be decomposed using Peter-Weyl decomposition as well. For the emerged universe to be isotropic, we require the individual quantas to be as isotropic as possible, enforcesσ(g v , φ) to only have support over equilateral tetrahedra [16], whose area of the four faces are equal. Since each area is determined by the associated spin j i of the face, we see that only the coefficients with j 1 = j 2 = j 3 = j 4 = j survive in the decomposition [16] where we write j for j = (j 1 , j 2 , j 3 , j 4 ) = (j, j, j, j), I j,ι + m is the intertwiner labeled by ι, and 72 D j l m l n l (g l ) are the Wigner matrix functions for SU(2). 74 The decomposition (3) indicates that the time dependence of the condensate is only encoded in σ j (φ), hence effectively the dynamics of the condensate can be given by the following action

Volume dynamics and equation of state
, ξ j is an effective parameter encodes the details of the kinetic term of the fundamental GFT action (in the isotropic restriction), and denote derivatives with respect to φ 0 . Finally, from a rather phenomenological approach, the interaction kernel U(σ,σ) can be modelled in a simple, rather general form [24]: where λ j and µ j are interaction couplings correspond to each mode j satisfy that |µ j | 75 |λ j | |ξ 2 j −π 2 0 | and we assume that constants n j > n j > 2. 76 Varying the action (4) with respect toσ j we can get the equation of motion [22,24]. For the purpose that will be clear later, we can decomposeσ , then the equation of motion splits into imaginary and real part respectively. The imaginary part corresponds to a conserved quantity whose derivative vanishes, Q j = 0. Then the real part becomes [22,24] where m 2 j = ξ 2 j −π 2 0 . This equation can be integrated once directly, provides another conserved quantity,which corresponds to "clock-time translation" invariance of the system [16,22,24], Now we are in a position to extract observables from the condensates. In the condensate, for each equilateral tetrahedra characterized by mode j we can associate a volume V j ∝ l 3 p j 3/2 , and the module square |σ j | 2 = ρ 2 j gives the number of such tetrahedra. Hence the total volume of the universe can be approximated by (ignoring the quantum fluctuations) [16,22] In the free case where λ j = µ j = 0, the usual FLRW equation can be reproduced when 77 the total volume is large [16].

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In a homogeneous and isotropic universe, the modified FLRW equations (10) and (11) are enough enough to track the evolution of our universe. However, the relevant features of the dynamics can be extracted more easily from a deduced quantity, the effective equation of state w = −1 − 2Ḣ/(3H 2 ), where H is the Hubble parameter and the dot represents derivative respect to the co-moving time t [3]. In the relational language, w can be rewritten as [10] where V is the total volume and the indicates the derivative with respect to the relational 79 time φ, and we choose the time gauge, in which the volume V = a 3 for scale factor a.

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In the following we will discuss the late time cosmological expansion using the  has only one interaction term and setting µ j = 0 in equations (7) and (8) for example.

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When ρ j is large, since n j > 2 and µ j = 0 we see that the λ term dominates in equation (8), which can be approximated to We require λ j < 0 so that ρ j stays real. Equation (13) can be easily solved and we obtain where φ j∞ is a constant of integration, determined by initial conditions and we have approximately [10] Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 10 January 2022 Since for each j the module ρ j will diverge at φ = φ j∞ , the total volume V = ∑ j V j ρ 2 j 91 will be dominated by the mode with smallest φ j∞ . Note that the divergence of volume 92 at finite relational time won't lead to singularities in our model, as the effective energy 93 density of the whole universe remains finite [10].

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On the other hand, other modes, even sub-dominated, can still modify the universe evolution. In fact, adding another mode will change the way of how the effective equation of state w approaches its asymptotic value [10]. To see this, we consider the case with two modes, and to save space, we use ρ 1,2 to represent ρ j 1 ,j 2 and similarly for other parameters. Keeping in mind that in this report we set µ 1 = µ 2 = 0, then at large volume w will be dominated by the λ terms as well as the volume. For simplicity we further assume that n 1 = n 2 = n, then w will only depend on the ratio r = ρ 2 /ρ 1 in the large volume limit which approaches to the asymptotic value w → 2 − n/2 when the universe volume is 95 large.

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When becomes a constant. Submitting shows that w approaches this asymptotic value from above [10]. While for φ 1∞ < φ 2∞ , 99 ρ 1 diverges before ρ 2 does, hence r → 0 and w will approach the asymptotic value from 100 below.

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The case with n = 6 is of particular interest since w = −1 corresponds to the equation of state of a cosmological constant. Without losing generality we can assume φ 1∞ < φ 2∞ , then equation of state (16) can be expanded with respect to small r, and to the next to leading order we get Therefore, w < −1 for φ < φ 1∞ and corresponds to some kind of phantom energy, whose 102 energy density increases when the universe grows [3,27,28]. This leads us to the so called 103 phantom analogue of de Sitter spacetime [29].

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This report discusses the possibility to reproduce the late time acceleration phase 116 in the universe expansion from a candidate theory of quantum gravity, the group field 117 theory. In this formalism, the universe is constituted by a large number of building 118 blocks, which are excitations of the GFT field. Taking into account the homogeneity, 119 the universe at a given time can be model by CPS, the condensate state peaked around 120 a relational time φ 0 . The observables, such as the total volume, can be extracted from 121 the CPS, and in particular the effective equation of state w can be constructed from the 122 volume and its derivatives.

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To get the dynamics, we first show that the wave function σ(g v , φ) can be decom-124 posed into different modes σ j (φ), and the evolution of the universe can be extracted by 125 considering single or multiple modes. With a suitable choice of the effective action, the 126 equation of motion for the module ρ j (φ) = |σ j (φ)| can be solved approximately at large 127 volume. We then use the solution (14) to investigate the behaviour of w and find that in