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Cosmological Complexity in K-Essence

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Cosmological Complexity in K-Essence Cosmological Complexity in K-essence a,b c d Ai-chen Li ,∗ Xin-Fei Li ,† Ding-fang Zeng ,‡ and Lei-Hua Liu a Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ıi Franqu`es 1, 08028 Barcelona, Spain b Departament de F´ısica Qu`antica i Astrof´ısica, Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain c School of Science, Guangxi University of Science and Technology, 545026 Liuzhou, China d Theoretical Physics Division, College of Applied Sciences, Beijing University of Technology, China and e Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University, 416000 Jishou, China We calculate the cosmological complexity under the framework of scalar curvature perturbations for a K-essence model with constant potential. In particular, the squeezed quantum states are de- fined by acting a two-mode squeezed operator which is characterized by squeezing parameters rk and φk on vacuum state. The evolution of these squeezing parameters are governed by the Schrodinger¨ equation, in which the Hamiltonian operator is derived from the cosmological perturbative action. With aid of the solutions of rk and φk, one can calculate the quantum circuit complexity between unsqueezed vacuum state and squeezed quantum states via the wave-function approach. One ad- vantage of K-essence is that it allows us to explore the effects of varied sound speeds on evolution of cosmological complexity. Besides, this model also provides a way for us to distinguish the different cosmological phases by extracting some basic informations, like the scrambling time and Lyapunov exponent etc, from the evolution of cosmological complexity. I. INTRODUCTION terests are focused on understanding the physics of quan- tum circuit complexity from the sides of quantum field theory [20–22] or quantum mechanics [23, 24]. As pointed In recent years, the association between quantum in- out in [23], some elementary information about a quan- formation/computation theory and gravity has attracted tum chaotic system, like the scrambling time and Lya- a great amount of attentions. At the begining, this idea punov exponent, can be captured by circuit complexity. was motivated by the Anti-de Sitter/conformal field the- The Fubini-Study approach is proposed in [20], in order ory (AdS/CFT) [1–3], especially the research about holo- to measure the complexity is identified as the geodesic graphic entanglement entropy [4]. Afterwards, another distance connecting the reference and the target states in complementary physical quantity called quantum circuit group manifold (both the reference and the target states complexity has been involved into the AdS/CFT dictio- should be the coherent states of a specific group). Be- nary. In particular, for a thermo-field double state which sides, for Gaussian quantum states, anther geometric way is dual to an eternal asymptotic-AdS black hole [5], [6] for calculating the quantum circuit complexity is given indicates that the entanglement entropy fails to depict by Nielsen [25–27], and has been generalized to the con- the growth behavior after reaching the thermal equilib- text of QFT by [21], including the wave-function and the rium for the Einstein-Rosen Bridge (ERB) behind the covariance matrix approachs respectively [28]. horizon. As an alternative way, [7, 8] propose that the growth behavior of ERB in black hole interior, particu- Recently, in field of cosmology, the application of larly on late time, corresponds to the evolution of quan- quantum circuit complexity to scalar curvature pertur- tum circuit complexity between reference state and tar- bation on an expanding Friedmann-Lemaitre-Robertson- get state on AdS boundary. Specifically, this conjecture Walker (FLRW) background has been investigated by called complexity-volume (CV) supposes that the circuit [29]. Similar to the definition of squeezed quantum states complexity derived from the CFT on boundary is dual in inverted harmonic oscillator [30, 31], the two-mode to the maximum volume of the ERB in bulk spacetime. squeezed state formalism and the corresponding differ- Subsequently, [9, 10] suggest another version of CV called ential equations in framework of cosmological perturba- complexity-action (CA) conjecture, which associates the tions are developed by [32–34]. In [29], the complexity of arXiv:2102.12939v1 [gr-qc] 25 Feb 2021 circuit complexity on spacetime boundary to the gravi- cosmological perturbations (hereafter we call this quan- tational action evaluated on a region of Wheeler-DeWitt tity as cosmological complexity) between unsqueezed vac- patch in the bulk spacetime. After that, many extensive uum state and squeezed quantum state is computated studies on CV and CA conjectures have produced many by using the wave-function approach. Their results un- profound results as shown in [11–19]. cover that during the inflation epoch the complexity is Inspired by the holographic duality, more and more in- frozen inside the horizon, while it grows in an exponen- tial way after the mode exits the horizon. And then the universe de-complexifies during the subsequent radiation epoch and eventually the complexity is frozen after hori- ∗Electronic address: [email protected], [email protected] zon re-entry. Since then, [35] explores the cosmological †Electronic address: xfl[email protected] complexity for both expanding and contracting FLRW ‡Electronic address: [email protected] backgrounds with varied equation of state w. Besides, 2 the cosmological complexity is also studied in some typi- II. K-ESSENCE MODELS AND THE cal cosmological models which are alternative theories to CORRESPONDING COSMOLOGICAL the cosmic inflation scenario, like ekpyrosis and bouncing PERTURBATIONS cosmology [36, 37]. A. K-essence cosmology The K-essence cosmology are described by coupling a scalar field to Einstein gravity [38, 39] 1 S = d4x√ gR + d4x√ gP (X, ϕ) (1) In this paper, our aim is to investigate the cosmolog- 2 − − ical complexity in K-essence cosmology models [38–40], Z Z which are typically described by a large class of higher- in which the Lagrangian P (X, ϕ) is allowed to have a order (non-quadratic) scalar kinetic terms. K-essence as dependence on higher-order powers of the kinetic term X = 1 gµν ∂ ϕ∂ ϕ. In(1), why we denote the La- an important model to drive cosmic inflation, manifesting − 2 µ ν the many aspects of advantages that inflationary evolu- grangian for the scalar field as P (X, ϕ) because it plays tion is driven by higher-order scalar kinetic terms only the role of pressure, as shown in (3). Note that we take and inflation starts from very generic initial conditions the convention 8πG = c = 1, for the convenience of cal- without the help of potential terms. Moreover, the dy- culations. namical attractor solutions derived from the K-essence The Einstein field equations could be obtained by vary- can avoid fine-tuning of parameters and anthropic ar- ing the Lagrangian (1) with respect to metric tensor, guments in explaining the accelerated expansion of the 1 universe at present [41–43]. Our motivations come from Rµν gµν R = Tµν (2) the following aspects: Firstly, in [29] the cosmological − 2 complexity is considered in scalar curvature perturba- ∂P (X, ϕ) 2 Tµν = ∂µϕ∂ν ϕ + gµν P (X, ϕ) tions with constant sound speed, i.e. cS = 1. As a step ∂X forward, it is valuable to consider the effects of varied sound speed on the evolution of cosmological complexity. Decomposing the above energy momentum tensor into This purpose could be achieved in perturbative theory the form of a perfect fluid, we find of K-essence[39], the varied c2 is included in scalar cur- S T = u u + P g + u u (3) vature perturbations since the higher order corrections µν E µ ν µν µ ν on the canonical momentum for scalar fields. Secondly, in which the 4-velocity is for the K-essence model given by [40], the enriched cos- mological phases could be observed in same physical pa- ∂ ϕ u = µ (4) rameters with different initial conditions. Although the µ (2X)1/2 differences of these cosmological phases are reflected by the equations of state and scale factor, we also expect and the energy density is given by that the evolution of the cosmological complexity could provide some information to distinguish these cosmolog- ∂P =2X P (5) ical phases. E · ∂X − In a background of flat Friedman-Lemaitre-Robertson- Walker (FLRW) universe, ds2 = dt2 + a(t)2δ dxidxj (6) − ij the following independent equations are found by plug- Our work is structured as follows. In section II, an ging (6) into the Einstein field equation (2), specific K-essence cosmology model and the correspond- ing perturbative theories are reviewed. In section III, by 3H2 = (7) combining the definition of squeezed quantum states with E 2H˙ = + P (8) the perturbative actions given in section II, we derive the − E differential equations governing the evolution of squeez- a˙ ing parameter rk and squeezing angle φk, and the cor- in which H = a represents the Hubble constant. And responding numerical solutions are obtained. The com- then, by varying the Lagrangian with respect to ϕ, the plexity between unsqueezed vacuum state and squeezed equation of motion for scalar field reads quantum states are computed through the wave-function ∂P (X, ϕ) µν ∂P (X, ϕ) approach in section IV. We give conclusions and some ν g ∂µϕ + = 0 (9) future directions in the last section. ∇ ∂X ∂ϕ 3 X X 0.8 expand (9) explicitly, we give X1 4 ∂P (X, ϕ) ∂P (X, ϕ) ϕ¨ +3Hϕ˙ 0.6 ∂X ∂X X0 2 d ∂P (X, ϕ) ∂P (X, ϕ) +ϕ ˙ = (10) 0.4 dt ∂X ∂ϕ 0.2 0.4 0.6 X 0.2 X0 Note that the (10) could also be derived from the continu- Xc X1 X+ -2 ity equation ˙ +3H( + P ) = 0.
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