Complexity of non-trivial sound speed in inflation
aa r X i v : . [ g r- q c ] F e b Complexity of non-trivial sound speed in inflation
Lei-Hua Liu ∗ and Ai-Chen Li
2, 3, † Department of Physics, College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, China Institut de Ci`encies del Cosmos, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain 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
We will consider the effects of non-trivial sound speed on the evolution of cosmological complexityin a method of squeezed quantum states. In the standard procedure, we will treat the vacuum stateof the curvature perturbation as the squeezed vacuum state referring to the Gaussian state. Squeezedquantum states are obtained by acting a two-mode squeezed operator which is described by angleparameter φ k and squeezing parameter r k on a squeezed vacuum state. Through Schr ¨ odinger equation, one can obtain the corresponding evolution equation of φ k and r k . Subsequently, thequantum circuit complexity between a squeezed vacuum state and squeezed states are evaluated inscalar curvature perturbation with a type of non-trivial sound speeds. Our result indicates that theevolution of complexity will not change dramatically at a late time, only by considering the effectsof the non-trivial sound speed in an inflationary de-Sitter spacetime. However, compared to thecase of c S = 1, the evolution of complexity at an early time shows the rapid oscillation. INTRODUCTION
With the development of AdS / CFT [1], the under-standing of the nature of spacetime has been made sig-nificant progress, in which a bulk gravitational theory isequivalent to a CFT at its corresponding boundary. Thisconjecture can be dubbed as a particular realization ofthe holographic principle, consequently one can considerthat gravity is an emergent phenomenon not as a fun-damental force in nature. For the further developmentof this methodology, another well-known conjecture wasproposed that the spactime comes via quantum entan-glement [2], especially from a realizion called ER = EPR[3]. In light of this logic, it motivates us for exploringthe nature of spactime from the perspective of quantumentanglement [4–8]. Thus, holographic entanglement hasbecome a key step for investigating the essence of space-time. However, Ref. [9] found that a boundary QFTwill reach a thermal equilibrium within a short time, theevolution of its corresponding wormhole will cost muchlonger time comparing with QFT. Therefore, it motivesthat there is another quantum quantity named by com-plexity for describing the evolution of wormhole [10, 11].More precisely, this evolution can be depicted by holo-graphic complexity. For calculating complexity in lightof holographic principle, Susskind and his collaboratorshave proposed two conjectures: CV conjecture (complex-ity=volum) and CA conjecture (complexity=action), inwhich the first one is translated into the computation ofmaximum volume of the wormhole in bulk space and thesecond is referred as the computation of action evaluatedon a bulk subregion called Wheeler-DeWitt patch. Af-ter that, extensive studies of such theories have producedmany profound results, see for example [12–19].Motivated by the AdS/CFT, a hotspot research direc-tion in the field of high energy physics is to investigate thephysics of circuit complexity from viewpoint of quantum field theory. Currently, there are mainly two methods forcalculating the complexity: a). Nielsen et al . are pioneer-ing to propose a geometrical method for computing thecomplexity in phase space of quantum gates [20–22]; b).Under the framework of information geometry, one canuse the ”Fubini-Study distance” to proceed [23]. Accord-ing to these two methods, one can use the wave functionsin position space to explicitly obtain the complexity [24–26]. One can also use the covariance matrix to calculatethe complexity [27–31]. Although the implication of com-plexity in high energy physics is still elementary, there aresome attempts are investigated, such as the definition ofcomplexity in QFT [32]. Even the Hawking radiation canbe treated by this kind of quantum circuit complexity[33, 34]. More striking, the spacetime in some sense canbe interpreted as the quantum circuit complexity [35].In light of Nielson’s geometrical method [20], it can beimplemented into the early Universe, namely for the in-flationary epoch. The essence of curvature perturbationis quantum perturbation, however, we are not capable ofdistinguishing that it is quantum or classical from obser-vations. The standard procedure to deal with the cir-cuit complexity is utilizing squeezed state satisfied withthe minimal uncertain relation, thus it naturally utilizesthis squeezed state as a ground state of curvature per-turbation. Ref. [36] investigates the relation betweenthe bounce Universe and complexity. Similar work hasbeen studied in [37], in which more inflationary modelsare studied and they found that the complexity in thematter dominant period (MD) is growing fastest. Oncetaking effects of background expansion into account [39],it is possible to study the evolution of circuit complex-ity for curvature perturbation where they found that theinflationary period has the most simple linear relation ofcomplexity evolution and its value will be enhanced inlater Universe. The complexity is an essential conceptin a chaotic system, and it is a possible diagnostic forquantum chaos, for which it has been dubbed as an inte-gral part of web of diagnostics for quantum chaos [40–45].Consequently, the complexity of a chaotic system couldprovide essential information about the scrambling time,Lyapunov exponent etc [42].The concept of complexity plays an essential role inmodern physics, i . g . for the computer science, quantuminformation e . t . c . With the development of its impor-tance, this concept has become active and dubbed as oneof the core intersect of theoretical physics. For the cos-mological field, it also motives us to explore the quantumcomplexity of quantum cosmology, especially for the veryearly universe [39]. In [36, 39], the quantum circuit com-plexity has been calculated for the scalar curvature per-turbations in simple inflation model with constant soundspeed, i.e. c S = 1. Alternatively, it is valuable to eval-uate the deviation from the trivial should speed, we willconsider the effects from non-trivial sound speed into thequantum circuit complexity, especially for the differentpartial differential equations of parameters of squeezedstate also comparing with Ref. [36, 39]. Thus, in this pa-per, our purpose is to study the effects of varying soundspeed on the evolution of quantum circuit complexity inbackground of inflationary de-Sitter spacetime. In partic-ular, we are interested in a type of resonant sound speedwhich is proposed as a key ingredient for generating thePBH in inflation epoch [46]. As indicated by [46], thisnew mechanism uses the instability derived from the res-onant sound speed to result in the amplification of thepower spectrum.This paper is organized as follows. In section 2, wewill review the non-trivial sound speed c S from the per-spective of resonant production of PBH. In section 3,the squeezed state of cosmological perturvations will begiven, in which there exists two parameters for thissqueezed states. In section 4, the complexity of these cos-mological perturbation will be obtained under the frame-work of geometrical method via Nielson’s work. In sec-tion 6, we give our main conclusions and discussions. THE EFFECTS OF NON-TRIVIAL SOUNDSPEED
Before recalling the non-trivial sound speed, wewill consider the Friedmann-Lemaitre-Robertson-Walkerbackground metric considered as our working metric, ds = a ( η ) ( − dη + d~x ) , (1)where a ( η ) is scale factor in conformal time and ~x denotesthe spatial vector. Under this background, we could de-fine the perturbation of some scalar field in inflationaryperiod φ ( x µ ) = φ ( η ) + δφ ( x µ ) with its correspondingmetric defined by ds = a ( η ) (cid:18) − (1 + ψ ( η, x )) dη + (1 − ψ ( η, x )) dx (cid:19) , (2) where ψ ( η, x ) is the perturvbation of metric. Beingarmed these two metrics and the perturbations of somescalar field, the perturbed action can be denoted by thecurvature perturbation S = 12 Z dtd xa ˙ φ H (cid:20) ˙ R − a ( ∂ i R ) (cid:21) , (3)where H = ˙ aa , R = ψ + H ˙ φ δφ . Action (3) can be trans-ferred into canonical normalized scalar field in light ofthe Mukhanov variable v = z R where z = a √ ǫ with ǫ = − ˙ HH = 1 − H ′ H , S = 12 Z dηd x (cid:20) v ′ − ( ∂ i v ) + z ′ z v − z ′ z v ′ v (cid:21) , (4)where prime implies that the derivative with respect tothe conformal time η even for H . In action (4), it clearlyindicates the perturbation of curvature is of trivial soundspeed namely, c S = 1. To evaluate the influence of non-trivial sound speed for the complexity, we will introducea simple mechanism of production of PBH from a reso-nant sound speed. In this new mechanism, the key in-gredient for producing the enhanced value of curvatureperturbation is the non-trivial sound speed c S . For ageneral non-trivial sound speed [47, 48], it is defined bya canonical variable v = zζ with z = √ ǫa/c S where ǫ has the same definition as the previous discussions. Forthe canonical variable, its corresponding Fourier mode v k satisfied with the following equation of motion, v ′′ k + ( c S k − z ′′ z ) v k = 0 , (5)This equation of motion is our starting point for con-structing the model. If the sound speed c S is one, it willnicely recover eom of curvature perturbation for slow-roll single-field inflation. As for the deviation from oneof c S , it experiences the non-canonical kinetic term, inwhich the inflation embedded into string theory [49, 50].Meanwhile, this mechanism can be realized in DBI infla-tion [51]. From the perspective of effective field theory,when integrating out the heavy modes, it could also leadto the non-trivial sound speed c S [52].Since the amplification of curvature perturbation ishighly relevant with this non-trivial c S , thus it can bedetermined by c S = 1 − ξ [1 − cos( k ∗ τ )] , (6)where ξ is the amplitude of oscillation and k ∗ is the fre-quency of oscillation. Combining eq. (6) and eq. (5) andkeeping the first order of ξ for z ′′ z since the value of ξ istiny compared with one. Subsequently, changing variableas x = k ∗ τ , then eq. (5) will become Mathieu equation, d v k dx + ( A k − q cos 2 x ) v k = 0 (7)where A k = k /k ∗ +2 q − ξ and q = 2 ξ − ( k /k ∗ ) ξ . Thereis a key feature of the Mathieu equation, since it existsan instability region of eq. (7) depicting its exponentialgrowth, in which it can be read by v k ∝ exp( ξk ∗ τ / . (8)This exponential growth could lead to the amplificationof curvature perturbation. Consequently, it will realizethe generation of PBH during inflation. Observing thateq. (7) belongs to the narrow resonance due to the smallvalue of ξ . And the resonance only occurs as k = nk ∗ with n is an integer. Lastly, we will evaluate the deviationfrom the standard sound speed, particularly the informa-tion from ξ . In figure (1), it clearly indicates that the de- c S ξ = ξ = ξ = ξ = FIG. 1: It shows the resonant sound speed varying with scalefactor a ( τ ) with various values of ξ . For a resonable input of ξ , we set its value are 0 , . , . , .
08, respectively. viation from the standard sound speed ( c S = 1) varyingwith scale factor a ( τ ), in which it tells that there exists anoscillation at the very beginning of inflation ( a ≈ . ξ , where we have used η = − Ha . From figure(1), we could also know that the parameter ξ cannot belarge due to the constraints of deviation from standardsound speed. Moreover, we will utilize this parameter toinvestigate its effects to the cosmological complexity. THE SQUEEZED QUANTUM STATES FORCOSMOLOGICAL PERTURBATIONS
In this section, we will evaluate the squeezed state forcosmological perturbations via non-trivial sound speed.For convenience, we will use the Mukhanov variable fordepicting the action. To be more precise, EOM (5) isour starting point for calculating the complexity. To ob-tain EOM (5), its corresponding action can be explicitlywritten as 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) (9)Here, the expression of c s is given by (6). The canonicalmomentum is obtained from (9) π ( η, ~x ) = δLδv ′ ( η, ~x ) = v ′ − z ′ z v (10)And then the Hamiltonian H = R d x ( πv ′ − L ) is H = 12 Z d x (cid:2) π + c S ( ∂ i v ) + z ′ z ( vπ + πv ) (cid:1)(cid:3) (11)Promoting the Mukhanov variable v to the quantum fieldand decomposing it into Fourier modesˆ 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 (12)Meanwhile, the corresponding ˆ π ( η, ~x ) isˆ 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 (13)where ˆ c †− ~k and ˆ c ~k represent the creation and annihila-tion operators respectively. By choosing an appropriatenormalization condition for mode functions u k ( η ) , v k ( η ),one can give the following Hamiltonianˆ 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) (14)In case of c S = 1, the (14) reduces to a Hamiltonianwhich is similar to the form of the inverted harmonicoscillator [38]. The unitary evolution operator acting on astate can be parameterized in the factorized form [53, 56]ˆ U ~k ( η, η ) = ˆ S ~k ( r k , φ k ) ˆ R ~k ( θ k ) (15)In (15), the ˆ R ~k is the two-mode rotation operator writtenin term of the rotation angle θ k ( η )ˆ 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) (16)Meanwhile, ˆ S is the two-mode squeeze operator writtenin term of the squeezing parameter r k ( η ) and the squeez-ing angle φ k ( η ) respectively.ˆ 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) (17)Since the rotation operators only give an irrelevantphase factor when acting on the initial vacuum state,we will not include it in subsequent analysis. By act-ing the squeeze operator on the two-mode vacuum state |
0; 0 i ~k, − ~k , a two-mode squeezed state is obtained | Ψ i sq = 1cosh r k ∞ X n =0 ( − n e inφ k tanh n r k | n ; n i ~k, − ~k (18)in which | n ; n i ~k, − ~k represents the two-mode excited state | 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 (19)Together (14) , (18) with Schr ¨ odinger equation i ddη | Ψ i sq = ˆ H | Ψ i sq (20)We give the differeitial equations which control the timeevolution of the squeezing paremeters r k ( η ) , φ k ( η ) − dr k dη = k c S −
1) sin(2 φ k ) + z ′ z cos(2 φ k ) (21) 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 (22)These two equations are difficult to solve analytically,thus we have to consider the numerical solutions. For thesake of simplicity for calculation, the variable log a areutilized to take place of the conformal time η . Under thisvariable transformation, together with the expression (6)and z = √ ǫac S , equations (22) are expanded as10 y H ln[10] · drdy = (cid:0) k ⋆ ξ sin( k ⋆ y H )1 − ξ + 2 ξ cos( k ⋆ y H ) − y H (cid:1) · cos(2 φ k )+ kξ (cid:0) − cos( 2 k ⋆ y H ) (cid:1) · sin(2 φ k ) (23)10 y H ln[10] · dφ k dy = k (cid:0) − ξ (cid:0) − cos( 2 k ⋆ H · y ) (cid:1)(cid:1) + (cid:0) y H − k ⋆ ξ sin( k ⋆ y H )1 − ξ + 2 ξ cos( k ⋆ y H ) (cid:1) · sin 2 φ k coth 2 r k + kξ (cid:0) − cos( 2 k ⋆ H · y ) (cid:1) · cos 2 φ k coth 2 r k (24)in which we have denoted y = log a . Being armed withthis variable transformation, one can obtain the numer-ical solution of φ k and r k depicted by figure (2). Theupper pannel of figure (2), the trend of numerical solu-tions of r k with various ξ are almost the same, partic-ularly for the case of ξ = 0 which will be recoverd thesolution of Ref. [36]. One could see that deviation ofdistinct r k are pratically negliable, only in the amplifiedregiem of r k reveils that the frequency of oscillation of will be enahnced as the value of ξ increases. And theslope of these cases will be approaching a same constant.Similar situation arises for the numerical solution of φ k ,the below pannel shows the evolution of different casesare convergent except the turning point of case ξ = 0 . ξ = 0 .
08) will be faster approaching aconstant. To conclusion, the generic evolution of thesesolutions are similar whose difference only occurs at thevery early time of inflation. Being armed these two nu- - - - - L(cid:0)(cid:1) ( a ) r k - - - - ξ = ξ = ξ = ξ = - - - Log ( a ) ϕ k ξ = ξ = ξ = ξ = FIG. 2: The numerical solutions of φ k and r k in terms oflog ( a ) with ξ = 0, ξ = 0 . ξ = 0 .
04 and ξ = 0 .
08. Ourplots adopts H = 1. merical solution of φ k and r k , one can investigate thecomplexity under the inflence of resonant sound speed. THE COMPLEXITY OF COSMOLOGICALSQUEEZED STATES
In this paper, we will evaluate the circuit complexity byusing Nielsen’s method [20–22]. 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 (25)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) (26)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 (27)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 ) (28)Then a cost f unctional is defined as follows C ( U ) = Z F ( U , ˙ U ) dτ (29)The complexity is obtained by minimizing the functional(29) 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 ) (30)In this work, the target state is the two-mode squeezedvacuum state (18). After projecting | Ψ i into the positionspace, the following wavefunction is implied [54, 55]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 (31)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) (32) B ( r k , φ k ) = 2 k (cid:18) e − iφ k tanh r k e − iφ k tanh r k − (cid:19) (33)By using a suitable rotation in vector space ( q ~k , q − ~k ), theexponent in (31) 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 (34)˜ 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 / (35)˜ m = (cid:18) ω ~k ω − ~k (cid:19) According to the definition (25), one can associate thetarget state (34) with the reference state (35) through aunitary transformationΨ τ ( q ~k , q − ~k ) = ˜ U ( τ )Ψ ( q ~k , q − ~k ) ˜ U † ( τ ) (36)Ψ τ =0 ( q ~k , q − ~k ) = Ψ ( q ~k , q − ~k ) (37)Ψ τ =1 ( q ~k , q − ~k ) = Ψ sq ( q ~k , q − ~k ) (38)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. As considered bythe work [24], ˜ U ( τ ) will take the form˜ U ( τ ) = exp[ X k =1 Y k ( τ ) M diagk ] (39)where the { M diagk } represent the 2 diagonal generatorsof GL (2 , C ) M diag = (cid:18) (cid:19) , M diag = (cid:18) (cid:19) Note that the off-diagonal components are set to zerosince they will increase the distance between two states.The complex variables { Y I ( τ ) } are constructed as [57] Y I ( τ ) = Y I ( τ = 1) · τ + Y I ( τ = 0) (40)From the boundary conditions (37) and (38), one obtainsIm( Y , ) (cid:12)(cid:12) τ =0 = Re( Y I ) (cid:12)(cid:12) τ =0 = 0 (41)Im( Y , ) (cid:12)(cid:12) τ =1 = 12 ln | Ω ~k, ~ − k | ω ~k, ~ − k (42)Re( Y , ) (cid:12)(cid:12) τ =1 = 12 arctan Im(Ω ~k, ~ − k )Re(Ω ~k, ~ − k ) (43)And then the complexity could be rewritten as the fol-lowing line length C ( ˜ U ) = Z dτ q G IJ ˙ Y I ( τ )( ˙ Y J ( τ )) ⋆ (44)in which G ij is an induced metric for the group mani-fold. As indicated by the work [24], there exists an ar-bitrariness in term of the choice of G IJ . For the sake ofsimplicity, we suppose that the operator space is a flatgeometry, namely G IJ = δ IJ . Substitute (40) into (44),the expression for the complexity is calculated as C ( k ) = 12 (cid:18)(cid:0) ln | Ω ~k | ω ~k (cid:1) + (cid:0) arctan Im(Ω ~k )Re(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:19) / (45) - - - - Log ( a ) C - - (cid:6)(cid:7)(cid:8) - - (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)1(cid:26)(cid:27) FIG. 3: Complexity is based on the numeical solution of r k and φ k . These colorful lines are corresponding to the previousfigure (2). Our plots adopt H = 1 From figure (3), the complexity is explicitly obtainedvia the numerical solution of r k and φ k , in which it indi-cates that the evolution of complexity is nearly the samewithin the inflationary period, particularly, the generictrend of complexity is enhancing. The only differenceoccurs that there exists a faster oscillation with a largervalue of ξ . Subsequently, all of these complexities withvarious ξ will be of the same slope, and the oscillationdisappears. In our plots, the case of ξ = 0 correspondingto Ref. [36], the other cases from non-vanishing ξ showthat the evolution of complexity is generic and robust,no matter what there is resonant sound speed or not. SUMMARY AND DISCUSSION
In this paper, we evaluate the influence of non-trivialsound speed (resonant sound speed) on the evolution ofcomplexity in the inflationary period for the cosmologicalperturbations in light of Nielson’s geometrical method.Under the framework of standard cosmological pertur-bation theory and combining with the non-trivial soundspeed, we utilize the squeezed state for investigations ofcurvature perturbation in the inflationary period. Toachieve this goal, the most essential squeezed parametersare r k and φ k in the phase space, respectively. Follow-ing the procedure, we obtain the evolution equation of r k and φ k in terms of conformal time depicted by Eq.(22), in which there is a non-trivial sound speed termwhose crucial information contained in the parameter ξ comparing with Ref. [39]. Through the numerical so-lution of r k and φ k , we could see the evolution of thissqueezed state in terms of r k and φ k . Through figure(2), it clearly indicates that the trend of r k and φ k , re-spectively. In figure (2), the case of ξ = 0 corresponds tothe case of Ref. [39], the other cases whose value of ξ isnon-vanishing only contains more oscillations for r k . As for φ k , the only difference occurs at the very early timeof inflation where the turning point of ξ = 0 .
08 for ap-proaching a constant. Physically speaking, the squeezedstates are only distinct with more frequency and variousangle parameter. In conclusion, our study shows that theevolution of complexity and r k is similar to trivial soundspeed, which means that the influence of resonant soundspeed for the evolution of complexity is not significantfor the late time. However, it is worthwhile for noticingthat this kind of resonant sound speed will lead to therapid oscillation of complexity at the very early time, asRef. [46] mentioned that the resonant sound speed is thekey ingredient for producing the primordial black hole.Thus, it can be conjectured that there some connectionbetween this PHB production mechanism and this rapidoscillation in some sense.In light of these parameters, one can investigate thecomplexity under the influence of resonant sound speed.However, we found that the complexity with various ξ almost has the same evolution trend well agreed withthe analysis of Ref. [39], especially for the inflationaryperiod. Being armed with the numerical solutions of r k and φ k , one can evaluate the complexity. Our resultsindicate that complexity will not be affected by the res-onant sound speed, in some sense that it reveals thatthe non-trivial sound speed will not bring much effectsto the complexity for the inflationary period. Therefore,the result of Ref. [39] shows that the evolution of com-plexity in inflationary period is generic and robust. How-ever, Ref. [37] analyzed that the complexity is stronglymodel dependent showing that matter domination hasthe largest complexity in inflation, in which the com-plexity of these models are varying with e-folding num-ber whose range locates from 0 to 30. If they translatedinto the non-trivial sound speed, there must exist extraterms in action (9) left to the future work. Furtherly, wecould investigate the complexity from the modified grav-ity, especially for f ( R ) gravity [58, 59] or the multifieldinflation [60, 61], transformed into Einstein frame, thereare non-trivial sound speed and possible extra terms ofaffecting the evolution of complexity. Acknowledgements
LH is funded by Hunan Natural Provincial ScienceFoundation NO. 2020JJ5452 and Hunan Provincial De-partment of Education, NO. 19B464. AC is supported byNSFC grant no.11875082 and the University of Barcelona(UB) / China Scholarship Council (CSC) joint scholar-ship. AC is also supported by National Natural ScienceFoundation of China under Grant Nos.11947086. We aregrateful for the numerical calculation via Prof. Ding-Fang Zeng from Institute of Theoretical Physics In Bei-jing University of Technology
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