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Evolution of domain wall networks: the PRS algorithm L. Sousa1,2,∗ and P.P. Avelino1,2,† 1Centro de F´ısica do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 2Departamento de F´ısica da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal The Press-Ryden-Spergel(PRS) algorithm is a modification to the field theory equations of motion, parametrized by two parameters (α and β), implemented in numerical simulations of cosmological domain wall networks, in order to ensure a fixed comoving resolution. In this paper we explicitlydemonstratethatthePRSalgorithmprovidesthecorrectdomainwalldynamicsinN+1- dimensional Friedmann-Robertson-Walker(FRW)universesif α+β/2=N, fully validatingitsuse innumericalstudiesofcosmicdomainevolution. Wefurthershowthatthisresultisvalidforgeneric thin featureless domain walls, independentlyof the Lagrangian of the model. I. INTRODUCTION are the PRS algorithmparameters of ref. [1]). Although thisclaimisstronglysupportedbynumericaltestsithas never been proven that the same Nambu-Goto effective 1 The dynamics of cosmological domain walls has been action is recovered in the thin wall limit. In this paper 1 investigatedusingbothhigh-resolutionnumericalsimula- 0 we eliminate this shortcoming, extending the analysis to tions and a semi-analitical velocity-dependent one-scale 2 generic thin featureless domain walls in FRW universes (VOS) model [1–7]. Most of these studies were moti- with an arbitrary number of spatial dimensions. n vated by the suggestion [8] that a frozen domain wall a network could be responsible for the observed accelera- J tion of the Universe (see also [9–12]). Although, current II. THE PRS ALGORITHM I 7 observationalconstraintsontheequationofstateparam- 1 eter of dark energy strongly disfavor domain walls as a ConsidertheGoldstonemodelwithasinglerealscalar ] singledarkenergycomponent[13,14],theyareunableto field φ described by the Lagrangian h ruleoutasubstantialimpactofafrustrateddomainwall t - network on the acceleration of the Universe around the =X V(φ), (1) p L − present time. However, analytical and numerical results he strongly support the conjecture that no frustrated do- where X = −φ,µφ,µ/2 and V(φ) is the potential. This [ main wall network, accounting for a significant fraction model admits domain wall solutions if the potential, of the energy density of the Universe today, could have V(φ), has, at least, two discrete degenerate minima. 1 emerged from realistic phase transitions. These results, Varying the action, v on their own, seem to rule out any significant contribu- 0 5 tion of domain walls to the dark energy budget. How- S = d4x√ g , (2) Z − L 3 ever, they rely heavily on the validity of the so-called 3 Press-Ryden-Spergel(PRS) algorithmusedin cosmolog- withrespecttothescalarfield, φ,oneobtainsthefollow- 1. ical domain wall network simulations. ing equation of motion 0 Domain walls have a constant physical thickness and, 1 ∂V 1 consequently,theircomovingthicknessdecreasespropor- √ gφ,µ = . (3) 1 √ g − ,µ ∂φ tionally to the inverse of the cosmological scale factor. − (cid:0) (cid:1) : v In numerical studies of cosmological domain wall evo- Here g = det(g ) and g is the metric tensor. In this αβ αβ i X lution the rapid decrease of the comoving domain wall paper the Einstein summation convention will only be thickness would be serious problem since it would im- used with greek indices (such as in Eq. (3)). r a ply that domain walls could only be resolved during a In a flat FRW universe, the line element is small fraction of the simulation dynamical range. The ds2 =a2(η)( dη2+dx dx), (4) PRSalgorithmisamodificationtothefieldtheoryequa- − · tionsofmotion,implementedinnumericalsimulationsof where a(η) is the scale factor, η =dt/a is the conformal cosmological domain wall evolution, allowing for a fixed time, t is the physical time and x are comoving coordi- comoving resolution. It has been argued that the PRS nates. Theequationofmotionforthescalarfieldφgiven algorithm[1],providesthecorrectdomainwalldynamics by Eq. (3) becomes in 3+1 dimensions, as long as α+β/2 = 3 (α and β ∂V φ¨+(N 1) φ˙ 2φ= a2 , (5) − H −∇x − ∂φ ∗Electronicaddress: [email protected] where a dot represents a derivative with respect to con- †Electronicaddress: [email protected] formal time, N is the number of spatial dimensions, 2 = a˙/a and 2 is the comoving Laplacian. A static a rapid change of φ occurs only in the direction orthog- H ∇x straight domain wall solution oriented along the x di- onal to the wall [15]. It is convenient to choose spatial rection can be obtained by choosing initial conditions coordinates(u,w,z)such that locallythe walls are coor- such that φ = φ(x) with φ˙ = 0 and φ¨ = 0 (we take dinate surfaces satisfying the condition u=constant. In x=(x ,x ,x )andx =x). Eq. (5)preservesthephys- this case, the domain wall is parameterized by the coor- 1 2 3 1 ical thickness of the domain walls so that the comoving dinates w and z and it moves along the u-direction. It thickness is proportional to a−1. This is a problem for is useful to choose an orthogonal coordinate system in cosmological domain wall network simulations since the which w = constant and z = constant are lines of cur- comoving thickness of the domain walls decreases very vature so that the coordinate curves coincide with the rapidly and can only be resolved during a small fraction principal directions of curvature of the surface defined of the simulation dynamical range. by u = constant. It is always possible to construct such The PRS algorithmconsists of the following modifica- a coordinate system in the vicinity of any non-umbilic tion to the equations of motion point (in which the two principal curvatures exist and are not equal)of a surface embedded in a flat space [16]. ∂V φ¨+α φ˙ 2φ= aβ , (6) Ifthedomainwallhasvelocityv,thenthedomainwall H −∇x − ∂φ solution is still be given by φ=φ (l) with s where α and β are constants. By taking β =0 it is pos- ∂l ∂l ∂l ∂l sibletofixthecomovingthicknessofthedomainwallsso = γv, =γ, = =0. (11) ∂ξ − ∂s ∂s ∂s u w z that they can be resolved throughout the full dynamical range of the simulations. Moreover, it was shown that whereds = d~r is the arclengthalongdirectionu and i i i if α+β/2 = 3 the dynamics of planar domain wall in d~r = h du uˆ| (uˆ| is the unit vector along the direction i i i i i a 3+1-dimensional FRW universe would be maintained u ). We shall use the gauge freedom to choose a coordi- i [1]. nateuwhichmeasuresthearc-lengthalongthe direction perpendicular to the domain wall, so that h = 1 and u ds =du. u III. THE PRS ALGORITHM II Therefore, one has Changingthe space-timecoordinates,in Eq. (6), from ∂φ dφs ∂φ dφs = γv , =γ , (12) (η,x) to (ξ,y), defined by ∂ξ − dl ∂u dl ∂2φ d2φ ∂(γv)dφ ∂ 1 ∂ = (γv)2 s s . (13) = , (7) ∂ξ2 dl2 − ∂ξ dl ∂ξ aβ/2∂η y = aβ/2x, (8) On the other hand, taking into account that φ=φ(ξ,u) and h =1, u yields 1 ∂ h h ∂φ ∂2φ + α+ β H∂φ 2φ= ∂V , (9) ∇2φ= huhwhz (cid:20)∂u(cid:18) whuz ∂u(cid:19)(cid:21)= (14) ∂ξ2 (cid:18) 2(cid:19) ∂ξ −∇y −∂φ 1 ∂h 1 ∂h ∂φ ∂2φ w z = + + . (cid:20)(cid:18)h ∂u h ∂u (cid:19)∂u ∂u2(cid:21) where 2 =a−β 2 and H=a−β/2 . w z ∇y ∇x H In Minkowski space-time (a = 1) a planar static do- Thecurvatureofacurveparameterizedbypisdefined main wall solution oriented along the y direction will be as k = k where k = deˆ /ds , eˆ is the unitary tan- p p p p p p given by φ=φ (l) with | | s gent vector to the curve and ds is the arc-length. The p principal curvatures of a surface, defined by a constant d2φ ∂V s = , (10) u=u0, are given by kw =kw uˆ and kz =kz uˆ with dl2 ∂φ · · 1 ∂wˆ with l = y (we take y = (y ,y ,y ) and y = y). If the k = , (15) 1 2 3 1 w h (cid:18)∂w(cid:19) domainwallisboostedalongthepositivey direction,the w z=z0 planar domain wall solution to eq. (9) is still φ = φ (l) 1 ∂ˆz s k = . (16) butnowl=γ(y vξ)wherevisthedomainwallvelocity z h (cid:18)∂z(cid:19) − z w=w0 and ξ =t. In this case ∂l/∂ξ= γv and ∂l/∂y=γ. Considerthemoregeneralcase−ofacurveddomainwall The vectors uˆ, wˆ and zˆ form an orthonormal but, in ina3+1dimensionalflatFRWuniverse. Thegeneraliza- general, non-coordinate basis. Their derivatives can be tiontoN+1dimensionsistrivialandforsimplicityitwill calculated using be made only at the end of the section. In the following ∂uˆ 1 ∂h 1 ∂h we shall assume that the thickness of the domain walls i = juˆ iuˆ . (17) j k is very small compared to their curvature radii, so that ∂uj hi ∂ui −Xk hk∂uk 3 Hence, IV. GENERIC DOMAIN WALL MODELS 1 ∂h 1 ∂h kw = wuˆ, kz = zuˆ. (18) Inthissectionweshowthatthemainresultofthepre- −h ∂u −h ∂u w z vioussection(Eq. (24))describesthedynamicsofgeneric The relevant curvature for domain wall dynamics is the thin domain walls, independently of the Lagrangian, extrinsiccurvature,i.e. the”bending”ofthewallinrela- (φ,X), of the model. We will follow closely the deriva- L tiontotheflatembeddinguniverse. Mathematicallythis tion presented in ref. [6] where the validity of Eq. (24) is measured by the curvature parameter hasbeendemonstratedforplanardomainwalls. Varying the action, 1 ∂h 1 ∂h =(k uˆ+k uˆ)= w + z . (19) K w· z · −(cid:18)hw ∂u hz ∂u (cid:19) S = dt d3x√ g (φ,X), (26) Z Z − L Therefore, Eq. (14) can be written as with respect to φ, one obtains ∂φ ∂2φ 1 2φ= + . (20) √ g φ,µ = , (27) ∇ −K∂u ∂u2 √ g − L,X ,µ −L,φ − (cid:0) (cid:1) Inserting this in Eq. (9), taking into account Eqs. (12) where =∂ /∂X and =∂ /∂φ. ,X ,φ and (13) and the fact that ∂2φ/∂u2 = γ2d2φs/dl2, one AssuLming a NL +1-dimeLnsionalLFRW metric given by obtains Eq. (4) and the transformations given by Eqs. (7) and d2φ dφ ∂V (8) one obtains s s + = , (21) − dl2 F dl −∂φ ∂ ∂φ β ∂φ + α+ H ,X ,X ∂ξ (cid:18)L ∂ξ(cid:19) (cid:18) 2(cid:19) L ∂ξ where φ 2φ= (,28) ∂ β − ∇yL,X ·∇y −L,X∇y L,φ = (γv) α+ Hγv+ γ. (22) F −∂ξ −(cid:18) 2(cid:19) K with α = N 1 and β = 2. Notice that, in Minkowski − spacetime, a planar static domain wall solution oriented Taking into account Eq. (10), we conclude that =0. along the y direction will be given by φ=φ (l) with F s Notice, however,that ds and ds are not the comov- w z ingarc-lengthssincethecomovingspacecoordinateshave d dφs = . (29) been scaled by a factor a−β/2. The comoving curvature − dl (cid:18)L,X dl (cid:19) L,φ parameter is instead with l=y. κ=aβ/2 . (23) Consider the coordinate system (u,w,z) as described K in section II. Suppose that the wall is moving along the Changingintotheoriginalvariables(η,x),wefinallyfind direction u with velocity v. Taking into account that that ∂ ∂φ ,X v˙ + 1 v2 α+ β v κ =0. (24) ∇yL,X ·∇yφ= ∂Lu ∂u, (30) − (cid:20)(cid:18) 2(cid:19)H − (cid:21) (cid:0) (cid:1) as well as Eqs. (11-13) and (20), the equation of motion (28) yields Bysettingtheparametersαandβtotheiroriginalvalues (α = β = 2) we find that, if the modified equations are d dφ dφ s s toyieldthecorrectdynamicsina3+1-dimensionalFRW ,X + ,X = ,φ. (31) − dl (cid:18)L dl (cid:19) FL dl L universe, we must have that α+β/2=3. InaN+1-dimensionalFRWuniversedomainwallsare Again, since φ(l) must be a solution of Eq. (29) one has defects with N 1spatialdimensions whosedynamics is = 0 and, consequently, Eq. (24) remains valid. Fur- − F stillgivenbyEq. (3)(seeref. [17]forananalyticalstudy) thermore, although only models with a single real scalar with field have been considered, it is straightforward to ver- ify that Eq. (24) describes the correct thin domain wall N−1 dynamics in the context of generic models with various κ=aβ/2uˆ k . (25) · i scalar fields. Xi=1 Here, k are the curvature vectors associated with the i N 1 coordinate curves of the domain wall. Hence, the V. DOMAIN WALL DYNAMICS IN 2+1 dyn−amics of thin domain walls is unaffected by the PRS DIMENSIONS algorithmaslongasα+β/2=N (theoriginalparameters were α=N 1 and β =2). In particular, the dynamics The world history of an infinitely thin domain wall planar(κ=0−)domainwallsissuchthatvγ a−α−β/2 in a flat FRW universe can be represented by a two- a−N. ∝ ∝ dimensional world-sheet with x = x(η,σ), obeying the 4 usualGoto-Nambuaction. Theequationsofmotiontake where we havetaken into account that ∂wˆ/∂s=κuˆ and the form the fact that the physical length along a 2-dimensional domain wall is given by ds = dx = Cdσ. Henceforth, x¨+2 1 x˙2 x˙ = ǫ−1 ǫ−1x′ ′ (32) the normal component of eq. (3|2)|yields: H − (cid:0) (cid:1) ǫ˙ = 2 (cid:0)ǫx˙2, (cid:1) (33) − H v˙ +(1 v2)(2 v κ)=0, (39) − H − with which confirms Eq. (24) in the particularcase with N = x˙ x′ = 0, (34) 2. · 1 x′2 2 ǫ = , (35) (cid:18)1 x˙2(cid:19) − VI. CONCLUSIONS where dots and primes are derivatives with respect to η and σ, respectively. InthispaperweexplicitlydemonstratedthatthePRS Let us define unit normal and tangent vectors as algorithm provides the correct dynamics of thin featureless domain walls in FRW universes with an arbitrary x˙ x′ uˆ = , wˆ = , (36) number, N, of spatial dimensions, if α+β/2 = N. Our v C results fully justify the use of the PRS algorithm in nu- wherev(η,σ)= x˙ andC(η,σ)= x′ . Eq. (35)cannow mericalstudiesofcosmologicaldomainwallnetworkevo- be written as ǫ =| |γC with γ = (1| |v2)−1/2. Therefore, lution. Although, fixing the comoving thickness of the − domain walls, using the PRS algorithm, increases artifi- the left hand side of Eq. (32) is given by cially the impact of the junctions on the overallnetwork x¨+2 1 x˙2 x˙ =v˙uˆ+vuˆ˙ +2 1 v2 vuˆ, (37) dynamicsduringthecourseofthesimulations,thiseffect H − H − is negligible for the light junctions usually considered in (cid:0) (cid:1) (cid:0) (cid:1) where uˆ˙ is proportional to wˆ. Moreover, the right hand such simulations. Acknowledgments side of Eq. (32) gives ǫ−1 ǫ−1x′ ′ = 1 ∂ wˆ (38) We thank Carlos Herdeiro, Roberto Menezes and γ∂s(cid:18)γ (cid:19) Joana Oliveira for useful discussions. This work (cid:0) (cid:1) was funded by FCT (Portugal) through contract κ ∂v = uˆ v wˆ , CERN/FP/109306/2009. (cid:18)γ2 − ∂s (cid:19) [1] W.H.Press,B.S.Ryden,andD.N.Spergel,Astrophys. (1999), astro-ph/9812022. J. 347, 590 (1989). [9] B. Carter, Int. J. Theor. Phys. 44, 1729 (2005), hep- [2] P. P. Avelino, C. J. A. P. Martins, J. Menezes, ph/0412397. R. Menezes, and J. C. R. E. Oliveira, Phys. Rev. D73, [10] R. A. Battye, B. Carter, E. Chachoua, and A. 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