Cosmological Complexity in K-essence
aa r X i v : . [ g r- q c ] F e b Cosmological Complexity in K-essence
Ai-chen Li a,b , ∗ Xin-Fei Li c , † Ding-fang Zeng d , ‡ and Lei-Hua Liu a Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ı i 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 perturbationsfor 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 r k and φ k on vacuum state. The evolution of these squeezing parameters are governed by the Schr ¨ odinger equation, in which the Hamiltonian operator is derived from the cosmological perturbative action.With aid of the solutions of r k and φ k , one can calculate the quantum circuit complexity betweenunsqueezed 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 ofcosmological complexity. Besides, this model also provides a way for us to distinguish the differentcosmological phases by extracting some basic informations, like the scrambling time and Lyapunovexponent etc, from the evolution of cosmological complexity. I. INTRODUCTION
In recent years, the association between quantum in-formation/computation theory and gravity has attracteda great amount of attentions. At the begining, this ideawas motivated by the Anti-de Sitter/conformal field the-ory (AdS/CFT) [1–3], especially the research about holo-graphic entanglement entropy [4]. Afterwards, anothercomplementary physical quantity called quantum circuitcomplexity has been involved into the AdS/CFT dictio-nary. In particular, for a thermo-field double state whichis dual to an eternal asymptotic-AdS black hole [5], [6]indicates that the entanglement entropy fails to depictthe growth behavior after reaching the thermal equilib-rium for the Einstein-Rosen Bridge (ERB) behind thehorizon. As an alternative way, [7, 8] propose that thegrowth behavior of ERB in black hole interior, particu-larly on late time, corresponds to the evolution of quan-tum circuit complexity between reference state and tar-get state on AdS boundary. Specifically, this conjecturecalled complexity-volume (CV) supposes that the circuitcomplexity derived from the CFT on boundary is dualto the maximum volume of the ERB in bulk spacetime.Subsequently, [9, 10] suggest another version of CV calledcomplexity-action (CA) conjecture, which associates thecircuit complexity on spacetime boundary to the gravi-tational action evaluated on a region of Wheeler-DeWittpatch in the bulk spacetime. After that, many extensivestudies on CV and CA conjectures have produced manyprofound results as shown in [11–19].Inspired by the holographic duality, more and more in- ∗ Electronic address: [email protected], [email protected] † Electronic address: xfl[email protected] ‡ Electronic address: [email protected] terests are focused on understanding the physics of quan-tum circuit complexity from the sides of quantum fieldtheory [20–22] or quantum mechanics [23, 24]. As pointedout in [23], some elementary information about a quan-tum chaotic system, like the scrambling time and Lya-punov exponent, can be captured by circuit complexity.The Fubini-Study approach is proposed in [20], in orderto measure the complexity is identified as the geodesicdistance connecting the reference and the target states ingroup manifold (both the reference and the target statesshould be the coherent states of a specific group). Be-sides, for Gaussian quantum states, anther geometric wayfor calculating the quantum circuit complexity is givenby Nielsen [25–27], and has been generalized to the con-text of QFT by [21], including the wave-function and thecovariance matrix approachs respectively [28].Recently, in field of cosmology, the application ofquantum circuit complexity to scalar curvature pertur-bation on an expanding Friedmann-Lemaitre-Robertson-Walker (FLRW) background has been investigated by[29]. Similar to the definition of squeezed quantum statesin inverted harmonic oscillator [30, 31], the two-modesqueezed state formalism and the corresponding differ-ential equations in framework of cosmological perturba-tions are developed by [32–34]. In [29], the complexity ofcosmological perturbations (hereafter we call this quan-tity as cosmological complexity) between unsqueezed vac-uum state and squeezed quantum state is computatedby using the wave-function approach. Their results un-cover that during the inflation epoch the complexity isfrozen inside the horizon, while it grows in an exponen-tial way after the mode exits the horizon. And then theuniverse de-complexifies during the subsequent radiationepoch and eventually the complexity is frozen after hori-zon re-entry. Since then, [35] explores the cosmologicalcomplexity for both expanding and contracting FLRWbackgrounds with varied equation of state w . Besides,the cosmological complexity is also studied in some typi-cal cosmological models which are alternative theories tothe cosmic inflation scenario, like ekpyrosis and bouncingcosmology [36, 37].In this paper, our aim is to investigate the cosmolog-ical complexity in K-essence cosmology models [38–40],which are typically described by a large class of higher-order (non-quadratic) scalar kinetic terms. K-essence asan important model to drive cosmic inflation, manifestingthe many aspects of advantages that inflationary evolu-tion is driven by higher-order scalar kinetic terms onlyand inflation starts from very generic initial conditionswithout the help of potential terms. Moreover, the dy-namical attractor solutions derived from the K-essencecan avoid fine-tuning of parameters and anthropic ar-guments in explaining the accelerated expansion of theuniverse at present [41–43]. Our motivations come fromthe following aspects: Firstly, in [29] the cosmologicalcomplexity is considered in scalar curvature perturba-tions with constant sound speed, i.e. c S = 1. As a stepforward, it is valuable to consider the effects of variedsound speed on the evolution of cosmological complexity.This purpose could be achieved in perturbative theoryof K-essence[39], the varied c S is included in scalar cur-vature perturbations since the higher order correctionson the canonical momentum for scalar fields. Secondly,for the K-essence model given by [40], the enriched cos-mological phases could be observed in same physical pa-rameters with different initial conditions. Although thedifferences of these cosmological phases are reflected bythe equations of state and scale factor, we also expectthat the evolution of the cosmological complexity couldprovide some information to distinguish these cosmolog-ical phases.Our work is structured as follows. In section II, anspecific K-essence cosmology model and the correspond-ing perturbative theories are reviewed. In section III, bycombining the definition of squeezed quantum states withthe perturbative actions given in section II, we derive thedifferential equations governing the evolution of squeez-ing parameter r k and squeezing angle φ k , and the cor-responding numerical solutions are obtained. The com-plexity between unsqueezed vacuum state and squeezedquantum states are computed through the wave-functionapproach in section IV. We give conclusions and somefuture directions in the last section. II. K-ESSENCE MODELS AND THECORRESPONDING COSMOLOGICALPERTURBATIONSA. K-essence cosmology
The K-essence cosmology are described by coupling ascalar field to Einstein gravity [38, 39] S = 12 Z d x √− gR + Z d x √− gP ( X, ϕ ) (1)in which the Lagrangian P ( X, ϕ ) is allowed to have adependence on higher-order powers of the kinetic term X = − g µν ∂ µ ϕ∂ ν ϕ . In (1), why we denote the La-grangian for the scalar field as P ( X, ϕ ) because it playsthe role of pressure, as shown in (3). Note that we takethe convention 8 πG = c = 1, for the convenience of cal-culations.The Einstein field equations could be obtained by vary-ing the Lagrangian (1) with respect to metric tensor, R µν − g µν R = T µν (2) T µν = ∂P ( X, ϕ ) ∂X ∂ µ ϕ∂ ν ϕ + g µν P ( X, ϕ )Decomposing the above energy momentum tensor intothe form of a perfect fluid, we find T µν = E u µ u ν + P (cid:0) g µν + u µ u ν (cid:1) (3)in which the 4-velocity is u µ = ∂ µ ϕ (2 X ) / (4)and the energy density is given by E = 2 X · ∂P∂X − P (5)In a background of flat Friedman-Lemaitre-Robertson-Walker (FLRW) universe, ds = − dt + a ( t ) δ ij dx i dx j (6)the following independent equations are found by plug-ging (6) into the Einstein field equation (2),3 H = E (7) − H = E + P (8)in which H = ˙ aa represents the Hubble constant. Andthen, by varying the Lagrangian with respect to ϕ , theequation of motion for scalar field reads ∇ ν (cid:0) ∂P ( X, ϕ ) ∂X g µν ∂ µ ϕ (cid:1) + ∂P ( X, ϕ ) ∂ϕ = 0 (9)expand (9) explicitly, we give¨ ϕ ∂P ( X, ϕ ) ∂X + 3 H ˙ ϕ ∂P ( X, ϕ ) ∂X + ˙ ϕ ddt (cid:0) ∂P ( X, ϕ ) ∂X (cid:1) = ∂P ( X, ϕ ) ∂ϕ (10)Note that the (10) could also be derived from the continu-ity equation ˙ E + 3 H ( E + P ) = 0. It should be indicatedthat there are only two independent equations among(7) , (8) , (9).In this paper, we will consider the quadratic purelykinetic Lagrangian with constant potential V ( ϕ ) [40],namely P ( X, ϕ ) = X + C X − V (11)Plug (11) into (7) and (10), we get the differential equa-tions which controll the evolution of scale factor a ( t ) andclassical scalar field ϕ ( t )3 H = X + 32 C X + V (12)˙ X = − HX C X C X (13)From (5), (11), the effective speed of sound is expressedas c S = dP/dtd E /dt = ∂P/∂X∂ E /∂X = 1 + C X C X (14)In (13), let us eliminate H by using (12)˙ X = ± X s(cid:0) X + C X + V (cid:1) C X C X (15)Note that the ”+” and ”-” branches in (15) correspondto the H <
H > V > , C < X = ( ˙ ϕ ) > X is well-defined only inregions [0 , √ − C V − C ]. As shown by the left panel ofFig.1, the right-hand side of (15) vanishes at three places X = 0, X = − C and X + = √ − C V − C . Meanwhile,this dynamical system is not defined at X c = − C ,which is a terminating singularity but not a curvaturesingularity. After solving this dynamical system (12)-(15) numerically, as shown by Fig.1, the various cosmo-logical phases are shown when X ( t ) is in different initialpositions. In the domain [ X , X c ), as plotted by the cyancurves, the usual inflationary de Sitter phase is observedas X ( t ) approaches the X . From Fig.2, it is easy tosee that this phase has a well-defined behavior, namely E + P > c S >
0. However, although the orangecurves whose initial value of X ( t ) is in domain ( X c , X ) X - X X X c X X +
10 15 20 25 30 35 t X X X X + t a ( t ) t - - H ( t ) FIG. 1: (color online). According to the expression of (15),we plot the variation of ˙ X with respect to X in case of therepresentative parameters V = 1 , C = −
2. While, thechange of
X, a, H with time are shown by solving (12) , (13)numerically. The corresponding initial conditions in domains[ X , X c ), ( X c , X ), ( X , X + ) are denoted with cyan, orange,purple points. has the de Sitter behaviour as well, it exhibits a nega-tive velocity of sound , c S <
0, which implies an exoticstate. Finally, as displayed by the purple curves, the X ( t )moves toward the X + from an initial position in the re-gion ( X , X + ), and then moves in opposite direction af-ter reaching this point. This motion mode correspondsto a bounce phase in cosmology which transits from aphantom state to another phantom state. It is neces-
10 15 t - - - - ℰ + P t - - - c S FIG. 2: (color online). The variation of E + P , c S as the timeincreases, which corresponds to the numerical results given inFig.1. sary to point out that the evolution of X ( t ) is controlledby the ”+” and ”-” branches respectively, in process ofapproaching and leaving the bouncing point X + respec-tively. B. Perturbative actions
Under the inhomogeneous quantum fluctuations, thescalar ϕ has the following ansatz ϕ ( t, ~x ) = ¯ ϕ ( t ) + δϕ ( t, ~x ) (16)Meanwhile, in the longitudinal gauge [44], the perturbedspacetime metric is written as ds = − (cid:0) t, ~x ) (cid:1) dt + a ( t ) (cid:0) − t, ~x ) (cid:1) δ ij dx i dx j (17)After substituting these perturbative ansatzs into (1), theperturbative action in second order is [39] S = Z dtd x (cid:8) ξ ˆ (cid:3) ˙ ζ − H c S a ( E + P ) ξ ˆ (cid:3) ξ + a ( E + P )2 H ζ ˆ (cid:3) ζ (cid:9) (18)where the new variables ξ and ζ are associated to the δϕ and Φ through the definitionsΦ a = 12 Hξ (19) δϕ ˙ ϕ = ζH − aξ (20)Besides, the perturbative Einstein equations in linearizedorder imply the two independent equations1 a ˆ (cid:3) Φ − H ˙Φ − H Φ = 12 δT (21) ∂ i (cid:0) ˙Φ + H Φ (cid:1) = 12 δT i (22) δT = ∂ E ∂X δX + ∂ E ∂ϕ δϕ = E + Pc S (cid:0) δϕ ˙ ϕ − Φ (cid:1) − H ( E + P ) δϕ ˙ ϕδT i = ( E + P ) ∂ i (cid:18) δϕ ˙ ϕ (cid:19) Substituting (19), (20) into (21), (22) and combiningwith the classical EOMs (7) − (10), we get the follow-ing equations for ξ and ζ ˙ ξ = a ( E + P ) H ζ (23)˙ ζ = c S H a ( E + P ) ˆ (cid:3) ξ (24)The action (18) could be further simplified via the equa-tion (24), S = 12 Z dηd x z (cid:0) ζ ′ + c S ζ ˆ (cid:3) ζ (cid:1) (25)in which we have used the conformal time η = R dta ( t ) andprime denotes the derivative with respect to η . Mean-while, the variable z is z = a ( E + P ) / c S H (26) By introducing the Mukhanov variable v = zζ , one canrewrite the action (25) as S = 12 Z dηd x (cid:18) v ′ + c S v ˆ (cid:3) v + z ′′ z v (cid:19) (27) III. THE SQUEEZED QUANTUM STATES FORCOSMOLOGICAL PERTURBATIONS
By using the integration by parts, the action (27) aretransformed into S = Z dηL = 12 Z dηd x (cid:18) v ′ − c S ( ∂ i v ) + (cid:0) z ′ z (cid:1) v − z ′ z v ′ v (cid:19) (28)In this paper, we shall restrict our attention to the caseof flat universe, so we have set K = 0 in (28). From (28),the canonical momentum is defined as π ( η, ~x ) = δLδv ′ ( η, ~x ) = v ′ − z ′ z v (29)So the Hamiltonian H = R d x ( πv ′ − L ) is constructedas H = 12 Z d x (cid:2) π + c S ( ∂ i v ) + z ′ z ( vπ + πv ) (cid:1)(cid:3) (30)By means of the method of second quantization,the field v ( η, ~x ) , π ( η, ~x ) are promoted to operatorsˆ v ( η, ~x ) , ˆ π ( η, ~x ). Similar to the quantum mechanics ofinverted harmonic oscillator [30], we suppose the ˆ v ( η, ~x )and ˆ π ( η, ~x ) have the following decomposition in Fourierspaceˆ v ( η, ~x ) = Z d k (2 π ) / √ k (cid:0) ˆ c †− ~k v ⋆k ( η ) + ˆ c ~k v k ( η ) (cid:1) e i~k · ~x (31)ˆ π ( η, ~x ) = i Z d k (2 π ) / r k (cid:0) ˆ c †− ~k u ⋆k ( η ) − ˆ c ~k u k ( η ) (cid:1) e i~k · ~x (32)in which ˆ c †− ~k and ˆ c ~k are the creation and annihilationoperators respectively. Through choosing an appropriatenormalization condition for mode functions u k ( η ) , v k ( η ),besides the necessary uncertainty relation[ v ( η, ~x ) , v ′ ( η, ~y )] (cid:12)(cid:12) η = η = iδ ( ~x − ~y ) (33)the Hamiltonian operator could be also simplified asˆ H = Z d k ˆ H k = Z d k (cid:8) k c S + 1)ˆ c †− ~k ˆ c − ~k + k c S + 1)ˆ c ~k ˆ c † ~k + (cid:0) k c S −
1) + iz ′ z (cid:1) ˆ c † ~k ˆ c †− ~k + (cid:0) k c S − − iz ′ z (cid:1) ˆ c ~k ˆ c − ~k (cid:9) (34)Just as the inverted harmonic oscillator, given thequadratic Hamiltonian, the unitary evolution operatorcould be factorized in the following form [32, 33]ˆ U ~k ( η, η ) = ˆ S ~k ( r k , φ k ) ˆ R ~k ( θ k ) (35)In (35), the ˆ R ~k is the two-mode rotation operator withthe following definitionˆ R ~k ( θ k ) = exp (cid:2) − iθ k ( η ) (cid:0) ˆ c ~k ˆ c † ~k + ˆ c †− ~k ˆ c − ~k (cid:1)(cid:3) (36)where the θ k ( η ) is the rotation angle. Meanwhile, ˆ S isthe two-mode squeeze operator defined asˆ S ~k ( r k , φ k ) = exp (cid:2) r k ( η ) (cid:0) e − iφ k ( η ) ˆ c ~k ˆ c − ~k − e iφ k ( η ) ˆ c †− ~k ˆ c † ~k (cid:1)(cid:3) (37)where r k ( η ) and φ k ( η ) represent the squeezing parameterand squeezing angle respectively. Note that we will ignorethe effects of rotation operators, because it only producesan irrelevant phase when acting on the initial vacuumstate.By using the operator ordering theorem given in [31],we could expand (37) in the following ordered formˆ S ~k ( r k , φ k ) = exp (cid:2) − e iφ k tanh r k ˆ c †− ~k ˆ c † ~k (cid:3) · exp (cid:2) − ln(cosh r k ) (cid:0) ˆ c †− ~k ˆ c − ~k + ˆ c ~k ˆ c † ~k (cid:1)(cid:3) · exp (cid:2) e − iφ k tanh r k ˆ c ~k ˆ c − ~k (cid:3) (38)After acting the squeeze operator (38) on the two-modevacuum state |
0; 0 i ~k, − ~k , a two-mode squeezed state willbe got | Ψ i sq = 1cosh r k ∞ X n =0 ( − n e inφ k tanh n r k | n ; n i ~k, − ~k (39)where the two-mode excited state | n ; n i ~k, − ~k is | n ; n i ~k, − ~k = 1 n ! (cid:0) ˆ c † ~k (cid:1) n (cid:0) ˆ c †− ~k (cid:1) n |
0; 0 i ~k, − ~k (40)After substituting (34) , (39) into the following Schr ¨ odinger equation i ddη | Ψ i sq = ˆ H | Ψ i sq (41)we can give rise to the time evolution of the squeezingparemeters r k ( η ) , φ k ( η ) − dr k dη = k c S −
1) sin(2 φ k ) + z ′ z cos(2 φ k ) (42) dφ k dη = k ( c S + 1)2 − k c S −
1) cos 2 φ k coth 2 r k + z ′ z sin 2 φ k coth 2 r k (43) Note that the variable z is given by (26), which is relatedto the scale factor a , the speed of sound c S , the energydensity E and the pressure P . For the sake of simplifi-cation in numerical calculations, the variable log ( a ) isused to take place of the conformal time η . After plug-ging the numerical solutions of a ( t ) , c S ( t ) , E ( t ) + P ( t )in Fig.1-Fig.2 into the differential equations (42)-(43),we could show the evolution of squeezing parameters inFig.3-Fig.6. The result in Fig.3 is analogous to the one - - - - Log [ a ] r k - - - - - - - - - Log [ a ] ϕ k FIG. 3: (color online). The squeezing parameters vs. the logof scale factor for the usual inflationary de Sitter phase. Theleft figure shows the squeezing parameter r k oscillates withsmall amplitude initially, and then it grows linearly with re-spect to log a . Note that the sub-figure is the amplificationof the evolution of r k in early time before horizon exit. Forthe right figure, the squeezing angle φ k grows intensively, thenit keeps constant. given in [29], the squeezing parameter r k stays small insub-horizon limit and then grows rapidly in an exponen-tial way after the mode exits the horizon. The Fig.4 - - - - Log [ a ] r k - - - - Log [ a ] - - - ϕ k FIG. 4: (color online). The squeezing parameters vs. the logof scale factor for the de Sitter expansion with the negative c S . The left figure shows that the squeezing parameter r k increases dramatically, and then grows linearly with respectto log a . For the right figure, the squeezing angle φ k growsdramatically at first until it reaches the minimal point, thenit grows slowly. Note that the sub-figure is the amplificationof φ k -log ( a ) in early time, which shows that the squeezingangle varies smoothly at the minimal point. corresponds to the numerical solutions of squeezing pa-rameters in background of de Sitter expanding phasewith negative c S , in which the most remarkable char-acter is that the squeezing parameter r k rises steeplywithout oscillatory behavior inside the horizon. This”fast-squeezed” behavior is originated from the negative c S . On the background of bounce phase, the evolutionsof squeezing parameters in contracting and expandingstages are displayed in Fig.5 and Fig.6 respectively. Thisresult is consist with [37], in which the circuit complexityof two well known bouncing cosmological solutions, i.e. - - - Log [ a ] r k - - - Log [ a ] ϕ k FIG. 5: (color online). Squeezing parameters vs. the log ofscale factor before bouncing. For the left figure, the magni-tude of the squeezing parameter is almost constant while itsfrequency is increasing with respect to log as seen from theright to the left. For the right figure, the squeezing angle isgradually decreasing with small oscillation. Cosinehyperbolic and
Exponential models of scale fac-tors, have been studied. Similar to [37], in contractingstage, we observe that the squeezing parameter r k is vig-orously oscillatory with decreasing amplitudes when ap-proaching the bouncing point. After crossing the bounc-ing point and entering the expanding stage, the r k is os-cillatory with slowly varying amplitudes at the beginningand then grows fast like the Fig.3. - - - - Log [ a ] r k - - - - Log [ a ] ϕ k FIG. 6: (color online). Squeezing parameters vs. the log ofscale factor after bouncing. For the left figure, the magnitudeof the squeezing parameter r k oscillates very fast at first, thenits amplitude increases linearly with respect to log a . Forthe right figure, the squeezing angle φ k increases dramatically,then it tends to be constant. IV. COMPLEXITY FOR SQUEEZEDQUANTUM STATES OF COSMOLOGICALPERTURBATIONS
In this paper, we will evaluate the circuit complexity byusing Nielsen’s method [25–27]. First, a reference state | ψ R i is given at τ = 0. And then, we suppose that atarget state | ψ T i could be obtained at τ = 1 by acting aunitary operator on | ψ R i , namely | ψ T i τ =1 = U ( τ = 1) | ψ R i τ =0 (44)As usual, τ parametrizes a path in the Hilbert space.Generally, the unitary operator is constructed from apath-ordered exponential of a Hamiltonian operator U ( τ ) = ←−P exp (cid:18) − i Z τ dsH ( s ) (cid:19) (45) where the ←−P indicates right-to-left path ordering. TheHamiltonian operator H ( s ) can be expanded in terms ofa basis of Hermitian operators M I , which are the gener-ators for elementary gates H ( s ) = Y ( s ) I M I (46)The coefficients Y ( s ) I are identified as the control func-tions that determine which gate should be switched on orswitched off at a given parameter. Meanwhile, the Y ( s ) I satisfy the Schr ¨ odinger equation dU ( s ) ds = − iY ( s ) I M I U ( s ) (47)Then a cost f unctional is defined as follows C ( U ) = Z F ( U , ˙ U ) dτ (48)The complexity is obtained by minimizing the functional(48) and finding the shortest geodesic distance betweenthe reference and target states. Here, we restrict ourattentions on the quadratic cost functional F ( U, Y ) = sX I ( Y I ) (49)In this work, the target state is the two-mode squeezedvacuum state (39). After projecting | Ψ i into the positionspace, the following wavefunction is implied [34, 45]Ψ sq ( q ~k , q − ~k ) = ∞ X n =0 ( − n e inφ k tanh n r k cosh r k h q ~k ; q − ~k | n ; n i ~k, − ~k = exp[ A ( r k , φ k ) · ( q ~k + q − ~k ) − B ( r k , φ k ) · q ~k q − ~k ]cosh r k √ π p − e − iφ k tanh r k (50)in which the coefficients A ( r k , φ k ) and B ( r k , φ k ) are A ( r k , φ k ) = k (cid:18) e − iφ k tanh r k + 1 e − iφ k tanh r k − (cid:19) (51) B ( r k , φ k ) = 2 k (cid:18) e − iφ k tanh r k e − iφ k tanh r k − (cid:19) (52)By using a suitable rotation in vector space ( q ~k , q − ~k ), theexponent in (50) could be rewritten by a form of diagonalmatrixΨ sq ( q ~k , q − ~k ) = exp[ − ˜ M ab q a q b ]cosh r k √ π p − e − iφ k tanh r k (53)˜ M = (cid:18) Ω ~k ′
00 Ω − ~k ′ (cid:19) = (cid:18) − A + B − A − B (cid:19) Meanwhile, the reference state is the unsqueezed vacuumstate, Ψ ( q ~k , q − ~k ) = h q ~k ; q − ~k |
0; 0 i ~k, − ~k = exp[ − ( ω ~k q ~k + ω − ~k q − ~k )] π / = exp[ − ˜ m ab q a q b ] π / (54)˜ m = (cid:18) ω ~k ω − ~k (cid:19) According to the definition (44), one can associate thetarget state (53) with the reference state (54) through aunitary transformationΨ τ ( q ~k , q − ~k ) = ˜ U ( τ )Ψ ( q ~k , q − ~k ) ˜ U † ( τ ) (55)Ψ τ =0 ( q ~k , q − ~k ) = Ψ ( q ~k , q − ~k ) (56)Ψ τ =1 ( q ~k , q − ~k ) = Ψ sq ( q ~k , q − ~k ) (57)where ˜ U ( τ ) is a GL (2 , C ) unitary matrix which give theshortest geodesic distance between the target state andthe reference state in operator space. In general, theoperators of GL (2 , C ) can be expressed as˜ U ( τ ) = exp[ X I =1 Y I ( τ ) M I ] (58)where the { M I } represent the 4 generators of GL (2 , C ),namely M = (cid:18) (cid:19) , M = (cid:18) (cid:19) M = (cid:18) (cid:19) , M = (cid:18) (cid:19) Therefore the geometry in this operator space is de-scribed by the metric [20], ds = G IJ dY I d ( Y J ) ⋆ (59)In language of geometry, together with definitions (48)and (49), the complexity is described by the followingline length C ( ˜ U ) = Z dτ q G IJ ˙ Y I ( τ )( ˙ Y J ( τ )) ⋆ (60)in which the dot denotes the derivative with respect to t .Due to the fact that the manifold of general linear group GL ( N, C ) could be an Euclidean geometry, it is simpleto set G ij = δ ij . It follows that the shortest geodesicbetween target state and reference state is a straight line,namely Y I ( τ ) = Y I ( τ = 1) · τ + Y I ( τ = 0) (61)Note that both (53) and (54) are diagonal, the off-diagonal generators will increase the distance betweentwo states in operator space; thus the components Y , Y is set to zero [21]. With the help of boundary conditions(56) and (57), one getsIm( Y , ) (cid:12)(cid:12) τ =0 = Re( Y I ) (cid:12)(cid:12) τ =0 = 0 (62) Im( Y , ) (cid:12)(cid:12) τ =1 = ln | Ω ~k, ~ − k | ω ~k, ~ − k Re( Y , ) (cid:12)(cid:12) τ =1 = arctan Im(Ω ~k, ~ − k )Re(Ω ~k, ~ − k ) (63)Finally, plugging (61)-(63) into (60), the complexity iscalculated as C ( k ) = 12 s(cid:0) ln | Ω ~k | ω ~k (cid:1) + (cid:0) ln | Ω − ~k | ω − ~k (cid:1) + (cid:0) arctan Im(Ω ~k )Re(Ω ~k ) (cid:1) + (cid:0) arctan Im(Ω ~ − k )Re(Ω ~ − k ) (cid:1) (64)where ω ~k = ω − ~k = | ~k | = k .The complexity is explicitly obtained in Fig.7-Fig.8 viathe numerical solutions of r k and φ k . As indicated by[23], some basic informations about a quantum chaoticsystem could also be captured by the circuit complexity.In particular, by studying an inverted harmonic oscillatormodel, they find that the time scale when the complexitystarts to grow could be identified as the scrambling time,while the Lyapunov exponent approximately equals tothe slope of the linear growth part. Basing on the analy-sis of [23], after calculating the cosmological complexity on background of inflationary de Sitter expansion, [29]finds that the scrambling time scale is the time of horizonexit while the dCdt is approximately equal to the Hubbleconstant (to be precise, dCdt < H ). Their results are con-sistent with the left panel of Fig.7, which describes theevolution of the cosmological complexity for K-essence inthe usual inflationary de Sitter phase. However, in case ofthe de Sitter expansion with negative c S , the scramblingtime appears in the subhorizon limit while the chang-ing rate of the complexity in linear growth part satisfies dCdt < H , as displayed by the right panel of Fig.7. Fi- - - Log ( a ) C - - - - - horizonexit - - Log ( a ) C horizonexit FIG. 7: (color online). Complexity vs. the log of scale factor.For the left figure in the de-Siiter inflation, the complexityoscillates at first, and its amplitude keeps constant. Then,the complexity grow linearly with respect to log a . For theright figure in the inflation with the squire of sound speed c s <
0, the complexity grows linearly. - - - Log ( a ) C t - - - - Log ( a ) C horizonexit FIG. 8: (color online). Complexity vs. the log of scale factor.For the left figure, before bouncing, the complexity oscillates,and its frequency keeps growing. For the right figure, afterbouncing, the complexity oscillates intensively at first, thenits complexity grows linearly with respect to log a nally, for the bouncing cosmology phase, one should notein Fig.8 that the scrambling time occurs after the timeof horizon exit, and the dCdt < H in linear growth regionis smaller than the value of Hubble constant. V. CONCLUSIONS AND DISCUSSION
In this paper, we have computated the cosmologicalcomplexity in a type of K-essence cosmology model withconstant potential [40]. After solving the dynamical sys-tem of this K-essence model, as shown in Fig.1-Fig.2, weobserve three kinds of cosmological phases in same phys-ical parameters with different initial conditions , namelythe usual inflationary de Sitter phase (cyan curves), thede Sitter expansion with negative c S (orange curves), thebouncing cosmology phase (purple curves). Besides, bas-ing on the work [39], the perturbative action describingthe perturbations of curvature scalar is given by (27),in which the varied c S is included due to the higher or-der corrections to the kinetic term of scalar field ϕ . Byutilizing the method of second quantization to the per-turbative action, together with the definition of squeezedquantum state which is characterized by squeezing pa-rameter r k and squeezing angle φ k , we obtain the differ-ential equations governing the evolution of r k and φ k in(42)-(43). The numerical solutions of these differentialequations in backgrounds of three kinds of cosmological phases are displayed in Fig.3-Fig.6 respectively. Subse-quently, the physical meaning of quantum circuit com-plexity and a type of computational method called wave-function approach are reviewed briefly. According to thisapproach, the complexity between unsqueezed vacuumstate and squeezed quantum states are calculated underthe framework of cosmological perturbations. The evolu-tion of cosmological complexity in different cosmologicalphases are shown from Fig.7 to Fig.8.For the evolution of cosmological complexity in theusual inflationary de Sitter phase, we observe the con-sistent results compared to the work [29], namely thescrambling time scale is just the time of horizon exitwhile the dCdt in linear growth portion is in same magni-tude as the Hubble constant (to be precise, dC/dt < H ).However, for the de Sitter expansion with negative c S ,the scrambling time occurs far earlier than the time ofhorizon exit. And the changing rate of the complexity inlinear growth part is bigger than the value of Hubble con-stant. Besides, it is easy to observe that the complexitygrows rapidly without any oscillatory behavior in earlystate inside the horizon. Actually, this fast-growing phe-nomenon is arisen from the negative c S which leads to the”fast-squeezed” behavior of the r k in subhorizon limit asshown by Fig.4. Finally, in regard to the bouncing cos-mology phase, the scrambling time is lag behind the timeof horizon exit while the changing rate of the complexityin linear growth part satisfies dC/dt > H . As a characterin bouncing cosmology phase, it should be noticed thatthe oscillatory behavior of the cosmological complexitywould last after the horizon exit.As discussions, we suggest the following extended top-ics. In this paper, we estimate the scarambling time andLyapunov exponent just basing on the analysis of [23].Actually, one can more precisely obtain these physicalquantities by applying the out-of-time-order correlator(OTOC) method [46] to this K-essence cosmology model.Besides, one could also apply the cosmological complexityto investige the multi-field inflation model. Especially, itis interesting to explore the possible associations betweenthe geometrical instability arising in multi-field inflationand the Lyapunov exponent estimated from the cosmo-logical complexity or OTOC. VI. ACKNOWLEDGEMENTS
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