Holographic complexity growth in an FLRW universe

Yu-Sen An,1,2,* Rong-Gen Cai,1,2,† Li Li ,1,2,‡ and Yuxuan Peng1,3,§1CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China2School of Physical Sciences, University of Chinese Academy of Sciences,

No.19A Yuquan Road, Beijing 100049, People’s Republic of China3East China University of Technology, Nanchang, Jiangxi 330013, People’s Republic of China

(Received 7 October 2019; accepted 21 January 2020; published 5 February 2020)

We investigate the holographic complexity growth rate of a conformal field theory in a Friedman-Lemaître-Roberstson-Walker (FLRW) universe. We consider two ways to realize an FLRW spacetime froman anti–de Sitter Schwarzschild geometry. The first one is obtained by introducing a new foliation of theSchwarzschild geometry such that the conformal boundary takes the FLRW form. The other one is toconsider a brane universe moving in the Schwarzschild background. For each case, we compute thecomplexity growth rate in a closed universe and a flat universe by using both the complexity-volume andcomplexity-action dualities. We find that there are two kinds of contributions to the growth rate: one is fromthe interaction among the degrees of freedom, while the other one from the change of the spatial volume ofthe universe. The behaviors of the growth rate depend on the details to realize the FLRW universe as wellas the holographic conjecture for the complexity. For the realization of the FLRW universe on theasymptotic boundary, the leading divergent term for the complexity growth rate obeys a volume law whichis natural from the field theory viewpoint. For the brane universe scenario, the complexity-volume andcomplexity-action conjectures give different results for the closed universe case. A possible explanation ofthe inconsistency when the brane crosses the black hole horizon is given based on the Lloyd bound.

DOI: 10.1103/PhysRevD.101.046006

I. INTRODUCTION

Anti–de Sitter/conformal field theory (AdS/CFT) corre-spondence has greatly deepened our understanding of thequantum gravity [1–4]. In particular, motivated by theholographic entanglement entropy [5], an intrinsic potentialconnection between quantum information theory and grav-ity physics has been uncovered. However, in the context ofthe thermofield double state (TFD state) which is dual tothe eternal black hole [6], it has been shown that entangle-ment entropy cannot capture all the information during theevolution of an AdS wormhole [7]. As a more refinedinformation quantity, complexity has been proposed todescribe the situation where entanglement entropy fails,such as the wormhole growth behavior far beyond thethermal equilibrium. Both the field theory definition and

the holographic definition of complexity have receivedgreat attention. Although there are many investigations onthe complexity from field theory side, such as [8–15], aunique and consistent definition is still lacking. From theholographic point of view, there are two proposals forcomplexity, known as complexity-volume (CV) duality[16] and complexity-action (CA) duality [17,18]. Thereare many investigations regarding holographic complexitygrowth properties, such as the seminal paper [19,20] for theEinstein AdS gravity. After that there are many general-izations to various gravity settings such as [21–24] forcharged dilaton black hole, [25–27] for warped-AdS blackhole, [28] for Taub-Nut-AdS black hole, [29] for geometryduring phase transition and [30–36] for black holes inhigher derivative gravity theory. Moreover, as the holo-graphic complexity is a divergent quantity at one spatialtime slice, the divergence structure was also investigated inRefs. [37,38].While most works on the complexity growth rate

considered the static case, the generalization to the time-dependent case is also quite interesting and it is worthwhileto study how the complexity evolves in a dynamicalprocess; for related studies on the Vaidya spacetime, seeRefs. [39–41]. The investigation of the complexity can alsobe generalized to states on other dynamical backgrounds

*anyusen@itp.ac.cn†cairg@itp.ac.cn‡liliphy@itp.ac.cn§yxpeng@itp.ac.cn

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI. Funded by SCOAP3.

PHYSICAL REVIEW D 101, 046006 (2020)

2470-0010=2020=101(4)=046006(14) 046006-1 Published by the American Physical Society

which correspond to different slices from gravity side,such as de Sitter boundary in Ref. [42]. Of particularinterest is the boundary metric that has the Friedman-Lemaître-Robertson-Walker (FLRW) form, which mightlead to the understanding of the nonperturbative aspects ofcosmology.In Ref. [43], starting with an AdS-Schwarzschild black

hole, one can choose a different foliation away fromthe black hole to make the metric time dependent and torealize the boundary with the form of an FLRW spacetime.The Friedman equation can also be obtained by consideringthe mixed boundary conditions on the new slice.Holographically, the dual field theory on the FLRWboundary may represent an expanding plasma and theauthors of Ref. [43] calculated its stress energy tensor andentropy production. The paper [43] adopted the Fefferman-Graham (FG) coordinates. Instead of going to the FGcoordinates, the author of Ref. [44] found a simple foliationof the AdS Schwarzschild black hole and got the sameFLRW metric on the boundary. It will be interesting toinvestigate the complexity behavior of the state on thistime-dependent boundary, and we hope the results couldhave some new phenomenon due to the nonequilibriumphysics.There are also other ways to realize the FLRW cosmol-

ogy from the bulk AdS-Schwarzschild black hole, such asintroducing a codimension one brane. The original moti-vation to consider this realization is from the holographicprinciple. The relation between cosmology and holographywas first raised by Fischler and Susskind [45]. After that,Verlinde [46] investigated the entropy bound and foundthat the entropy formula called Cardy-Verlinde formula in aCFT can reproduce the Friedman equation, which impliespossible connection between CFT and FLRW universe.The various arguments proposed in Ref. [46] have beennaturally realized in the brane-world scenario in Ref. [47],where the authors embedded the Randall-Sundrum type IIbrane in the Schwarzschild-AdS black hole. Randall-Sundrum brane world was first proposed as a solution tothe hierarchy problem [48,49]. Maldacena first pointed outthat the field theory on the brane should be seen as a CFTcoupled to gravity. This idea has been summarized inRefs. [50,51]. The spacetime ends on the brane, and thebrane can be seen as a time-dependent boundary withconformal radiation on it. It is also interesting to investigatethe complexity evolution on the brane universe.This paper is organized as follows. In Sec. II, we

compute the growth rate of the holographic complexityfor the FLRW type boundary theory and show the effect ofthe time dependence on the complexity growth rate. InSec. III, we investigate the complexity growth rate on thebrane using both the CV and CA duality conjectures. Weconsider two cases: a spherical black hole which corre-sponds to a closed universe and a planar black brane whichdescribes a flat universe. We show the time evolution of the

complexity growth rate. In Sec. IV, we summarize ourresults and discuss possible future directions.

II. COMPLEXITY GROWTH ONTHE FLRW TYPE BOUNDARY

This section explores the holographic complexity ofsome particular FLRW universe which lives on the asymp-totic AdS boundary. We briefly introduce the backgroundsolution following the setup of Ref. [44]. Then we study thecomplexity growth with both the CA and CV conjectures indetails.

A. The metric

The (dþ 1)-dimensional static asymptotically AdSblack hole is described by the metric

ds2 ¼ −fðrÞdt2 þ fðrÞ−1dr2 þ ΣðrÞ2dΩ2k;d−1; ð1Þ

where fðrÞ ∼ r2=L2 and ΣðrÞ ∼ r=L at large r with L theAdS radius. dΩ2

k;d−1 denotes the line element of thecodimension two maximally symmetric subspace whichcan be spherical ðk ¼ þ1Þ, planar (k ¼ 0), or hyperbolicðk ¼ −1Þ, and we will use Ωk;d−1 to represent the spatialvolume of this subspace. Going to the Eddington-Finkelstein coordinates fv; r;…g via dv ¼ dtþ dr=fðrÞ,we write the metric as

ds2 ¼ 2dvdr − fðrÞdv2 þ ΣðrÞ2dΩ2k;d−1: ð2Þ

We introduce the new radial coordinate R ¼ raðVÞ and the

new time coordinate V, dv ¼ dV=aðVÞ. Here aðVÞ is somepositive function of V. After plugging dv and dr ¼aðVÞdRþ R _aðVÞdV into the metric Eq. (2), and takingthe large R limit, one obtains the following time-dependentmetric:

ds2 ∼ 2dVdRþ R2

L2½−dV2 þ aðVÞ2dΩ2

k;d−1�: ð3Þ

It is obvious that the new conformal boundary at R → ∞has precisely the desired FLRW form with the timecoordinate V. Note that such a cosmological boundary isnot the same as the commonly used AdS boundary atr → ∞ where one has a static boundary metric. Theentropy density is given by the area of the apparent horizon

s ¼ ΣðRhaÞd−14G

; ð4Þ

where G is the Newton constant and Rh is determined bythe equation

�∂VΣþ

�fðRaÞ2a2

− R_aa

�∂RΣ

�����R¼Rh

¼ 0: ð5Þ

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We can also associate a local temperature to the blackhole as

TðVÞ ¼ TH

aðVÞ ; ð6Þ

where TH is the Hawking temperature of the black holeof Eq. (1). In the present paper, we will focus on theSchwarzschild-AdS black hole, for which the blackeningfactor is

fðrÞ ¼ kþ r2

L2−

16πGMðd − 1ÞΩk;d−1rd−2

ð7Þ

and ΣðrÞ ¼ r2=L2. Here M is the mass of the black hole.The energy density can be calculated by using the holo-graphic renormalization procedure, and when d ¼ 4 theresult is

E ¼ 3ð _a2 þ kÞ2 þ 12M̃64πGa4

; ð8Þ

with

M̃ ¼ 16πGML2

3Ωk;3r2: ð9Þ

In the following, we will compute the complexitygrowth rate associated with this FLRW foliation of theSchwarzschild-AdS black hole using both the CV and CAconjectures.

B. The complexity growth with CV conjecture

The CVand CA methods for computing the holographiccomplexity are shown schematically in the Penrose dia-grams in Fig. 1. The CV proposal is described in the leftpanel: the maximal volume of the codimension one surfaceconnecting the end points on both time-dependent bounda-ries is proportional to the complexity of the boundary state,

CV ¼ max½Volume�Gl

; ð10Þ

where l is a dimensional constant which is usually chosento be equal to the AdS radius L. In the original CV proposal,

the CFT lives on a static boundary, while we will extendthe original definition to a time-dependent boundary. Thecalculation method mainly follows the procedure providedin Refs. [19,20]. Note that the dual state depends on twotimes tL and tR with subscripts L and R representing the leftand right boundary times, respectively. We are interested inthe symmetric configuration with tL ¼ tR.As the maximal surface is symmetric with respect to

the innermost point of the surface located at the radialcoordinate rmin, we only need to focus on the right side ofthe maximal surface from rmin to an UV cutoff, say at rmax.Furthermore, the maximal surface has the same symmetryas the horizon, and the volume of the maximal surface canbe expressed as

Volume ¼ 2Ωk;d−1W ð11Þwhere

W ¼Z

rmax

rmin

dλrd−1

Ld−1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−fðrÞv02 þ 2v0r0

q; ð12Þ

with the prime denoting a derivative with respect to λwhichis the parameter describing the surface. Note that the UVcutoff rmax will be taken to be infinity finally. By solvingthe parameter equations of vðλÞ and rðλÞ, one can deter-mine the maximal surface and calculate the complexitygrowth rate. For more details about the computation, onecan consult Ref. [20].The end point at the right cutoff boundary is covered by

the Schwarzschild coordinates ftR; rg, and coordinatesfVR; Rg simultaneously. The complexity growth rate ofour FLRW universe is proportional to the quantity∂W=∂VR. According to the chain rule of differentiation,it is given by

∂W∂VR

����ðVR;RmaxÞ

¼ ∂Wðrmax; tRÞ∂tR

∂tR∂VR

����ðVR;RmaxÞ

þ ∂Wðrmax; tRÞ∂rmax

∂rmax

∂VR

����ðVR;RmaxÞ

: ð13Þ

The partial derivatives are given by

∂Wðrmax; tRÞ∂tR ¼ −E;

∂tR∂VR

¼ 1

aðVRÞ−

Rmax

fðrmaxÞ_aðVRÞ;

∂Wðrmax; tRÞ∂rmax

¼ 1

fðrmaxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðrmaxÞr2ðd−1Þmax þ E2

q;

∂rmax

∂VR¼ _aðVRÞRmax; ð14Þ

where the dots denote the derivative with respect to VR andrmax ¼ RmaxaðVRÞ. We have introduced a quantity E which

r = 0

r=0

timedependentboundary

ABERB

r = 0

r=0

r = rm

time dependentboundary

AB

FIG. 1. Penrose diagram for the two-sided eternal AdS blackhole. The CV conjecture is related to the size of an Einstein-Rosen bridge (ERB) to the computational complexity of the dualquantum state (left). The CA conjecture relates the action of theWheeler-Dewitt patch to the complexity of the CFT state (right).

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is the conserved charge on the ERB asW is independent ofthe coordinate v. While E is constant for a given ERB, itchanges when the ERB evolves. It approaches 8πGLM

ðd−1ÞΩ0;d−1at

the late time limit for k ¼ 0 case and has deviations from itdue to curvature corrections for k ¼ 1 and k ¼ −1 cases.The quantity E is fixed by

fðrminÞr2ðd−1Þmin þ E2 ¼ 0: ð15Þ

Since the maximal surface is symmetric, the innermostpoint has the time coordinate t ¼ 0. Moreover, E < 0 in theupper half (black hole region) of the Penrose diagram.Taking the limit Rmax → ∞ with VR fixed, for the

Schwarzschild-AdS black hole, Eq. (7), we arrive at

GL2Ωk;d−1

∂CV

∂VR

����ðVR;RmaxÞ

≈ −Ea−1

2kL3ad−4 _aRd−3

max

þ _aLRd−1maxad−2 þ � � �

¼ −Eaþ LRd−1

max

d − 1

ddVR

ðad−1Þ

−1

2kL3ad−4 _aRd−3

max þ � � � ; ð16Þ

where we have used fðrmaxÞ ∝ R2maxa2=L2 at large Rmax.

Apart from the finite term −E=a, the result above contains aleading divergent term (the second term) proportional to thegrowth rate of the volume of the universe on the boundary.The third term of Eq. (16) is due to the spatial curvature ofthe horizon and is vanishing for the planar case, i.e., k ¼ 0.Other subleading divergent terms are denoted by “� � �.” Inparticular, there are no such subleading divergent termswhen d ¼ 4 as Rmax → ∞.

C. The complexity growth with CA conjecture

The CA conjecture is schematically shown in the rightpanel of Fig. (1). The red lines are actually null sheetsstarting from the two end points, A and B, on the boundary,and the complexity is proportional to the action in theregion surrounded by these sheets, which is called the“Wheeler-DeWitt (WDW)” patch,

CA ¼ Action of WDW patchπℏ

; ð17Þ

with ℏ the reduced Planck constant. The system we willconsider is described by the Einstein-Hilbert action with anegative cosmological term, and therefore the blackeningfactor of Eq. (1) is given by Eq. (7). The method for thecalculation of the action in the presence of null boundaryhas been developed by Refs. [19,53,54], where the actionreads

I ¼ 1

16πG

ZM

ddþ1xffiffiffiffiffiffi−g

p ðR − 2ΛÞ

þ 1

8πG

ZBddx

ffiffiffiffiffiffijhj

pK þ 1

8πG

ZΣdd−1x

ffiffiffiσ

pη

−1

8πG

ZB0dξdd−1x

ffiffiffiγ

pκ þ 1

8πG

ZΣ0dd−1x

ffiffiffiσ

paþ Icount:

ð18ÞTerms in the expression above are, respectively, bulk term,Gibbons-Hawking-York boundary term for spacelike ortimelike boundary, Hayward joint term [55], null boundaryterm, null joint term, and counter term needed to cancelthe dependence of arbitrary normalization parameter. Thejoints η and a are constructed by the rules summarized inRef. [19]. h,σ,γ is the determinant of the induced metric ofthe corresponding hypersurface, and ξ is the parameter ofnull hypersurface. For simplicity, we will choose affineparametrization and set κ ¼ 0 in the following, so thecontribution of null boundary vanishes.Following the analysis in the previous subsection, see in

particular Eq. (14), the complexity growth rate is given by

∂CA

∂VR

����R¼Rmax

¼�

1

aðVRÞ−

Rmax

fðrmaxÞ_aðVRÞ

� ∂CA

∂tRþ _aðVRÞRmax

∂CA

∂rmax: ð19Þ

The term ∂CA∂tR has been already obtained in the literature[17,18,20] and the equivalence between method inRefs. [17,18] and method in Refs. [19,20] was givenin [52]. Note that at the boundary Rmax → ∞,fðrÞ → ðRmaxa=LÞ2. So, the second term in parenthesesvanishes. If we consider the late time limit, the first termwill reduce to 2M

aðVÞ, where M is the energy of the bulk static

spacetime.All we need to do is to calculate the second term, i.e., the

derivative of the complexity with respect to the r coor-dinate. In order to do it, we first fix the boundary to belocated at a finite position R ¼ Rmax and then take the limitRmax → ∞. According to Ref. [56], the UV cutoff r ¼ rmaxwill also induce a corresponding cutoff surface at r ¼ r0near the singularity at r ¼ 0, and as rmax → ∞, r0 goes to 0.Below we will follow this prescription. We present theformal derivation for general d and take d ¼ 4 in the finalexpression.The bulk term of the action consists of three parts and we

denote the cutoff by rmax,

IIbulk ¼ −dΩk;d−1

8πGL2

Zrh

r0

rd−1�tR2þ r�ðrmaxÞ − r�ðrÞ

�dr;

ð20Þ

IIIbulk ¼ −dΩk;d−1

8πGL2

Zrmax

rh

rd−12ðr�ðrmaxÞ − r�ðrÞÞdr; ð21Þ

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IIIIbulk ¼ −dΩk;d−1

8πGL2

Zrh

rm

rd−1�−tR2þ r�ðrmaxÞ − r�ðrÞ

�dr;

ð22Þ

where rh is the horizon radius, r� denotes the tortoisecoordinate defined by r� ¼ R

drfðrÞ, and rm is the radius of the

point where the two past null sheets meet with each other,as shown in the right plot of Fig. 1.The surface term of the cutoff surface inside the

horizon is

If ¼ −rd−1Ωk;d−1

8πG

�∂rfðrÞ þ

2ðd − 1ÞfðrÞr

�

×

�tR2þ r�ðrmaxÞ − r�ðrÞ

�����r¼r0

: ð23Þ

There are also various joint terms. The joint term at thepoint rm reads

Ijnt ¼ −Ωk;d−1rd−1m

8πGlog

jfðrmÞjα2

; ð24Þ

where α is a constant normalization parameter of the nullnormal vector. The joint term at the surface inside thehorizon is

Ijnt;sing ¼ −Ωk;d−1

8πGrd−1 log jfðrÞjjr¼r0 : ð25Þ

Moreover, the two joint terms on the cutoff surfaces are

Ijnt;cut ¼rd−1maxΩk;d−1

4πGlog

fðrmaxÞα2

: ð26Þ

In order to eliminate the dependence of the arbitrary choiceof the reparametrization, we also add a counterterm to thenull boundary

Icount ¼ −2Z

dd−1xdξffiffiffiγ

pΘ log jL̃Θj; ð27Þ

where L̃ is an arbitrary length scale, and Θ ¼ð1= ffiffiffi

γp Þð∂ ffiffiffi

γp

=∂ξÞ is the expansion. Although this termcan modify the full time dependence of the complexitygrowth, it has no effect on the late time result [24].After that, we may take the derivative with respect to

rmax and then take rmax to infinity. We see that both thesurface term and the joint term inside the horizon arevanishing. So only the bulk term and another three jointterms contribute. The first bulk term is

dIIbulkdrmax

¼ dΩk;d−1

8πGL2rd−10

�tR2þ r�ðrmaxÞ − r�ðr0Þ

�dr0drmax

−Ωk;d−1

8πGL2

rdh − rd0fðrmaxÞ

: ð28Þ

Taking the limit rmax → ∞, we find that this term vanishesdue to the relation between r0 and rmax [56]. The secondbulk term at the boundary reads

dIIIbulkdrmax

¼ −dΩk;d−1

4πG

Zrmax

rh

rd−1

fðrmaxÞ¼ −

Ωk;d−1

4πGrd−2max: ð29Þ

The third bulk term is given by

dIIIIbulk

drmax¼ −

dΩk;d−1

8πGL2

Zrh

rm

rd−1

fðrmaxÞdr; ð30Þ

where we have used the relation −tR=2þ r�ðrmaxÞ−r�ðrmÞ ¼ 0. This term also vanishes by taking rmax → ∞.Next, we consider the derivative of the boundary joint

terms.

dIjnt;cutdrmax

¼ ðd − 1Þrd−2maxΩk;d−1

4πGlog

fðrmaxÞα2

þ rd−1maxΩk;d−1

4πG1

fðrmaxÞdfðrmaxÞdrmax

: ð31Þ

The joint term at rm is given by

dIjntdrmax

¼ dIdrm

drmdrmax

¼ −Ωd−1rd−1m

8πGfðrmaxÞdfðrmÞdrm

−ðd − 1ÞΩd−1rd−2m

8πGfðrmÞfðrmaxÞ

logjfðrmÞjα2

; ð32Þ

where we have used the relation drmdrmax

¼ fðrmÞfðrmaxÞ. We find that

this term also vanishes by taking the boundary limitrmax → ∞. Now, the result depends on the choice α, andone needs to consider the counterterm which eliminatessuch arbitrariness. We can take a special parametrizationξ ¼ r=α. It is worth noting that the final expression doesnot depend on the choice of α. Following the result ofRef. [20], the counterterm contribution is given by

Icount ¼Ωk;d−1

2πGrd−1max

�log

ðd − 1ÞαL̃rmax

þ 1

d − 1

�

−Ωk;d−1

4πGrd−1m

�log

ðd − 1ÞαL̃rm

þ 1

d − 1

�: ð33Þ

By taking the derivative with respect to rmax, we find thatthe dependence of α cancels precisely with Eq. (31). Then,we obtain the result

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∂CA

∂rmax

����rmax→∞

¼ ðd − 1ÞΩk;d−1

4πGrd−2max log

ðd − 1Þ2L̃2fðrmaxÞr2max

þ rd−2maxΩk;d−1

4πG

¼�2ðd − 1Þ log ðd − 1ÞL̃

Lþ 1

�rd−2maxΩk;d−1

4πG

þ � � � ; ð34Þ

where we have used the boundary behavior of fðrmaxÞ asrmax → ∞, and the subleading divergent terms are denotedby “� � �.” Therefore, we arrive at the final result,

∂CA

∂VR

����R¼ 1

a∂CA

∂tR þ ddVR

ðRd−1maxad−1ÞC0; ð35Þ

with C0 ¼ 2ðd−1Þ logðd−1ÞL̃L þ1

d−1Ωk;d−14πG .

One finds that the complexity growth using the CAconjecture gives very similar behavior as the CV con-jecture at leading order of rmax. The first term in Eq. (16)is of just the same form as the first term in Eq. (35). Thesecond terms are the time derivatives of the spatialvolume of the universe. For both conjectures, we willrefer to the first term as the “interaction part” as it comesfrom the interaction of the field theory degrees offreedom (d.o.f.) on the boundary and the second termas the “volume part” since it comes from the change ofthe spatial volume. Note that when aðVÞ ¼ 1, the FLRWtype boundary at R ¼ ∞ reduces to the static AdSasymptotic boundary. Equations (16) and (35) reduceto dCV

dtR¼ − 2Ωk;d−1

GNLE and dCA

dtR, respectively, so the evolution

is the same as in Ref. [20].

III. COMPLEXITY GROWTH OFTHE BRANE COSMOLOGY

A. Brane in the AdS-Schwarzschildblack hole background

The above section focuses on the complexity growth of aCFT on an FLRW-like background. One disadvantage isthat the scale factor aðτÞ can be an arbitrary function andis given by hand. There is another way to realize theFLRW cosmology from the AdS-Schwarzschild black hole,inspired by the Randall-Sundrum model and the holo-graphic principle. The idea is to introduce a lower dimen-sional brane with a constant tension in the background ofthe (dþ 1)-dimensional AdS-Schwarzschild black hole.The movement of the brane is described by the followingboundary action [47]:

Lb ¼1

8πG

Z∂M

ddxffiffiffih

pK þ κ

8πG

Z∂M

ddxffiffiffih

p; ð36Þ

where K is the trace of the extrinsic curvature Kab, κ isrelated to the tension of the brane, and h is the determinantof the induced metric hab on the surface of the brane ∂M.By varying this action, we obtain the equation of motion ofthe brane,

Kab ¼κ

d − 1hab: ð37Þ

We begin with the bulk geometry

ds2 ¼ 1

fðaÞ da2 − fðaÞdt2 þ a2dΩ2

k;d−1; ð38Þ

where fðaÞ is just the function of Eq. (7) with the radius rreplaced by a. Next, we introduce a new time parameter τand take t and a to be τ dependent with the followingconstraint:

1

fðaÞ�dadτ

�2

− fðaÞ�dtdτ

�2

¼ −1; ð39Þ

which ensures that τ is the proper time on the brane. So, thebrane is described by the parameter τ and the (d − 1)-dimensional cross section. On the brane, the induced metrictakes the form

ds2d ¼ −dτ2 þ aðτÞ2dΩ2k;d−1; ð40Þ

which describes a standard FRW universe with aðτÞ thescale factor. From the brane equation of motion and theconstraint, we can get the relation between ft; ag and τ,

dtdτ

¼ aLfðaÞ ; ð41Þ

�dadτ

�2

¼ a2

L2− fðaÞ; ð42Þ

where we have set κ ¼ 1=L as Ref. [47].In this case, we can see that the scale factor on the brane

can be deduced from the equation of motion, Eq. (42), oncethe background is fixed. It has been argued by Maldacenathat the brane world should be interpreted as a CFT on thebrane coupled to gravity. As the conformal field is coupledto gravity, the behavior of complexity will be morecomplicated than the previous case on the asymptoticAdS boundary. In contrast to the previous section, thebrane is now located at finite radius. So, the result is freefrom divergence.By considering the time coordinate on the right brane,

the growth rate can be obtained by the chain rule

dCdτ

¼ ∂C∂tR

dtRdτ

þ ∂C∂a

dadτ

: ð43Þ

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There are two different effects in the equation above. Thefirst term is just the same as the first term in Eq. (16) or (35).So, we also call it the interaction part. The second termincludes the contribution of the volume change, just like thesecond term in Eq. (16) or (35), so we also refer to it as thevolume part.In the following, we still stick to the picture of symmetric

objects (maximal surface/WDW action) for both CV andCA proposals, and obtain the final results by consideringthe right side of the object and the end point ðtR; aÞ on theright side brane. We first examine the k ¼ 1 case corre-sponding to a closed universe that first expands and thencontracts on the brane (see Fig. 2). Then, we consider thek ¼ 0 case which corresponds to an ever-expanding openuniverse on the brane, shown in Fig. 3.

B. Complexity evolution for the closed universe:the CV conjecture

According to the chain rule Eq. (43), the growth rate isgiven by

dCV

dτ¼ 2Ωk;d−1

GL

�−E

dtRdτ

þ 1

fðaÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðaÞa2ðd−1Þ þ E2

qdadτ

�:

ð44Þ

On the end point of the maximal surface, the timecoordinate tR is expressed by (see Ref. [34] for moredetails)

tRðτÞ ¼Z

aðτÞ

rmin

drE

fðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðrÞr2ðd−1Þ þ E2

q ;

¼ −Z

aðτÞ

rmin

dr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi−fðrminÞ

prd−1min

fðrÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifðrÞr2ðd−1Þ − fðrminÞr2ðd−1Þmin

q :

ð45Þ

On the other hand, tRðτÞ should also satisfy the differentialequation (41). Therefore, one can find out the relationbetween rmin and τ by combining Eqs. (41) and (45). Thenthe complexity growth rate can be obtained by putting rminand τ into Eqs. (15) and (44).First, we should determine the location and shape of the

brane. As we can see from Eqs. (41) and (42), they dependon the spatial curvature, spacetime dimension, as well asthe location of the horizon. In the present section, we willconsider the closed universe in four dimensions, so we fixk ¼ 1 and d ¼ 4. We will show that the complexity growthrate exhibits distinct behaviors for small and large values ofthe horizon radius.As a concrete example, let us first consider the case with

the horizon radius at rh ¼ L. By integrating Eq. (42), we

obtain aðτÞL ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2 − τ2

L2

qwith τ from −

ffiffiffi2

pL to

ffiffiffi2

pL, which

corresponds to the universe that first expands and thencontracts. The relation between t and τ is

t ¼Z

τ

−ffiffi2

pL

adτLfðaÞ þ ti; ð46Þ

with ti the integration constant. As the evolution of thebrane is symmetric, it will be convenient to choose ti bythe condition that t ¼ 0 when τ ¼ 0. We are interested inthe contraction phase of this closed universe starting fromτ ¼ 0 with the initial state being jTFDðτ ¼ 0Þi. Comparingwith Ref. [20], here we consider the evolution of TFD statein the contracting FLRW background. When τ=L > 1, thebrane begins to cross the horizon of the AdS black hole andour description would become untrustable due to quantumcorrections [46]. So, in the following discussion, we will

FIG. 2. AdS domain wall for a spherical Schwartzchild blackhole which corresponds to a closed universe that expands to acertain size and then contracts.

FIG. 3. AdS domain wall for a planar Schwartzchild blackbrane which is related to a flat universe that will expand forever.

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restrict ourselves to the time range before the brane crossesthe horizon. We shall return to this point later. The time-evolution behavior of the complexity growth rate is shownin Figs. 4 and 5.The nonmonotonic behavior of the complexity growth is

observed in this case. The growth rate first rises as the timeevolves, arrives at its maximum at a certain time, and then itdecreases monotonously. It becomes negative at late time.One finds that the complexity will first increase and thendecrease, even though the universe is in a contractingphase. However, if one increases the radius of the blackhole, there will be no such nonmonotonic behavior. As onecan see from Fig. 6, the complexity growth rate is negativeand decreases all the time from τ ¼ 0. So, the complexity ofsuch contracting universe decreases faster and faster.

C. Complexity evolution for the closed universe:the CA conjecture

In this section, we turn to the complexity growth rate forthe closed universe by using the CA conjecture. Thestructure of the WDW patch is time dependent. There isa critical time at τcðtcÞ before which the WDW patchintersects two singularities and above which an additionaljoint term forms due to the intersection of the past two nullsegments. The critical time is can be obtained by therelation tc ¼

R ac0

dafðaÞ, where ac is the corresponding posi-

tion of the brane at that critical time.Before the critical time, the action contains the bulk term,

the past and future surface terms, and two joint terms at Aand B (see Fig. 2). The bulk part consists of three portions,

I1bulk ¼ −dΩ1;d−1

4πGL2

Zrh

0

rd−1ðtþ r�ðaÞ − r�ðrÞÞdr; ð47Þ

I2bulk ¼ −dΩ1;d−1

2πGL2

Za

rh

rd−1ðr�ðaÞ − r�ðrÞÞdr; ð48Þ

I3bulk ¼ −dΩ1;d−1

4πGL2

Zrh

0

rd−1ð−tþ r�ðaÞ − r�ðrÞÞdr: ð49Þ

Note that here we denote t to be the time on one sideboundary, and the total time is 2t. The surface term isgiven by

Ipastsurf ¼ −rd−1Ω1;d−1

8πG

�∂rfðrÞ þ

2ðd − 1Þr

fðrÞ�

× ð−tþ r�ðaÞ − r�ðrÞÞjr¼ϵ; ð50Þ

Ifuturesurf ¼ −rd−1Ω1;d−1

8πG

�∂rfðrÞ þ

2ðd − 1Þr

fðrÞ�

×ðtþ r�ðaÞ − r�ðrÞÞjr¼ϵ; ð51Þ

FIG. 5. The complexity growth rate of the brane cosmology forthe closed universe in the CV conjecture. The dimensionlessquantities are τ=L and ðd − 1Þ=ð8πMÞdCV=dτ. Here we haveconsidered the case with rh=L ¼ 1 and d ¼ 4. This figure showsthe range 0.054 < τ=L < 1.

FIG. 6. The complexity growth rate of the brane cosmologyfor the closed universe in the CV conjecture. We choose a largervalue of the horizon radius with rh=L ¼ 5 and dimensiond ¼ 4. The dimensionless quantities are τ=L and ðd − 1Þ=ð8πMÞdCV=dτ. This figure shows the range 0 < τ=L < 5.

FIG. 4. The complexity growth rate of the brane cosmology forthe closed universe in the CV conjecture. The dimensionlessquantities are τ=L and ðd − 1Þ=ð8πMÞdCV=dτ. Here we havechosen d ¼ 4 and rh=L ¼ 1. This figure shows the range0 < τ=L < 0.054.

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where ϵ is the infinitesimal cutoff near r ¼ 0. Finally, thejoint term at the brane reads

IAþBjnt ¼ ad−1Ω1;d−1

4πGlog

jfðaÞjα2

; ð52Þ

where α is the normalization constant. As one can see thatthe joint term depends on the affine parameter whosechoice is quite general. To cancel this ambiguity, one needsto add the counterterms given by Eq. (27); for simplicity,we take L̃ ¼ L. For the time before tc,

Icount ¼Ω1;d−1

2πGad−1

�log

ðd − 1ÞαLa

þ 1

d − 1

�: ð53Þ

Now, we are ready to calculate the complexity growthrate. The bulk contribution is

dIbulkdτ

¼ −Ω1;d−1

2πGL2

ad

fðaÞdadτ

: ð54Þ

The contribution from the two surface terms is

dIFsurfacedτ

¼ drd−2h Ω1;d−1

8πG

�1þ r2h

L2

��dtdτ

þ 1

fðaÞdadτ

�; ð55Þ

dIPsurfacedτ

¼ drd−2h Ω1;d−1

8πG

�1þ r2h

L2

��−dtdτ

þ 1

fðaÞdadτ

�:

ð56Þ

The part from the joint term at the brane reads

dIjntðaÞdτ

¼ ðd − 1ÞΩ1;d−1ad−2

4πGlog

fðaÞα2

dadτ

þ ad−1Ωk;d−1

4πG1

fðaÞdfðaÞda

dadτ

: ð57Þ

The counterterm contribution is

dIcountdτ

¼ ðd − 1ÞΩ1;d−1

2πGad−2 log

αLðd − 1Þa

dadτ

; ð58Þ

and the contribution from the joint term at the singularityvanishes. One can see that the whole time dependence ofthe complexity comes from aðτÞ. We show the complexitygrowth rate with respect to the time τ before τc in the toppanel of Fig. 7. The rate is negative and decreases as τ isincreased from τ ¼ 0 to τ ¼ τc.After the critical time but before crossing the horizon, the

action has an additional joint term which is due to theintersection of two past null surface, say at rm. Such jointterm contribution is given by

Ijnt ¼ −Ωd−1rd−1m

8πGlog

jfðrmÞjα2

; ð59Þ

with rm determined by

tþ r�ðrmÞ − r�ðaÞ ¼ 0: ð60ÞSo, the growth rate of this joint term is

dIjntðrmÞdτ

¼ −ðd − 1ÞΩ1;d−1rd−2m

8πGlog

jfðrmÞjα2

drmdτ

−rd−1m Ωk;d−1

8πG1

fðrmÞdfðrmÞdrm

drmdτ

: ð61Þ

The bulk parts read

I1bulk ¼ −dΩ1;d−1

4πGL2

Zrh

0

rd−1ðtþ r�ðaÞ − r�ðrÞÞdr; ð62Þ

I2bulk ¼ −dΩ1;d−1

2πGL2

Za

rh

rd−1ðr�ðaÞ − r�ðrÞÞdr; ð63Þ

I3bulk ¼ −dΩ1;d−1

4πGL2

Zrh

rm

rd−1ð−tþ r�ðaÞ − r�ðrÞÞdr; ð64Þ

FIG. 7. Time dependence of complexity for a closed universeon a brane. We have adopted the CA conjecture and set rh=L ¼ 1with the critical time τc=L ¼ 0.11. Top panel: the growth rate ofthe complexity before the critical time τc, Bottom panel: thegrowth rate after the critical time.

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and therefore,

dIbulkdτ

¼ −Ω1;d−1rdm4πGL2

dtdτ

þ Ω1;d−1

4πGL2

rdm − 2ad

fðaÞdadτ

: ð65Þ

Now, there is only one future boundary term, Eq. (55). Thejoint term at the brane does not change, but the countertermand its growth rate become different.

Icount ¼Ω1;d−1

2πGad−1

�log

ðd − 1ÞαLa

þ 1

d − 1

�

−Ωk;d−1

4πGrd−1m

�log

ðd − 1ÞαLrm

þ 1

d − 1

�; ð66Þ

dIcountdτ

¼ ðd − 1ÞΩ1;d−1

2πGad−2 log

αLðd − 1Þa

dadτ

−ðd − 1ÞΩk;d−1

4πGrd−2m log

αLðd − 1Þrm

drmdτ

; ð67Þ

where drm=dτ can be obtained from Eq. (60). Combiningthem together, we obtain the complexity growth rate.We show the time evolution of the growth rate in Fig. 7.

As one can see, although the volume of the universedecreases, the growth rate is first negative but then suddenlybecomes positive after the critical time τc. Here we havechosen rh ¼ L, but we have similar behaviors for othervalues of rh. Note that the contribution from the interactionpart and the volume part to the complexity growth hasopposite effects. The interaction part contribution maybecome dominant, and the complexity continues to groweven if the volume is contracting. In particular, when thebrane moves close to the horizon (after τ ≈ 0.8 in Fig. 7), thecomplexity growth rate increases very quickly and tends todiverge. Such unnatural behavior motivates us to conjecturethat there might be some inconsistency when the brane isvery close to the horizon.Wewill show some evidence basedon the Lloyd bound [57] in the discussion section.

D. Complexity evolution for the spatially flat universe

For the AdS spacetime, apart from the black hole withspherical topology, there are also black hole solutions withflat or hyperbolic horizons. It turns out that, for the k ¼ 0black brane geometry, the codimension one braneembedded in this background represents an expanding flatuniverse and some of its thermodynamic behavior wasdiscussed in Ref. [58]. In this section, we want to inves-tigate the complexity behavior on this flat universe. Most ofthe steps for the calculation will be the same as the closeduniverse case, so we will skim over the details and willshow the main results only.First, we need to determine the evolution of the brane

universe. We focus on the four-dimensional universe withd ¼ 4, for which the blackening factor reads fðaÞ ¼a2

L2 ð1 − r4h=a4Þ with rh the location of the horizon. The

differential equation (42) now becomes

daðτÞdτ

¼ r2h=LaðτÞ ; ð68Þ

and the relation between the time t and τ is also given byEq. (41). As a typical example, we set rh=L ¼ 1, then thescale factor is given by aðτÞ ¼ ffiffiffiffiffiffiffiffi

2τLp

. So, the universe willexpand forever and never contract. For the present case, thetime when the brane universe crosses the horizon is atτ ¼ 1

2L. There is a free parameter t0 when the brane crosses

the horizon by solving Eq. (41); this parameter just labelshow far the two flat universes are from each other. Below,we choose a specific case with t0 ¼ −6L.The time-evolution behavior from the CV proposal is

shown in Fig. 8. One can see that the complexity growthrate is positive and increases all the time. It is quite differentfrom the case for the closed universe in Fig. 4, where thecomplexity growth rate first increases and then decreases.For the CA case, we first need to determine the WDW

patch, which depends on the two functions r�ðaÞ − r�ð0Þ −t and r�ðaÞ − r�ð0Þ þ t. We plot both functions in Fig. 9.The WDW patch forms the past null joint when r�ðaÞ −r�ð0Þ − t < 0 and the future null joint when r�ðaÞ−r�ð0Þ þ t < 0. From Fig. 9, we can find that the criticaltime τc when the past joint term forms is τc ¼ 2.97L.In order to compare with the static case, we consider the

evolution from t ¼ 0. At t ¼ 0, we prepare the TFD statejTFDð0ÞiFRW and then evolve it along the τ direction.According to Eq. (41), when t ¼ 0, τ starts fromτ ¼ 2.16L. Before the critical time τc, the complexityevolution totally comes from the expansion of the volume,which is shown in the top panel of Fig. 10. After the criticaltime, as one can see from the bottom panel of Fig. 10, themain contribution of the growth rate is also from thevolume expansion.

FIG. 8. The complexity growth rate of the CV proposal for ad ¼ 4 flat universe in the background of a black brane. The figurestarts from t ¼ 0 with τ ≈ 2.16L. The green curve is the totalcomplexity growth rate, and the blue curve is the one due to thevolume expansion. We see that the contribution comes mainlyfrom the volume expansion.

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IV. SUMMARY OF RESULTS AND DISCUSSION

In this paper, we have studied the behaviors of thecomplexity growth rate during some kinds of cosmologicalevolution in the context of the AdS/CFT correspondenceand the brane-world framework in an AdS-Schwarzschildblack hole background. We have considered a closed uni-verse and a flat universe by using then CV and CA con-jectures. Here we summarize our analysis and main results.In Sec. II, we have investigated the complexity growth

rate of a TFD state defined on the FLRW type slice locatedat the asymptotic AdS boundary. Both the CV and CAconjectures give a similar result. We also note that for theCV case, there are also additional subleading terms and oneparticular contribution due to the spatial curvature. Thecomplexity growth rate consists of two parts. For the firstpart of Eq. (35), which we called interaction term, thecomplexity growth rate is decreasing for an expandingbackground and vice versa. On the other hand, the secondpart is proportional to the growth rate of the spatial volumeof boundary and is called volume term.The behavior of the interaction term can be under-

stood as follows. For the TFD state, after preparing thejTFDð0Þi state by Euclidean path integral at t ¼ 0, oneconsiders that there are two localized operators OLðxÞ andORðxÞ at left and right boundaries, respectively. As timeevolves, because of the interaction in the right boundarysystem,OR affects and correlates with more and more d.o.f.in the right side. The original correlation between OL andOR is distributed among many d.o.f., and OL correlateswith many other operators on the right besides OR. So, thecorrelation betweenOLðt; xÞ andORðt; xÞ decreases, whichcan be easily seen from the decrease of mutual informationand correlation function. Meanwhile, the system becomesmore complex because the original OR is scrambling intomany d.o.f. These explain the complexity growth for theTFD state. When the space expands or contracts, thespreading of OR will decrease or increase, respectively.So, while the complexity still grows, the growth rate willslow down or increase, depending on the evolution of thebackground.For the volume term of Eq. (35), we see that there is

some divergence in the complexity growth rate whenRmax → ∞. To be more concrete, we introduce the UVcutoff ϵ ¼ L=Rmax, and the volume term can be rewritten as

1

d − 1

ddτ

�Ld−1ad−1

ϵd−1

�: ð69Þ

One finds that such divergent term obeys a volume law.This result is quite natural from the field theory point ofview. Note that from the field theory definition of complex-ity, the leading contribution of complexity is indeed thevolume law. For example, with an appropriate choice of thecost function and the reference frequency, the complexityof free field theory is given by [8]

FIG. 10. The complexity growth rate in the CA conjecture forthe flat brane universe. Top panel: complexity growth rate forτ < τc. Bottom panel: complexity growth rate for τ > τc. Thegreen curve is the total complexity growth, and the blue curve isthe complexity growth due to the volume expansion. We see thatthe contribution comes mainly from the volume expansion part.

FIG. 9. The green line represents the function r�ðaÞ−r�ð0Þ − t, and the red line is for r�ðaÞ − r�ð0Þ þ t. We considerthe evolution after the brane crosses the horizon. We see clearlythat when 0.5L < τ < 1.73L, there is one future joint term andwe denote the position to be rm1. For 1.73L < τ < 2.97L, there isno joint term, and the WDW patch intersects the two singularities.For τ > 2.97L, the WDW patch has a past joint term with itslocation at rm2.

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C ¼ Vδd−1

; ð70Þ

with δ the UV cutoff and V the volume of the space wherethe field theory is defined. A simple physical picture is asfollows. As the volume expands, there appears many newd.o.f. which also appear in the computation process. So, thecomplexity will increase and be proportional to the growthof the background volume.In Sec. III, we focus on the brane cosmology for which

the FLRW universe lives in a brane located at finite radiusof an AdS black hole. We have considered both the closeduniverse k ¼ 1 and the flat universe k ¼ 0. Now, as theconformal radiation field is coupled to gravity, there aresome modifications to the above two terms. It is worthypointing out that in the brane cosmology setup, thecomplexity growth rate is free of UV divergence. Thebehavior of the complexity growth using CV duality andCA duality is different. For the CV duality, the maincontribution always comes from the volume part. For theclosed universe with k ¼ 1, the complexity growth behav-ior also depends on the horizon radius rh. When rh=L issmall, the growth rate first increases and then decreases.When rh=L is large, the growth rate decreases monoton-ically and is always negative due to the contraction of thevolume. For the flat case with k ¼ 0, the universe on thebrane expands. It has been found that the complexitygrowth rate is positive due to the expansion of the volume,and in both cases the interaction part plays little role. But,for the CA duality, there is some competition betweenthe interaction part and the volume part. For the k ¼ 1 case,the complexity growth rate is at first negative, but after thecritical time τc, it becomes positive, which is quite differentfrom the CV calculation. For the k ¼ 0 case, the complex-ity growth rate always grows with its contribution mainlyfrom the volume expansion.In this work, we have denoted the mass of the black hole

by M. But we should note that M is not the energy of theexpanding/contracting universe on the brane. So, it does notrelate to the Lloyd bound. The physical energy of the braneuniverse was given by the author of Ref. [46],

E ¼ MLa; ð71Þ

which depends on the evolution of the universe. Therelation between total boundary time t ¼ τL þ τR ¼ 2τand the complexity growth rate divided by the physicalenergy, 1

2EdCAdt , is presented in Fig. 11 for k ¼ 1 case.

Interpreting complexity growth as computation, there is aphysical bound for the growth rate conjectured by Lloyd[57]. Previous studies considered Lloyd bound in a staticboundary [17,18,20]. Here we would like to discuss theLloyd bound when the background is changing over time,more specifically during the cosmic evolution we havestudied. The k ¼ 1 case is of particular interest. The

universe is now in a contracting phase, and therefore thecontribution from the volume part is negative. So, the vastgrowth of complexity growth rate after the critical time τcis due to the interaction among the d.o.f. However, asshown in Fig. 11, when the brane is near the horizon, thecomplexity growth rate becomes badly divergent; hence,the Lloyd bound is violated. As this complexity growthdivergence is due to the first interaction part of Eq. (35), itbecomes very strange why the computation can be so fast,which is far beyond the physical constraint of the energy-time uncertainty relation. In Ref. [47], the author arguedthat when the brane crosses the horizon, the Casimir energydue to quantum corrections will no longer be small suchthat this period may not be trustworthy. In our work, basedon the complexity growth rate, we give further evidencethat there might be some inconsistency when the branecrosses the horizon.Many open questions and challenges remain. The

authors of Ref. [56] investigated the complexity growthrate for the TT̄ deformed CFT on the boundary at finiteradius. They found that in order to make the late timecomplexity growth rate to satisfy the Lloyd bound, one hasto introduce a corresponding cutoff surface inside thehorizon with its position determined by the outside cutoffsurface. For k ¼ 0 case, the position of the brane inside thehorizon, say at r0, and the cutoff radius, say at rc, have asimple relation for the AdS5 case,

r0r2c ¼ r3h: ð72Þ

It means that in the construction of the bulk from theboundary evolution, when the evolution is changed, forexample, from H to HTT̄ , the bulk should be changedaccordingly, and an additional brane inside the horizon isformed. In our brane-world scenario, similar thing may

FIG. 11. The complexity growth rate in the CA conjecture forthe closed brane universe with rh ¼ L and d ¼ 4. Before timet ¼ 1.6L, the complexity growth satisfies the Lloyd bound, whileafter that time as the brane cross the horizon, the complexitygrowth rate increase quickly again and thus violates theLloyd bound.

AN, CAI, LI, and PENG PHYS. REV. D 101, 046006 (2020)

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happen. It will be interesting to investigate where the newbrane is. The introduction of the inside brane might providea mechanism to prevent the brane from entering the horizonand to cure the inconsistency we found in Fig. 11.Note that in this work we have only investigated the

complexity growth rate on the expanding/contracting uni-verse from the holographic side. It will be also important tounderstand our results from the field theory point of view.We could use the Fubini-Study metric [9] to define thecomplexity and to check if the complexity of free fieldtheory defined on the expanding/contracting backgroundwill exhibit a similar behavior as our holographic results.We leave the analysis to the future work. In the presentpaper, we have only considered a simple case to obtain thebrane cosmology in the Schwarzschild black hole. Thereare many complicated constructions based on other blackholes, such as Refs. [59–61]. It would also be interesting tostudy the complexity for a generic FLRW background andto see if there are new features.

ACKNOWLEDGMENTS

Y.-.S. An would like to thank Zhuo-Yu Xian forvaluable discussion on the result. L. Li was supportedby the National Natural Science Foundation of China(Grant No. 11991052) and by the Chinese Academy ofSciences (CAS) Hundred-Talent Program. Y. Peng issupported in part by the National Postdoctoral Programfor Innovative Talents with Grant No. BX201700259. R.-G.Cai was supported in part by the National Natural ScienceFoundation of China Grants No. 11435006, No. 11647601,No. 11821505, No. 11851302, and No. 11847612 and bythe Key Research Program of Frontier Sciences of CAS.

Note added.—Recently, there has been a related paperstudying the holographic complexity in FLRW spacetimes[62]. In contrast to our setup, that paper considereda holographic screen in the FLRWuniverse and investigatedthe complexity growth rate of a CFT defined on the screen.

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