On the Transverse-Traceless Projection in Lattice Simulations of Gravitational Wave Production
PPrepared for submission to JHEP
On the Transverse-Traceless Projection in LatticeSimulations of Gravitational Wave Production
Daniel G. Figueroa a Juan Garc´ıa-Bellido b Arttu Rajantie c a Physics Department, University of Helsinki and Helsinki Institute of PhysicsP.O. Box 64, FI-00014, Helsinki, Finland b Instituto de F´ısica Te´orica UAM/CSIC, Universidad Aut´onoma de MadridCantoblanco 28049 Madrid, Spain c Theoretical Physics Group, Department of Physics, Imperial College London,London SW7 2AZ, United Kingdom
E-mail: [email protected] , [email protected] , [email protected] Abstract:
It has recently been pointed out that the usual procedure employed in orderto obtain the transverse-traceless (TT) part of metric perturbations in lattice simulationswas inconsistent with the fact that those fields live in the lattice and not in the continuum.It was claimed that this could lead to a larger amplitude and a wrong shape for thegravitational wave (GW) spectra obtained in numerical simulations of (p)reheating. Inorder to address this issue, we have defined a consistent prescription in the lattice forextracting the TT part of the metric perturbations. We demonstrate explicitly that theGW spectra obtained with the old continuum-based TT projection only differ marginallyin amplitude and shape with respect to the new lattice-based ones. We conclude thatone can therefore trust the predictions appearing in the literature on the spectra of GWproduced during (p)reheating and similar scenarios simulated on a lattice. a r X i v : . [ a s t r o - ph . C O ] O c t ontents Gravitational waves (GW) are expected to be produced copiously in the early universe inprocesses like (p)reheating after inflation [1–7], phase transitions [8–11], during the turbu-lent motion of plasmas [12] and from the self-ordering dynamics of Goldstone fields [13, 14].As opposed to the GW background generated during inflation from the quantum fluctua-tions of the metric [15], these post-inflationary processes correspond to classical mechanismsof GW production, due to the motion of large overdensities.Such backgrounds of GW could open a new window into the early universe by providingprecious information about the dynamics responsible for its production, much before pri-mordial nucleosynthesis. The very violent dynamics of the fields sourcing the GW cannotbe described in linear perturbation theory, and usually takes place very far from equi-librium. This is the reason why the study of the GW production in the early universe isusually done with the help of lattice simulations. It is therefore of great importance to havea precise handle on those simulations to be sure that the predictions made on the amplitudeand shape of the GW spectra is accurate enough for the future GW observatories [16] todetect and constrain these backgrounds.In homogeneous and isotropic background spaces, the six (independent) physical de-grees of freedom of the metric split into irreducible representations of rotations. There aretwo scalar, two vector and two tensor perturbations. The two tensor components corre-spond to the two polarizations of the GW, the transverse and traceless (TT) degrees of– 1 –reedom ( d.o.f. ) of the metric perturbations. The flat Friedman-Robertson-Walker (FRW)line element with metric perturbations in the synchronous gauge, can be written as ds = − dt + a ( t ) (cid:16) δ ij + h ij ( x , t ) (cid:17) dx i dx j , (1.1)The equations of motion of TT d.o.f. of h ij are ∂ µ ∂ µ h TT ij = 16 πG Π TT ij , (1.2)with Π TT ij the transverse-traceless part of the full anisotropic-stress tensor Π ij .The transverse-traceless tensor tensor Π TT ij is obtained by applying a projector Λ ij,lm in momentum space ˜Π TT ij ( k , t ) = Λ ij,lm (ˆ k ) ˜Π lm ( k , t ) (1.3)where Λ ij,lm (ˆ k ) ≡ P il (ˆ k ) P jm (ˆ k ) − P ij (ˆ k ) P lm (ˆ k ) , (1.4) P ij (ˆ k ) ≡ δ ij − ˆ k i ˆ k j , (1.5)and ˆ k = k / | k | .Because the projector is non-local in space, applying it to the source Π ij at every timestep is computationally expensive. In practice, it is therefore more convenient and stillmathematically equivalent (see [4] for details) to consider a tensor h ij which satisfies theequation of motion ∂ µ ∂ µ h ij = 16 πG Π ij , (1.6)and apply the projector ˜ h TT ij ( k , t ) = Λ ij,lm (ˆ k ) ˜ h lm ( k , t ) (1.7)only when calculating the output. Fourier transforming back ˜ h TT ij ( k , t ) to coordinate space,one finds that the metric perturbation h TT ij ( x , t ) = (cid:90) d k (2 π ) e − i kx Λ ij,lm (ˆ k )˜ h lm ( k , t ) (1.8)verifies the required conditions h TT ji = h TT ij , (Symmetry) (1.9) (cid:88) i h TT ii = 0 , (Tracelessness) (1.10) (cid:88) i ∇ i h TT ij = 0 , (Transversality) (1.11)at all x , t , necessary for identifying h TT ij ( x , t ) with the gravitational wave d.o.f. It is alsoeasy to check that the projector (1.4) is maximal in the sense that it leaves any tensor a TT ij that satisfies the conditions (1.9)–(1.11) unchanged,Λ ij,lm a TT lm = a TT ij . (Maximality) (1.12)This guarantees that we capture the whole transverse-traceless component.– 2 –he above procedure to obtain the TT d.o.f. is well-defined in the continuum. How-ever, it has recently been pointed out in Ref. [17] that on a lattice one needs to pay moreattention to the definition of the projector (1.4), since one can define the momentum k inmany different ways. In particular, Ref. [17] claims that if one applies the wrong projector,a significant leak of scalar modes into the tensor modes (GW) might occur, significantlymodifying the amplitude of the GW spectrum. In the context of (p)reheating, all latticesimulations carried out in recent years by the different groups [3–7], see also Refs. [18–20],filtered the TT metric d.o.f. with the projector defined above. Therefore, whether suchprojector is or not appropriate for the lattice, challenges the validity of the GW spectrashown in the literature.In this paper we investigate this issue in detail. We will review first, in Section 2, someideas about the discretization aspects in a lattice, and then in Section 3 we will presentour method for obtaining a TT-projector consistent with the symmetries of the lattice.In Section 4 we will compare analytically and numerically several GW spectra obtainedwith the continuum-based projection and with different lattice-based projections. Finallyin Section 5 we will summarize and conclude. When one simulates the dynamics of non-equilibrium fields like in (p)reheating, the fieldequations are discretized on a lattice. We consider a lattice with N points (representingspatial comoving coordinates) labeled as n = ( n , n , n ), with n i = 0 , , ..., N −
1. Afunction f ( x ) in space is represented by a lattice function f ( n ) which has the same valueas f ( x ) at x = n δx , with δx ≡ L/N the lattice spacing, and L the length of the lattice.To discretize the equations of motion, one has to replace the continuum derivative witha lattice expression that has the same continuum limit. A simple and symmetric definitionof a lattice derivative is the neutral one,[ ∇ i f ]( n ) = f ( n + ˆ ı ) − f ( n − ˆ ı )2 δx , (2.1)where ˆ ı is the unit vector in direction i . This has the drawback that it is insensitive tovariations at the smallest scale of one lattice spacing. In this sense, a definition involvingthe nearest neighbors may be preferable. A common way to do this, is to define the forwardand backward derivatives [ ∇ ± i f ]( n ) = ± f ( n ± ˆ ı ) ∓ f ( n ) δx , (2.2)but these definitions lack the symmetry of Eq. (2.1). This issue can be solved by definingthe derivative at half-way between the lattice sites, as[ ∇ i f ]( n + ˆ ı/
2) = f ( n + ˆ ı ) − f ( n ) δx . (2.3)To improve accuracy, one can also consider lattice derivatives which involve more points,but in practice the definitions have a symmetry either around a lattice site as Eq. (2.1) orhalf-way between lattice sites as (2.3). – 3 –n order to extract the transverse-traceless component of h ij , one needs to apply thelattice version of the projector (1.4). Since the projector is defined in Fourier space, on thelattice one has to use the discrete Fourier transform (DFT) ˜ f (˜ n ), defined as f ( n ) = 1 N (cid:88) ˜ n e − πiN ˜ nn ˜ f (˜ n ) , ˜ f (˜ n ) = (cid:88) n e + πiN ˜ nn f ( n ) , (2.4)where the index ˜ n = (˜ n , ˜ n , ˜ n ) labels the reciprocal lattice, with ˜ n i = − N + 1 , − N + 2 , ... − , , , ..., N . Imposing periodic boundary conditions in coordinate space, i.e. f ( n +ˆ iN ) = f ( n ), there will be necessarily a minimum infrared (IR) momentum k IR = πL in the Fourierspace, such that ˜ n will be representing the continuum momentum k = (˜ n , ˜ n , ˜ n ) k IR .Consequently, there will also be a maximum ultraviolet (UV) momentum k UV = N k IR perdimension, and periodic boundary conditions like ˜ f (˜ n + ˆ iN ) = ˜ f (˜ n ).In the calculation of GW production, there are four separate places where one needs totake care of the discretization details of derivatives: in the equations of the fields sourcingthe GW, in the two sides of Eq. (1.6), and in the transverse-traceless projection (1.7) itself.Ideally one should use a consistent choice of a lattice derivative everywhere, but in somecases there are restrictions that make this very difficult, for example when dealing withgauge fields [7]. It is enough in any case to have a consistent choice among Eqs. (1.6)and the equations for the GW sources, whilst the discretization details in the transverse-traceless projection can be considered separately. In this paper we focus precisely on suchdetails of the transverse-traceless projection in a lattice. With the transverse-traceless projection, we want to obtain a tensor h TT ij that satisfiesthe three conditions (1.9)–(1.11) on the lattice, with respect to the appropriate latticederivative. In the literature , this projection has been done by taking Λ ij,lm (˜ n ) as inEq. (1.4) evaluated at k = ˜ n k IR , h TT ij ( n ) = 1 N (cid:88) ˜ n e − πiN ˜ nn Λ ij,lm (˜ n ) ˜ h lm (˜ n ) . (2.5)However, as highlighted in Ref. [17], the resulting quantity is not transverse with respectto any of the usual lattice derivative operators ∇ i , i.e. ∇ i h TT ij (cid:54) = 0.In particular, due to this lack of transversality, Ref. [17] claims that a significant leakof scalar modes into the tensor modes might occur in such a way that the amplitude of theGW spectrum extracted with the above continuum-based projector (2.5) could be severalorders of magnitude higher than it should be.We therefore need a general and consistent procedure in order to define a TT-projectionin the lattice, i.e. a projector Λ ij,lm that restores the transversality with respect to a givenlattice derivative. Only then we will be able to quantify the potential distortion of theGW spectra with respect to the results obtained with the continuum-based projector. Inorder to construct the lattice equivalent of Eq. (1.4), we need a lattice momentum k . Suchmomentum will depend, of course, upon the choice of the lattice derivative with respect towhich the transversality condition will be attained. Some papers, for instance Ref. [3] and Ref. [5], considered the projection at the level of the source,whereas others like Ref. [4], [6] and [7], considered the projection at the level of the metric perturbations,as we are discussing here. – 4 – .3 The lattice momentum
The lattice momentum is given by the Fourier transform of the lattice derivative ∇ i . Tokeep the discussion general, we do not assume for the moment a specific form for thederivative, but simply assume that it is given by a linear operator in the space of latticefunctions. Therefore, the value of the derivative [ ∇ i f ] is a linear combination of the fieldvalues at different lattice sites,[ ∇ i f ] ( n ) = (cid:88) m D i ( n , m ) f ( m ) , (2.6)where D i ( n , m ) is a real-valued function of two variables on the lattice. For example, forthe neutral derivative (2.1), we have D i ( n , m ) = δ m , n +ˆ ı − δ m , n − ˆ ı δx . (2.7)Because we want the derivative to be translation invariant, D i ( n , m ) is only a function ofthe difference n − m , i.e. D i ( n , m ) = D i ( n − m ), and we can write[ ∇ i f ] ( n ) = (cid:88) m D i ( n − m ) f ( m ) = (cid:88) m (cid:48) D i ( m (cid:48) ) f ( n − m (cid:48) ) . (2.8)For the neutral derivative (2.1), we have D i ( m (cid:48) ) = δ m (cid:48) , − ˆ ı − δ m (cid:48) , ˆ ı δx . (2.9)For the nearest-neighbor derivative (2.3), m (cid:48) is half-integer, and one finds D i ( m (cid:48) ) = δ m (cid:48) , − ˆ ı/ − δ m (cid:48) , ˆ ı/ δx . (2.10)More generally, any odd function with compact support will give a meaningful definitionof a lattice derivative.The Fourier transform of the derivative ∇ i f is (cid:103) ∇ i f (˜ n ) = (cid:88) n e πiN ˜ n · n [ ∇ i f ]( n ) = (cid:88) n e πiN ˜ n · n (cid:88) m D i ( n − m ) f ( m )= (cid:88) n (cid:48) e πiN ˜ n · n (cid:48) D i ( n (cid:48) ) (cid:88) m e πiN ˜ n · m f ( m ) ≡ − i k eff (˜ n ) ˜ f (˜ n ) , (2.11)where the effective momentum k eff (˜ n ) is given by k eff (˜ n ) = i (cid:88) n e πiN ˜ n · n D i ( n ) . (2.12)Conversely, any function k eff (˜ n ) that has the correct leading behaviour in the Taylor ex-pansion of the IR limit | ˜ n | (cid:28) N , i.e. k eff (˜ n ) ≈ ˜ n k IR , defines a lattice derivative throughthe inverse Fourier transform.For example, the neutral derivative (2.1) gives k ,i = sin(2 π ˜ n i /N ) δx . (2.13)– 5 –he forward/backward derivatives (2.2) give k ± eff ,i = 2 e ± iπ ˜ n i /N sin( π ˜ n i /N ) δx = sin(2 π ˜ n i /N ) δx ± i − cos(2 π ˜ n i /N ) δx , (2.14)and the symmetric nearest-neighbor derivative (2.3) gives k eff ,i = 2 sin( π ˜ n i /N ) δx . (2.15)In general, if the lattice derivative is anti-symmetric, i.e. D i ( − n ) = − D i ( n ), then thelattice momentum k eff is real. This is the case also for the derivative used in Ref. [17]. In this Section we will define a lattice projection operator that satisfies the condition (1.9)–(1.12), and which therefore gives the TT d.o.f. of metric perturbations living on a lattice.Since the transversality notion on a lattice is associated to the choice of a lattice derivative ∇ i , we will introduce a projector that will guarantee tracelessness and transversality withrespect to any ∇ i chosen. Let us start with the simpler case of a real momentum, for example k in Eq. (2.13). Inthis case, the projector can be defined in the same way as in continuum.In analogy with Eqs. (1.4) and (1.5), we define P ij (˜ n ) = δ ij − k ,i k ,j ( k ) , (3.1)and Λ ij,lm (˜ n ) ≡ P il (˜ n ) P jm (˜ n ) − P ij (˜ n ) P lm (˜ n ) . (3.2)Using the properties k ,i P ij (˜ n ) = 0 , P ij (˜ n ) P jl (˜ n ) = P il (˜ n ) , P ij ( n ) = P ji ( n ) , (3.3)it is then straightforward to prove that ˜ h TT ij (˜ n ) = Λ ij,lm (˜ n )˜ h lm (˜ n ) satisfies the requiredconditions (1.9)–(1.12):Symmetry:Λ ji,lm (˜ n ) = Λ ij,ml (˜ n ) ⇒ ˜ h TT ji (˜ n ) = ˜ h TT ij (˜ n ) . ⇒ h TT ji ( n ) = h TT ij ( n ) . (3.4)Tracelessness: Λ ii,lm (˜ n ) = 0 ⇒ ˜ h TT ii (˜ n ) = 0 ⇒ h TT ii ( n ) = 0 , ∀ n (3.5)– 6 –ransversality: k ,i Λ ij,lm (˜ n ) = 0 ⇒ k ,i ˜ h TT ij (˜ n ) = 0 ⇒ ∇ i h TT ij ( n ) = 0 , ∀ n (3.6)It is also easy to see that the resulting tensor h TT ij is real in coordinate space. If h ij ( n )is real, its Fourier transform satisfies h ∗ ij (˜ n ) = h ij ( − ˜ n ), and then because Λ ij,lm (˜ n ) is realand even, h TT ∗ ij (˜ n ) = Λ ij,lm (˜ n ) h ∗ ij (˜ n ) = Λ ij,lm ( − ˜ n ) h ij ( − ˜ n ) = h TT ij ( − ˜ n ) ⇒ h TT ∗ ij ( n ) = h TT ij ( n ) . (3.7)Finally, to prove the maximality (1.12) of the operator, we assume a tensor a TT ij thatsatisfies the conditions (3.4)–(3.6), and note that then it also satisfies P ij a TT jk = δ ij a T T jk .Therefore we haveΛ ij,lm a TT lm = (cid:18) δ il δ jm − P ij δ lm (cid:19) a TT lm = a TT ij − P ij a TT ll = a TT ij , (3.8)Of course, all properties just discussed apply, not only to k in Eq. (2.13), but to anylattice momentum k eff as long it is real. The case of a derivative with an associated reallattice momentum, is therefore a simple generalization of the continuum case. In the more general case, the lattice momentum is complex. For example, this is the casewith the forward/backward derivatives (2.2) and the associated momenta k ± eff . Thus wewill be forced to take a projector P ij that is also complex.Thus, we look for a projector P ij that satisfies k eff ,i P ij ( k ) = 0. In order to do this, wedefine P ij (˜ n ) = δ ij − ( k eff ,i ) ∗ k eff ,j | k eff | , (3.9)with | k eff | = k eff ,i ∗ k eff ,i . This projector is complex and satisfies1) k eff ,i P ij ( k ) = 0 2) k ∗ eff ,i P ij ( k ) (cid:54) = 03) k eff ,j P ij ( k ) (cid:54) = 0 4) k ∗ eff ,j P ij ( k ) = 04) P ∗ ij (˜ n ) = P ji (˜ n ) 6) P ij ( − ˜ n ) = P ji (˜ n )7) P ij (˜ n ) P jl (˜ n ) = P il (˜ n ) 8) P ij (˜ n ) P lj (˜ n ) (cid:54) = P il (˜ n ) (3.10)In words, this projector is Hermitian, symmetric under Parity transformations n ↔ − n ,transverse to k but not to k ∗ , and idempotent ( P = P ) but with no inverse ( (cid:64) P − ) andnon-idempotent modulus ( P P ∗ (cid:54) = P ). Demanding property 1) we arrived at the form (3.9),and then properties 2) −
8) simply followed from such form.If Λ ij,lm (˜ n ) was built as in the real case (3.2), then property 3) would prevent h TT ij (˜ n ) ≡ Λ ij,lm (˜ n ) h lm (˜ n ) from being traceless. We are thus forced to redefine also Λ ij,lm in orderto guarantee the desired TT properties. Moreover, since P ij is now complex, so is Λ ij,lm .Therefore we must also ensure that h TT ij ( n ) = DF T { Λ ij,lm (˜ n ) h lm (˜ n ) } is real. From the– 7 –roperties of the Fourier transform and demanding h TT ∗ ij ( n ) = h TT ij ( n ), the latter conditioncan be achieved if and only if Λ ij,lm (˜ n ) satisfiesΛ ∗ ij,lm (˜ n ) = Λ ij,lm ( − ˜ n ) (3.11)This condition suggests how to build the new projector. We can defineΛ ij,lm (˜ n ) = P il (˜ n ) P ∗ jm (˜ n ) − P ij (˜ n ) P ∗ lm (˜ n ) , (3.12)which verifies Λ ∗ ij,lm (˜ n ) = Λ ij,lm ( − ˜ n ) = Λ ji,ml (˜ n ) = Λ ∗ ml,ji (˜ n ) = Λ lm,ij (˜ n ) , (3.13)From here it is easy to prove that h TT ij ( n ) ≡ DF T { Λ + ij,lm (˜ n ) h lm (˜ n ) } is traceless and trans-verse, as well as real. For completeness, let us show explicitly how we obtain these condi-tions using (3.12) and its properties:Tracelessness (1.10): h TT ii (˜ n ) = P il (˜ n ) P ∗ im (˜ n ) h lm (˜ n ) − P ii (˜ n ) P ∗ lm (˜ n ) h lm (˜ n )= P ml (˜ n ) h lm (˜ n ) − P ∗ lm (˜ n ) h lm (˜ n ) = 0 (3.14)Transversality (1.11): k eff ,i h TT ij (˜ n ) = k eff ,i P il (˜ n ) P ∗ jm (˜ n ) h lm (˜ n ) − k eff ,i P ij (˜ n ) P ∗ lm (˜ n ) h lm (˜ n )= 0 − h TT ∗ ij ( n ) = (cid:88) ˜ n e + ik IR δx n ˜ n Λ ∗ ij,lm (˜ n ) h ∗ lm (˜ n ) = (cid:88) ˜ n e + ik IR δx n ˜ n Λ ∗ ij,lm (˜ n ) h lm ( − ˜ n )= (cid:88) ˜ n e − ik IR δx n ˜ n Λ ∗ ij,lm ( − ˜ n ) h lm (˜ n ) = (cid:88) ˜ n e − ik IR δx n ˜ n Λ ij,lm (˜ n ) h lm (˜ n ) ≡ h TT ij ( n )(3.16)However, we find that the resulting tensor h TT ij is not symmetric , i.e. h TT ij ( n ) (cid:54) = h TT ji ( n ).Similarly, the following properties, which distinguish between the first and the second indexof h TT ij ( n ), are also verified1) k eff ,i h TT ij ( n ) = 0 , k ∗ eff ,i h TT ij ( n ) (cid:54) = 0 , k eff ,j h TT ij ( n ) (cid:54) = 0 , k ∗ eff ,j h TT ij ( n ) = 0 . (3.17)All these asymmetry aspects are simply a consequence of the properties 1)-4) of P ij , listedabove, which reflects the fact that P ij is not symmetric but rather Hermitian.Related to these issues we also encounter a subtle aspect about the maximality con-dition (1.12). We find that Λ ij,lm ( n ) A TT lm ( n ) = A TT ij ( n ) only holds for those transverse-traceless symmetric rank-2 tensors A TT ij which are transverse, not only with respect thelattice derivative ∇ i (with lattice momenta k eff used to build Λ ij,lm ), but also with respect– 8 –he conjugate derivative ∇ ∗ i defined through the lattice momentum k ∗ eff . For example, ifone builds the projector (3.12) with the lattice momenta associated to forward derivatives ∇ + i , then Λ ij,lm is only maximal with respect those tensors which are transverse both withrespect to forward and backward derivatives, i.e. ∇ + i A ij = ∇ − i A ij = 0.In the IR limit | ˜ n | (cid:28) N , both k eff ,i and k ∗ eff ,i approach the same momentum, ˜ n k IR ,and thus the full maximality condition and the symmetry under the exchange i ↔ j ,are recovered. Thus for arbitrarily big lattices these caveats should not be relevant. Inreality, we are of course limited by computer memory and the lattice sizes we can typicallyconsider have no more than N = 128, 256 or 512 points per dimension, depending upon thefield content. Nevertheless, despite these two caveats about the maximality and the even-symmetry, the projector defined by eqs. (3.9), (3.12), is one which generically guaranteesreality, transversality and tracelessness on a lattice, and recovers maximality and even-symmetry in the IR limit. Thus, any GW spectra obtained by this method should bereliable at least in the IR region of the Fourier space. To understand the difficulties faced in the complex case, let us note that the conditions ofsymmetry (1.9), tracelessness (1.10) and transversality (1.11) on a lattice involve comparingand adding together different components of the tensor h TT ij and its derivatives. For thisto be meaningful, these components should be arranged in a symmetric way on the lattice.This is not an issue for derivatives defined on lattice sites, such as the neutral derivativeof Eq. (2.1), which is why the TT projector defined with real lattice momenta verifiesnicely all of the required conditions. For those derivatives defined halfway between twolattice sites, with a complex lattice momentum, it is however something to take care of.To illustrate this, let us consider the forward derivative ∇ + i . The transversality conditionbecomes (cid:88) i (cid:2) h TT ij ( n + ˆ ı ) − h TT ij ( n ) (cid:3) = 0 , (3.18)which is not symmetric under parity, because it involves neighboring points only in thepositive directions.Thus, instead of separate asymmetric ∇ i and ∇ ∗ i derivatives with complex latticemomenta, such as the forward and backward derivatives (2.2), we should use the symmetricversion (2.3). In order to make the lattice transversality condition symmetric under parity,we should then define the tensor h TT ij not on the lattice site n , but at the point n +ˆ ı/
2+ ˆ / ı and ˆ startingat point n . More precisely, off-diagonal components are defined at plaquettes, but thediagonal components, for which i = j , live on lattice sites. To avoid confusion, we denotethe tensor defined in this way by h TT ij ( n + ˆ ı/ / h TT ji ( n + ˆ ı/ /
2) = h TT ij ( n + ˆ ı/ / , (3.19)is a meaningful concept because the two sides of the equation are defined on the sameplaquette n + ˆ ı/ /
2. The same is true for the trace h ii ( n ) because all the terms in thesum are now defined at the same location, (cid:88) i h TT ii ( n ) = 0 , (3.20)– 9 –nd similarly in the transversality condition for the tensor h ij ,[ ∇ i h ij ]( n + ˆ /
2) = (cid:88) i h ij ( n + ˆ ı/ / − h ij ( n − ˆ ı/ / δx = 0 , (3.21)all the terms are defined at the same location n + ˆ / k i ( ˜n )˜ h ij ( ˜n ) = 0, where k ( ˜n ) is thereal momentum in Eq. (2.15) and˜ h ij ( ˜n ) = (cid:88) n e πiN ˜n · ( n +ˆ ı/ / h ij ( n + ˆ ı/ / . (3.22)Because the momentum k ( n ) is real, we can now build the transverse-traceless projec-tion in the standard way (1.4)–(1.5), with Λ ij,lm ( ˜n ) = P il ( ˜n ) P jm ( ˜n ) − P ij ( ˜n ) P lm ( ˜n ) , (3.23) P ij ( ˜n ) = δ ij − k i k j k . (3.24)The projected field ˜ h TT ij ( ˜n ) = Λ ij,lm ( ˜n )˜ h lm ( ˜n ) (3.25)then satisfies obviously all the requirements:1. Symmetry: Λ ji,lm h lm = Λ ij,lm h lm if h ml = h lm .2. Reality: Λ ∗ ij,lm ( ˜n ) = Λ ij,lm ( − ˜n ).3. Tracelessness: Λ ii,lm = 0.4. Transversality: k i Λ ij,lm = 0.5. Maximality: Λ ij,lm h lm = h ij for any symmetric, transverse and traceless h ij .since the definition of (3.23) is the same as in Section 3.1.In actual lattice simulations, it is easier to use a tensor h TT ij defined on lattice sites.However, now that we have obtained the projector, we can shift the field to lattice sites bya simple translation h TT ij ( n ) ≡ h TT ij ( n + ˆ ı/ / , (3.26)and derive the form of the equivalent projector in that formulation. The Fourier transformsof h TT ij and h TT ij are related by˜ h TT ij ( ˜n ) = (cid:88) n e πiN ˜n · n h TT ij ( n ) = (cid:88) n e πiN ˜n · n h TT ij ( n + ˆ ı/ / e − πiN (˜ n i +˜ n j ) ˜ h TT ij ( ˜n ) . (3.27)This implies that ˜ h TT ij ( ˜n ) = e − πiN (˜ n i +˜ n j ) Λ ij,lm ( ˜n ) e + πiN (˜ n l +˜ n m ) ˜ h lm ( ˜n ) , (3.28)– 10 –nd therefore, the equivalent projector Λ ij,lm for h TT ij defined on lattice sites is related to Λ ij,lm , by Λ ij,lm ( ˜n ) ≡ e − πiN (˜ n i +˜ n j ) Λ ij,lm ( ˜n ) e + πiN (˜ n l +˜ n m ) (3.29)Noting the relation between P lm defined as a function of the complex lattice momentum k eff , see Section 3.2, with P ij defined previously as a function of the real momentum char-acteristic of the symmetrized derivative, P ij (˜ n ) = e − i πN ˜ n i P ij (˜ n ) e + i πN ˜ n j , (3.30)then we can write the new projector asΛ ij,lm ( ˜n ) = P il ( ˜n ) P jm ( ˜n ) − e πiN (˜ n l − ˜ n j ) P ij ( ˜n ) P lm ( ˜n ) , (3.31) P ij ≡ δ ij − k ∗ eff ,i k eff ,j | k eff | (3.32)At the same time, the shift of coordinates also turns Eqs. (3.19)-(3.21) into a less symmetric,but equivalent set of conditionsTracelessness : (cid:88) i h TT ii ( n − ˆ ı ) = 0 , Transversality : (cid:88) i (cid:2) h TT ij ( n ) − h TT ij ( n − ˆ ı ) (cid:3) = 0 , Symmetry : h TT ji ( n ) = h TT ij ( n ) (3.33)Note that Eq. (3.31), together with Eq. (3.32), define a projector that guarantees maxi-mality and reality of the Fourier transform of the projected d.o.f. ˜ h ij (˜ n ) = Λ ij,lm (˜ n )˜ h lm (˜ n ),as well as transversality, tracelessness and even-symmetry, in the way stated in Eqs. (3.33).To be more precise, Eqs. (3.31), (3.32) will guarantee all these conditions with respectto the forward/backward derivatives defined in (2.2), which is the one that we will im-plement in lattice simulations. For other asymmetric derivatives with lattice momenta k eff ,j (˜ n ) = e + iϕ j p eff ,j (˜ n ) , ϕ j ∈ [0 , π ) , p eff ,j (˜ n ) ∈ Re + , but with e + iϕ j (cid:54) = e + iπN ˜ n j , theprojector readsΛ ij,lm (˜ n ) = e + i (Φ i − Φ l ) e + i (Φ j − Φ m ) (cid:18) P il ( ˜n ) P jm ( ˜n ) − e − ϕ j − ϕ l ) P ij ( ˜n ) P lm ( ˜n ) (cid:19) , (3.34)with Φ i ≡ ϕ i − πN ˜ n i and P ij (˜ n ) defined as in Eq. (3.32). Of course, in the case ofthe forward derivative, e + iϕ j = e + i πN ˜ n j , so Φ i = 0 ∀ i , and then eq. (3.34) reduces toeq. (3.31). Eq. (3.34) will guarantee in general the reality and maximality conditions, theeven-symmetry and tracelessness defined as in eqs. (3.33), and transversality with respectto the lattice derivative defined through the momentum k eff ,j (˜ n ) = e + iϕ j p eff ,j (˜ n ). Here we refer to the complex momentum k eff ,i of the asymmetric derivative ∇ i , which differs from thereal momentum p eff ,i of the symmetrized version of ∇ i , just by a complex phase ϕ i , i.e. k eff ,i /p eff ,i = e + iϕ i .For the forward derivative this is just e + iϕ i = e + i πN ˜ n i . – 11 – Comparison of the GW spectra obtained with different projections
In this Section we will show the time evolution of the the GW spectra during (p)reheating indifferent inflationary models. For each model, we will superimpose the spectra obtained bytaking Λ ij,lm (˜ n ) in Eq. (1.4) evaluated at k = ˜ n k IR (i.e. the continuum-based projector),together with the spectra obtained by using the projectors defined in: 1) Eq. (3.2) evaluatedat the momenta of neutral derivatives, 2) Eq. (3.12) evaluated at the lattice momenta offorward derivatives, and 3) Eq. (3.31) evaluated also at the lattice momenta of forwardderivatives. Thus, we will show explicitly how the several procedures defined to extractthe GW spectra, compare with each other at different times. We will repeat this exercisefor different chaotic and hybrid models of inflation. Before describing the results, however,we must define the GW spectrum.The energy density of a homogeneous and isotropic GW background is described inthe continuum by [21] ρ GW ( t ) = (cid:90) dρ GW d log k d log k = 132 πG (cid:104) h TT (cid:48) ij ( x , t ) h TT (cid:48) ij ( x , t ) (cid:105) = 132 πG V (cid:90) V d x ˙ h ij ( x , t ) ˙ h ij ( x , t )= 132 πG (cid:90) d k d k (cid:48) (2 π ) ˙ h ij ( k , t ) ˙ h ij ( k (cid:48) , t ) 1 V (cid:90) V d x e − i x ( k − k (cid:48) ) (4.1)with (cid:104) ... (cid:105) a spatial average over a sufficiently large volume V encompassing all the relevantwavelengths of the background. In the limit V / → ∞ , the spectrum in the continuum(per logarithmic interval) of GW is obtained as dρ GW d log k = k (4 π ) G V (cid:90) d Ω k π ˙ h ij ( k, ˆ k , t ) ˙ h ∗ ij ( k, ˆ k , t ) (4.2)where d Ω k represent a solid angle element in k -space.In the lattice, we must be more careful since clearly we cannot consider the infinitevolume limit. Assuming the volume of the lattice [ V = ( N dx ) ] encompasses sufficientlywell the characteristic wavelengths of the simulated GW background, then it follows ρ GW ( t ) ≡ πG N (cid:88) n ˙ h TT ij ( n , t ) ˙ h TT ij ( n , t )= 132 πG N (cid:88) n (cid:88) ˜ n (cid:88) ˜ n (cid:48) e + ik IR dx n (˜ n − ˜ n (cid:48) ) ˙ h TT ij (˜ n , t ) ˙ h TT ∗ ij (˜ n (cid:48) , t )= 132 πG N (cid:88) ˜ n ˙ h TT ij (˜ n , t ) ˙ h TT ∗ ij (˜ n , t ) , (4.3)where we have used (cid:80) n e + ik IR dx (˜ n − ˜ n (cid:48) ) n = N δ (˜ n − ˜ n (cid:48) ). Binning the momentum-lattice in– 12 –pherical layers of radii | ˜ n | and width ∆˜ n , R (˜ n ) ≡ { ˜ n (cid:48) / | ˜ n | ≤ | ˜ n (cid:48) | < | ˜ n | + ∆˜ n } , then ρ GW ( t ) = 132 πG N (cid:88) | ˜ n | (cid:88) ˜ n (cid:48) ∈ R (˜ n ) ˙ h TT ij (˜ n (cid:48) , t ) ˙ h TT ∗ ij (˜ n (cid:48) , t )= 132 πG N (cid:88) | ˜ n | π | ˜ n | (cid:68) ˙ h TT ij ( | ˜ n | , t ) ˙ h TT ∗ ij ( | ˜ n | , t ) (cid:69) R (˜ n ) = (cid:88) | ˜ n | (cid:26) dx (4 π ) G L k ( | ˜ n | ) (cid:68) ˙ h TT ij ( | ˜ n | , t ) ˙ h TT ∗ ij ( | ˜ n | , t ) (cid:69) R (˜ n ) (cid:27) ∆ log k (4.4)where k (˜ n ) ≡ | ˜ n | k IR , ∆ log k ≡ k IR k (˜ n ) , k IR ≡ π/L , L = N dx , and (cid:68) ˙ h TT ij ( | ˜ n | , t ) ˙ h TT ∗ ij ( | ˜ n | , t ) (cid:69) R (˜ n ) is an average over configurations with lattice momenta ˜ n (cid:48) ∈ [ | ˜ n | , | ˜ n | + δ ˜ n ).The spectrum of GW in a lattice of volume V = L , is therefore defined as (cid:18) dρ GW d log k (cid:19) (˜ n ) ≡ k ( | ˜ n | )(4 π ) G L (cid:68)(cid:104) dx ˙ h TT ij ( | ˜ n | , t ) (cid:105) (cid:104) dx ˙ h TT ij ( | ˜ n | , t ) (cid:105) ∗ (cid:69) R (˜ n ) (4.5)In the continuum limit, one identifies DF T { f ( n ) dx } → CF T { f ( x ) } , and thus, ex-pression (4.5) matches the continuum expression (4.2), as it should be. Expression (4.5)highlights that the natural momenta in terms of which to express the lattice GW spectrum,is the continuum one k = ˜ n k IR , and not any of the lattice-ones defined from the choice ofa lattice derivative.Having defined the appropriate discretized spectrum of GW, we will now show numer-ical results from lattice simulations of several scenarios. The key question here will be toshow the difference in the GW spectra when considering the different projectors definedin the previous Section. In particular, we will consider chaotic and hybrid models of infla-tion, since the details of the GW production during reheating in those scenarios have beenstudied extensively in the recent years. First we define D i h TT ij as the sum of h TT ij over the lattice sites involved in a particularchoice of a derivative scheme ∇ h ij . For example, D i h TT ij ( n ) ≡ (cid:88) i ( h TT ij ( n ) + h TT ij ( n + ˆ i )) /δx, for forward derivatives (4.6) D i h TT ij ( n ) ≡ (cid:88) i ( h TT ij ( n + ˆ i ) + h TT ij ( n − ˆ i )) / (2 δx ) , for neutral derivatives (4.7)In the left panel of Fig. 1 we show the evolution in time of the dimensionless ratio δ ( t ) ≡ (cid:104)|∇ i h TT ij |(cid:105)(cid:104)|D i h TT ij |(cid:105) (4.8) One is forced to make this identification since the Parseval theorem in the continuum, i.e. the factthat (2 π ) (cid:82) d x f ( x ) = (cid:82) d k | ˜ f ( k ) | , gets translated into (2 π ) dx (cid:80) n f ( x ) = dk (cid:80) ˜ n | dx ˜ f ( k ) | in thelattice, with dx the lattice spacing and dk = k IR . In the text we refer to DF T and
CF T as the discreteand continuous Fourier transforms respectively. – 13 – δ ( t ) t Continuum-based Projector (Forward Derivatives)Continuum-based Projector (Neutral Derivatives)Complex Lattice-based Projector (Section 3.2)Real Lattice-based Projector (Section 3.1)General Lattice-based Projector (Section 3.3) λ ( t ) t Continuum-based ProjectorComplex Lattice-based Projector (Section 3.2)Real Lattice-based Projector (Section 3.1)General Lattice-based Projector (Section 3.3)
Figure 1 . Left:
The time evolution of the degree of transversality, δ ( t ), obtained for the neutraland forward derivatives, both for the continuum- and lattice-based projectors. In the latter case,the outcome is clearly limited only by the round-off machine errors, δ ( t ) ∼ O (10 − ), while in theformer case, δ ( t ) for the continuum can be more than 10 orders of magnitude larger. Right:
The timeevolution of λ ( t ) for the same projectors used in the left figure. Here the degree of transversalityis well achieved for all cases, including the continuum-based projector. Note that these plots wereobtained for a chaotic model, λ φ + g φ χ , with λ/g = 120. In other models of (p)reheatingthe curves look very similar, with amplitudes of the same order of magnitude. for different derivative schemes, where | · · · | is the absolute value and (cid:104)· · · (cid:105) represents anaverage over all the lattice points. Obviously the value chosen for j here is irrelevant. Theevolution of δ ( t ) gives an idea of how well the transversality condition ∇ i h TT ij is preservedin the lattice. It is clearly appreciated that with the old-continuum projector Λ ij,lm , thetransversality is not that well achieved. Using the projectors which guarantee transversalitywith respect to the neutral and forward derivatives, Eqs. (3.2), (3.12) and (3.31), we seethat transversality is very well preserved (as it should be, by construction), since δ ( t ) is oforder O (10 − ). Such amplitude simply represents the unavoidable round-off errors of themachine-precision. Thus, it is very clear that the new lattice-based projectors give raise toa well defined notion of transversality for h TT ij in the lattice.We also define the quantity λ ( t ) ≡ (cid:104)| (cid:80) i h TT ii |(cid:105)(cid:104) (cid:80) i | h TT ii |(cid:105) (4.9)and plot it in the right panel of Figure 1, as obtained for all the same projectors used inthe left panel of the same Figure. As expected, for all cases the degree of tracelessness isalso very high, only limited again by round-off machine errors. In summary, left and rightpanels of Fig. 1 demonstrate explicitly, and very clearly, that all TT-projectors definedin Section 3, effectively filter correctly in the lattice the transverse and traceless d.o.f. oftwo-rank symmetric tensors. Next we will discuss how the new lattice-based projectors modify the GW spectra ascompared to the spectra obtained with the old continuum-based projector. As we showexplicitly in Figs. 2, 3 and 4, the spectra of GW in different models is only modified in thelarge-momenta region, i.e. in the ultraviolet (UV) tail. The infrared (IR) features at low– 14 –omenta, including the shape and amplitude of the spectra, and the position of the peak,are not modified by such UV distortion. The UV region corresponds precisely to thosemodes for which the GW spectral amplitude should be exponentially suppressed, if theGW spectrum is to be trusted. This is because only the IR modes are excited initially viaexponential instabilities during (p)reheating [22, 24–26], whereas the UV tail of the spectrasimply grows by scattering and turbulence [23], see for instance Ref. [3–5] for details. Asimilar behavior occurs also in the context of gauge fields [7, 27–31].That the overall shape and final amplitude of the GW spectra does not change muchwhen using the new lattice-based projectors, might seem at first sight surprising, given thefact that the degree of transversality changes several orders of magnitude when replacingthe continuum-based projector by the lattice-based ones. However, the lattice-momentum k eff ,i (˜ n ) from which the lattice-based projectors are made, only differ significantly from themomentum used to build up the continuum-based projector, k i = ˜ n i k IR , for the highest˜ n i ’s. For instance, Re { k ± eff ,i (˜ n ) } = k ,i (˜ n ) = sin(2 π ˜ n i /N ) /dx ≈ k IR ˜ n i + O (2 π ˜ n i /N ) aslong as ˜ n i /N < / π . Thus, as long as ˜ n i is not close to the UV boundary of the Fourier-lattice ˜ n i = ± N/
2, and since the P ij operators from which Λ ij,lm ’s are made are quadraticin k eff ,i , the difference between the continuum- and the lattice-based projectors can onlybe proportional to the difference | k eff ,i (˜ n ) | − | k IR ˜ n i | , i.e. | Λ cont ij,lm (˜ n ) − Λ latt ij,lm (˜ n ) | ∼ O ( | k eff (˜ n ) | − | k IR ˜ n | ) ∼ O (2 π | ˜ n | /N ) . (4.10)From this point of view, the GW spectra obtained with the lattice-based projectors arenot expected to differ much from the spectra obtained with the continuum-based projector.In particular, in the IR region, say | ˜ n | < N/
4, they should be pretty much coincident, thebetter the smaller | ˜ n | . Of course, this IR reasoning is still not enough to conclude thatthe GW spectra with continuum- and lattice-based projectors will not be very different.As small as it might be such difference, if spurious non-TT modes were incorrectly filteredin the continuum-based projector, the difference in amplitude of the two spectra could beenhanced during the dynamical evolution of the fields responsible for the GW production.That is why implementing in a lattice code the new lattice-based projectors is fundamentalin order to check whether it makes a difference or not. In Figs. 2, 3 and 4 we quantify thisaspect, showing the outcome of numerical simulations in which the TT d.o.f. are filteredout with the different projectors discussed in Section 3.These figures show very nicely the IR aspect just mentioned. Independently of theprojector, they all tend asymptotically to the same shape in the IR region, the GW spectracoincide in shape and amplitude at every step of the evolution, independently of the modelanalyzed. However, the differences in the UV region can be more noticeable, depending onthe model. But this also depends on the TT-projector used. For instance, in the left panelsof Figures 2, 3 and 4, we compare the GW amplitude as obtained with the continuum-based projector and with the real lattice-based one defined in Eq. (3.2). The Discrepanciesin all models considered are just a factor O (1) in the final amplitudes reached, as seen inthe left panels of all the Figures. Thus, we can conclude that the difference in the UV tailsare totally irrelevant in this case.Let us look now at the right hand side panels of the same Figures, where we compare theGW spectral amplitude between the spectra obtained with the continuum-based projector The GW production becomes inefficient in all these models of (p)reheating when the fields enter intothe turbulent regime, see [3–5] for details, so the spectrum amplitude stops growing and saturates to aconstant and final shape. – 15 – Ω G W k Real Lattice-based Projector (Section 3.1)Continuum-based Projector Ω G W k Complex Lattice-based Projector (Section 3.2)Continuum-based ProjectorGeneral Lattice-based Projector (Section 3.3)
Figure 2 . The amplitude of the evolution of the GW spectra as obtained with the differentprojectors defined in Section 3, during (p)reheating in the model λ φ , with λ = 10 − and nocoupling to other fields. Left:
Here we compare the spectra obtained with the continuum-basedprojector versus the one obtained with the real lattice-based projector defined in Eq. (3.2). One canappreciate that the IR part of both spectra are identical during the whole evolution, whereas theUV region shows some difference in amplitude. Such difference smooths out at the end, when thedifferent spectra saturate to their final amplitude. Once the GW production becomes inefficient, andthe spectra of GW does not grow further, the difference among the UV tails between the two spectraare simply a factor O (1). Right : Here we compare the spectra obtained with the continuum-basedprojector versus the ones obtained with the complex lattice-based projectors defined in Eq. (3.12)and Eq.(3.31). Again one appreciates that the IR part of both spectra are identical during the wholeevolution. The difference in the UV region when the spectra saturate to their final amplitude,however, is just a factor O (10), at least at some of the peaks of such UV tail. Note that suchdiscrepancy only appears in a region in k -space where the amplitude of the GW spectrum is alreadya factor O (10 − ) smaller than the maximum amplitude. In any case, the difference between thetwo complex lattice-based projectors is only a factor of order one. and with the complex lattice-based projectors defined in eqs. (3.12), (3.31). There we see amore noticeable difference in the UV regions. In particular, we observe that in large rangeof momenta, the amplitude of the spectra obtained with the lattice-based projectors canbe of the order O (10) bigger than the amplitude for the continuum-based projector.In the model λ φ with no other couplings to secondary fields , it is worth noting thatsuch discrepancy only appears at a region in k -space where the amplitude of the GWspectrum is already suppressed a factor O (10 − ) compared to the maximum amplitude.Therefore, in that respect, it is still a marginal discrepancy.In a chaotic model λ φ + g χ φ , with φ the inflaton and χ just another field, thedifference in the UV region is however more visible, see right panel of Figure 3. Thedifference is appreciable at scales in which the GW spectra begin to fall off exponentially,but it is not yet suppressed, as compared to the maximum. Such a difference in theamplitude of the GW spectra is appreciable by eye, however it only represents an overallshift of a factor O (1) in the location of the scale where the UV tail begins to fall. A similarsituation arises in the case of a Hybrid model λ (Φ − v ) + g Φ χ , with Φ a waterfallfield coupled to the inflaton χ . In this scenario, see right panel of Figure 4, we find again a In this model, the GW spectra retain the characteristic peaks of the scalar field power spectra, seeRef. [4] for more details. – 16 – Ω G W k Real Lattice-based Projector (Section 3.1)Continuum-based Projector Ω G W k Complex Lattice-based Projector (Section 3.2)Continuum-based ProjectorGeneral Lattice-based Projector (Section 3.3)
Figure 3 . The amplitude of the GW spectra obtained with the different projectors defined inSection 3. The plots correspond to the evolution of (p)reheating in a Chaotic coupled model λ φ + g χ φ , with λ/g = 120. As in figure 2, we can appreciate that the IR parts of the spectraare identical during the whole evolution, both in the left and right panels. The discrepancies in theUV region are only a factor O (1) when comparing the output obtained with the continuum-basedprojector versus the one with the lattice-based real projector of Eq. (3.2), see left panel. Whencomparing the GW spectra obtained with the continuum-based versus the complex lattice basedprojectors of eqs. (3.12), (3.31), we see that the difference is more pronounced, a factor O (10),around the scale at which the UV tail begins to fall exponentially. Nevertheless, the difference inthe UV between the two outputs for the different complex projectors, are still less than a factor 3. discrepancy between the lattice-based and the continuum-based projectors at scales wherethe spectra is about to fall off exponentially. Such discrepancy represents neverthelessagain, only a shift of a factor O (1) of the scale where the UV tail of the GW spectrumbegins to fall.In any case, both in the hybrid and coupled chaotic models, the difference in the GWspectral amplitude between the output obtained with the two complex lattice-based pro-jectors (3.12) and (3.31), amounts only to a factor O (1). So the discrepancies in amplitudeamong spectra obtained with the two complex projectors are again marginal.We can conclude that in all the different models considered, the final amplitude, thespectral shape of the IR region and the overall shape of the spectra, are not significantlymodified. There is no leak of scalar modes into the amplitude of the GW spectra even if thisis obtained with the continuum-based TT-projection, as done repeatedly in the literature.The GW spectra show some difference in the UV region when the spectra are extractedwith the complex lattice-based projectors (3.12) and (3.31). However, those projectorspresent certain caveats, as discussed in Section 3, precisely in the the UV region. Whencomparing the GW spectra obtained with the lattice-based real projector (3.2) and theContinuum-based one, the spectral difference in the UV region is indeed really marginal.In this latter case, the differences in amplitude in all models considered are just a factor O (1), and only show up in the UV region, where the large-momenta tail of the spectra isalready exponentially suppressed. – 17 – Ω G W k Real Lattice-based Projector (Section 3.1)Continuum-based Projector Ω G W k General Lattice-based Projector (Section 3.3)Complex Lattice-based Projector (Section 3.2)Continuum-based Projector
Figure 4 . The amplitude of the GW spectra obtained, for a Hybrid model, with the differentprojectors defined in Section 3. The plots correspond to the evolution after a period of Hybridinflation described by the model λ (Φ − v ) + g Φ χ , with v = 10 GeV and λ = 2 g . As inprevious Figs. 2 and 3, we can appreciate that the IR parts of the spectra are again identical duringthe whole evolution, both in the left and right panels. When comparing the output obtained withthe continuum-based versus lattice-based real projector of Eq. (3.2), see left panel, the discrepanciesin the UV region are only a factor of order one, so again unimportant. However, when comparing theGW spectra obtained with the continuum-based versus the complex lattice-based projectors (3.12),(3.31), the difference in the UV is more noticeable. The discrepancy is a factor O (10) around thescale at which the UV tail begins to fall exponentially. Nevertheless, the difference in the UVbetween the two outputs for the different complex projectors, are still less a factor three, similar tothe Chaotic coupled scenario. In this paper we try to respond to the criticisms made in Ref. [17] with respect to thevalidity of the projectors used in numerical simulations of gravitational wave production at(p)reheating. It was pointed out that the usual procedure employed in order to obtain theTransverse-Traceless part of metric perturbations in lattice simulations was inconsistentwith the fact that those fields live in the lattice and not in the continuum. It was claimedthat this could lead to a larger amplitude and the wrong shape for the gravitational wavespectra obtained in numerical simulations of (p)reheating, due to the leakage of scalarmodes into the tensor (GW) modes. In order to address this issue, we have developed aconsistent prescription in the lattice for extracting the TT part of the metric perturbations.We have defined a general complex TT projector based on lattice momenta, as well as areal projector in the lattice. All these projectors satisfy the required symmetry propertiesassociated with gravitational wave amplitudes.We then run specific numerical simulations of GW production at (p)reheating withthe implementation of the various projectors, and demonstrate explicitly that the GWspectra obtained with the old continuum-based TT projection only differ marginally inshape with respect to the new lattice-based projectors. Therefore, we have been able toanswer the criticisms of Ref. [17] by showing explicitly that the numerical results obtainedwith the lattice-based projectors do not change appreciably with respect to those with thecontinuum-based projector. We thus confirm that all previous results in the literature,concerning the spectra of GW coming from lattice simulations of (p)reheating, should betrusted to the extent to which such simulations are trusted (i.e. within lattice artifacts’– 18 –odel / projector continuum lattice real latt. complex latt. generalChaotic Mixed 7 . × − . × − . × − . × − Chaotic Pure 1 . × − . × − . × − . × − Hybrid 4 . × − . × − . × − . × − Table 1 . The total energy density in gravitational waves in units of the total energy density for thedifferent models and computed using the various projectors. Note that they differ by less than afew percent, except in the hybrid case that can reach 20% between the continuum and the complexlattice projectors. effects and time evolution). Introducing different lattice momenta in the TT-projector, itonly gives rise to differences in the (exponentially suppressed) spectral amplitudes in theUV, or at most to small shifts (or order unity) in the scale where the spectra begin to fallexponentially. The overall shape, frequency and total amplitude do not change.Finally, in order to quantify by a single number the deviations induced by the choice ofprojector, to assess the validity of the lattice projector approximation, we have computedthe fraction of the total energy density in GW to the total energy density available during(p)reheating. We have included those ratios in Table 1. Assuming that all the energy at theend of inflation went into radiation at reheating, then this fraction represents the fractionof GW to radiation today, since they both redshift equally during the subsequent evolutionof the universe. Therefore the quantity that appears in the table must be multiplied byΩ rad h = 3 . × − in order to give the observable quantity today, Ω GW h , which couldeventually be measured in gravitational wave observatories. Note that the entries in Tab. 1differ by less than a few percent, except in the hybrid case that can reach 20% betweenthe continuum and the complex lattice projectors. These numbers reinforce again the ideathat the requirement of using the lattice-based projectors does not invalidate the previousresults found in the literature on GW production in lattice simulations. Acknowledgments
DGF would like to express his gratitude to the Theoretical Physics Group at ImperialCollege London and to the Instituto de F´ısica Te´orica in Madrid, for the hospitality receivedduring spring/summer 2011, when this project was initiated. This work was supported atHelsinki by the Academy of Finland grant Ref. 131454. AR was supported by the STFCgrant ST/G000743/1, and DGF and AR by the Royal Society International Joint ProjectJP100273. We also acknowledge financial support from the Madrid Regional Government(CAM) under the program HEPHACOS S2009/ESP-1473-02, and MICINN under grantAYA2009-13936-C06-06. DGF and JGB participate in the Consolider-Ingenio 2010 PAU(CSD2007-00060), as well as in the European Union Marie Curie Network “UniverseNet”under contract MRTN-CT-2006-035863.
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