Scalar field dark matter in hybrid approach
aa r X i v : . [ g r- q c ] S e p Scalar Field Dark Matter in HybridApproach
Pavel Friedrich ∗ , Tomislav Prokopec † Institute for Theoretical Physics, Spinoza Institute and the Center for Extreme Matter andEmergent Phenomena (EMME Φ ),Utrecht University, Buys Ballot Building, Princetonplein 5, 3584 CC Utrecht, TheNetherlands Abstract
We develop a hybrid formalism suitable for modeling scalar field dark matter, in which thephase-space distribution associated to the real scalar field is modeled by statistical equal-time two-point functions and gravity is treated by two stochastic gravitational fields in thelongitudinal gauge (in this work we neglect vector and tensor gravitational perturbations).Inspired by the commonly used Newtonian Vlasov-Poisson system, we firstly identify a suit-able combination of equal-time two-point functions that defines the phase-space distributionassociated to the scalar field and then derive both a kinetic equation that contains relativisticscalar matter corrections as well as linear gravitational scalar field equations whose sourcescan be expressed in terms of a momentum integral over the phase-space distribution function.Our treatment generalizes the commonly used classical scalar field formalism, in that it allowsfor modeling of (dynamically generated) vorticity and perturbations in anisotropic stressesof the scalar field. It also allows for a systematic inclusion of relativistic and higher ordercorrections that may be used to distinguish different dark matter scenarios. We also provideinitial conditions for the statistical equal-time two-point functions of the matter scalar fieldin terms of gravitational potentials and the scale factor. ∗ Electronic address:[email protected] † Electronic address: [email protected] ontents
A.1 Wigner Tranformation of 2-Point Function Dynamics . . . . . . . . . . . . . 34A.2 Einstein Equations in Longitudinal Gauge with Scalar Perturbations . . . . . 41A.3 Energy-Momentum Conservation for Composite Fluids . . . . . . . . . . . . 46A.4 Continuity and Euler Equation for Real Scalar Field Fluid . . . . . . . . . . 48
The standard model of cosmology attributes roughly one third of the universes energy todark matter, a particle or field whose nature is mostly unknown except for the effect that itinteracts with gravity [1]. There has been success in studying large-scale structures of theuniverse by modeling dark matter as non-relativistic particles that can be described by apressureless fluid. Linear perturbation theory can be used up to the scale of non-linearity k > k nl ∼ . − to predict the distribution of galaxy clusters and the perturbationtheory may be used to study higher-order effects [2]. On the other hand, interest has recently[3] [4] been shown in the study of axion or fuzzy dark matter [5] [6] [7] [8] [9] [10] which inthe end is a real scalar field with a certain mass range minimally coupled to gravity with self-interaction terms playing a minor role. Common to most minimal scalar field dark models isthat the mass is much bigger than the Hubble rate. It has been studied in linear perturbationtheory in different gauges [11] [12]. The non-relativistic limit of the Klein-Gordon equation ofa classical scalar field yields the Schr¨odinger equation. By defining energy-density and fluid1elocity via the so-called Madelung transformation, one can reproduce non-relativistic, non-linear hydrodynamic equations in FLRW-space-time for real [3] and complex [13] classicalscalar field theories. From a quantum field theory point of view, we think of the classical fieldsentering these models as Bose condensates that are obtained by coherent quantum stateswhose one-point function defines the classical field. In view of the semi-classical Einsteinequations, it is natural to extent the analysis to the statistical limit of the full two-pointfunctions where the additional degrees of freedom can account for all features of a fluid ofmassive collisionless particles in the classical limit. In fact, it is the expectation value ofsquares of (non-composite) field operators at equal times that couples to the Einstein tensorin semi-classical gravity. Thus, we should think of these two-point functions as buildingblocks of the fluid. In the classical limit these equal-time two-point functions reduce to thestatistical or Hadamard two-point function. It is a priori not clear why they should reduceonly to the product of classical fields, i.e. expectation values of one field operator insertion.Despite that one has to argue on how such condensates are generated in a quadratic potentialin late-time cosmology, working only with classical fields cuts down degrees of freedom thatmight be important for cold dark matter models. We underpin the later point by derivingthat statistical two-point functions are in a gradient approximation related to phase-spacedensities whose position and momentum dependence is initially generic by means of theconnected piece of the two-point function, i.e. the part which does not reduced to a productof expectation values. This makes it clear that they contain more features of the scalar fieldfluid than the one-point functions or classical fields are able to describe. From the perspectiveof a classical particle that is coupled to gravity, the studies of phase-space dynamics areinevitable when a single-stream fluid Ansatz breaks down due to what is called shell-crossing.The kinetic theory underlying dark matter is summarized in the Vlasov equation [14], whichrepresents a phase-space description that does not break down in the non-linear regimesince there is no shell-crossing in phase-space. Phase-space densities and the correspondingVlasov equation have previously been derived by using the Wigner transformation of the non-relativistic Schr¨odinger-Poisson system [15] [16] [17] and also for a relativistic scalar field [18].Once again, only one-point functions have been considered and the richness of the connectedpart of the statistical two-point function is lost.In this paper, we want to put forward the discussion about real scalar field dark matterfrom the perspective of phase-space dynamics which is according to us still incomplete at themoment. We show that instead of using classical fields, the more natural objects are statisticaltwo-point functions which via the additional space-time-dependence can be used to derivea momentum-dependence as it occurs in kinetic theory. Integrating out this momentum2ependence still leaves us with a non-homogeneous space-time dependence that is inducedby the stochastic gravitational fields. Furthermore, two-point functions naturally arise inquantum field theory, whose broad apparatus might even be used to simplify non-linearcalculations once a mapping to observables in cosmology is established as we do in this paper.Defining phase-space densities from two-point functions is a known business in Minkowki-space [19], it is a generalization of non-relativistic Wigner transformation [20] to specialrelativity. The idea is to change the coordinates to a collective and difference coordinateand to Fourier-transform with respect to the difference coordinate to obtain a momentumdependence. However, there are few publications on a generalization of this idea to curvedspace-times. Two independent works in [21] and in [22] postulate off-shell curved space-time Wigner transformations in a mathematical complicated expansion by using geodesicsand Riemann normal coordinates, respectively. The transformation is done with respectto a coordinate-independent physical distance between the two points on the space-timemanifold. A similar approach has again been discussed by [23]. However, in this paper wemake use of a simpler transformation that allows us to write down exact equations in a one-step transformation. The idea is to think of the two-point function as an object that dependson one point of the space-time manifold and on another point that belongs to the tangentspace over that point on the space-time manifold. Consequently, the momentum is a variableof the cotangent space over that point on the space-time manifold. This approach was usedin [24] to define particle densities in an unperturbed FLRW-univserse where the authorsstarted with off-shell equations and projected them onto on-shell quantities via integration.A similar approach was already proposed in [25] for general space-times and its implicationwere studied for Fermionic systems, however, no on-shell projection was discussed. As far aswe know, none of the previous works makes an attempt to clearly derive a set of equationthat reduce in the classical limit to the Newtonian on-shell Vlasov-Poisson system that isused in kinetic theory of dark matter. We consider this as an important gap in the theory ofscalar field or axion dark matter and it is the task of our paper to close it. Once we have aclear pictures on how the dynamics of dark matter is embedded in quantum field theory oncurved space-time, we might discover more ways to calculate cosmological quantities in thenon-linear regime.We call the approach in this paper a hybrid approach for the reason that we start inprinciple from a quantum field theory for the real scalar field but do not properly integrateout the gravitational constraint. Thus, we approximate the self-interaction terms that wouldbe generated by this procedure in terms of the gravitational potentials treated as externalsources which are by means of the semi-classical Einstein equations related to the statistical3wo-point functions themselves. The source of stochasticity is in part in the quantum originof scalar field fluctuations and in part in the fact that making an initial Gaussian state
Ansatz neglects interactions of dark matter with other matter fields and with gravity, which in generalwill create higher order (non-Gaussian) correlations that are neglected (coarse-grained) in ourformalism.The paper is structured as follows. We start by deriving a dynamical system of on-shelltwo-point functions that is converted from a pure space-time dependence to a dependence onphase-space variables. We specialize to a scalar linearized longitudinal gauge without vectorperturbations and without gravitons. But we keep the gravitational slip (defined as thedifference between the two gravitational potentials), which induces high-order correctionsin the fluid dynamics of scalar field dark matter that have not been captured so far inthe one-point function approach. We derive Einstein’s equations in that gauge and rewritethe energy-momentum tensor as momentum integrals over two-point functions. This allowsus in turn to define scalar hydrodynamic variables like energy density, rest-mass densityand pressure. However, hydrodynamic variables containing the four-velocity can only bedefined as composite operators leaving space for anisotropy. We then consider a gradientexpansion by introducing a variety of perturbation parameters on top of the linearizationin the gravitational potentials and show that we indeed recover the generalization of thecontinuity and Euler equation in the FLRW-space-time. We also use the energy-momentumto identify even and odd phase-space densities which brings us finally to the derivation ofthe on-shell Vlasov equation by making use of the unintegrated dynamical equations for thestatistical two-point functions.In the hybrid approach that we put forward in this paper we have two types of two-point functions involved. We have a statistical two-point function (also called Hadamardfunction) which consists of the expectation value of anti-commutators of the scalar fieldoperators evaluated with respect to some initial density matrix. In our hybrid approach,this initial density matrix is taken to depend on the stochastic gravitational potentials asthey appear in the context of cosmological perturbation theory. This dependence makesthe Hadamard two-point function (which is a c-number) itself a stochastic quantity arisingfrom the initial conditions. Thus, we can now take the expectation value of the product ofHadamard two-point functions with respect to the stochastic initial conditions and integratedversions thereof will correspond for example to density-density correlators in cosmologicalperturbation theory. Unless, we do not make a clear distinction, we mean statistical orHadamard two-point functions whenever we speak generically of two-point functions.4
Phase-Space Distribution From 2-Point Functions
Let us start by writing down the microscopic theory that captures the fundamental dynamics.It is a real scalar quantum field theory that indirectly self-interacts via a minimal and semi-classical coupling to gravity.We work in units where c = 1 and write down the action for the system S [ φ, g µν ] = S g [ g µν ] + S m [ φ, g µν ] , (1)where S m [ φ, g µν ] = − Z d D x √− g (cid:20) g µν ∂ µ φ∂ ν φ + m ~ φ (cid:21) , (2)is the matter action and S g [ g µν ] = M P ~ Z d D x √− gR (3)is the classical gravity action with R being the Ricci scalar. We will work with a metric forlinearized scalar perturbations in Newtonian (longitudinal) gauge which is specified by thetwo gravitational potentials Φ G and Ψ G , g ( η, x i ) = − a ( η ) (cid:2) G ( η, x i ) (cid:3) , g ij ( η, x i ) = a ( η ) δ ij (cid:2) − G ( η, x i ) (cid:3) . (4)We drop all quadratic terms Φ G , Ψ G , Φ G · Ψ G as higher-order corrections from the verybeginning. We also drop vector and tensorial perturbations for simplicity although in generalwe expect them to be generated due to non-linear evolution. Inflation generates gravitationalpotentials that can be to a good approximation treated as classical stochastic fields that areat large redshifts approximated by a Gaussian distribution. We note that the metric (4)is particularly useful to study the Newtonian limit of general relativity. It generalizes thelongitudinal metric that has been used in the classical real scalar field theory approach todark matter in [3] by allowing for a non-zero gravitational slip that we define in D space-timedimensions as gravitational slip := Φ G − ( D − G . (5)The quantum theory in the operator formalism is specified by the time-evolution or Hamil-ton operator ˆ H which is a functional of the field operator ˆ φ and its canonical momentumfield operator ˆΠ. We work in the Heisenberg picture and the canonical momentum operatorevaluates to ˆΠ( x ) = a ( D − ( η ) h − Φ G ( η, x i ) − ( D − G ( η, x i ) i ˆ φ ′ ( η, x i ) , (6)5here x := (cid:0) η, x i (cid:1) and (cid:16) . (cid:17) ′ := ∂∂η (cid:16) . (cid:17) . (7)The field operators obey equal-time commutation relations h ˆ φ ( η, x i ) , ˆΠ( η, e x i ) i = i ~ δ D − ( x i − e x i ) , h ˆ φ ( η, x i ) , ˆ φ ( η, e x i ) i = 0 , h ˆΠ( η, x i ) , ˆΠ( η, e x i ) i = 0 . (8)Since we are working in semi-classical gravity the Hamiltonian ˆ H additionally depends onthe gravitational potentials Φ G , Ψ G that act as external, stochastic fields,ˆ H ( ˆ φ, ˆΠ , g µν ) ≡ Z d D − x ˆΠ ˆ φ ′ − Z d D − x ˆ L m h ˆ φ, ˆ φ ′ , g µν i = − Z d D − x ˆΠ √− gg + 12 Z d D − x √− g (cid:20) g ij ∂ i ˆ φ∂ j ˆ φ + m ~ ˆ φ (cid:21) . (9)Using the Heisenberg equations we find the following time evolution of the canonical opera-tors, ˆ φ ′ ( x ) = a − ( D − ( η ) h G ( x ) + ( D − G ( x ) i ˆΠ( x ) , (10)ˆΠ ′ ( x ) = a D − ( η ) δ ij ∂ i "h G ( x ) − ( D − G ( x ) i ∂ j ˆ φ ( x ) − m ~ a D ( η ) h G ( x ) − ( D − G ( x ) i ˆ φ ( x ) . (11)We stress that the constraint fields Φ G , Ψ G do not evolve independently. Thus, we are notfully fixing the gauge by integrating out the constraint fields. We do this in order to makethe connection to the Einstein-Vlasov system clearer. This means, at the same time that weare approximating scalar interactions that are induced via gravity by stochastic two-pointfunctions. However, examining the fully gauged fixed theory at the quantum level is plannedfor the future. One might object that the quantum field theory framework we presented so far is a com-pletely exaggerated tool to describe effects that arise in the classical treatment of late-timecosmology. However, this description has on the one hand the advantage of being based ona fundamental theory which permits a Lagrangian description and in which in our simplemodel contains one parameter, the scalar field mass m . On the other hand, it is related to thetypical classical non-relativistic particle description by imposing conditions that approximatea classical stochastic rather than a quantum description as well as a gradient expansion thatcontains relativistic corrections. We note that quantum path integrals generalize classical6tochastic path integrals where the quantum commutators (8) are replaced by Poisson brack-ets [26]. For us it is important to inherit the stochastic correlations of two-point functionsfrom the quantum field theory framework since late-time cosmology is a classical stochastictheory whose stochastic seeds are given by the gravitational potentials. That quantum ef-fects are potentially present in this approach is a completely negligible add-on rather than acrucial ingredient. Still, this perspective has the advantage that we always keep the bridgeto non-equilibrium quantum field theoretic techniques as for example the Schwinger-Keldyshformalism [27] [28] that might be useful when studying non-linear evolution. Our formalismis a hybrid formalism in the sense that we use a mixture of 2-PI formalism [29] for the scalarfield matter and 1-PI formalism for gravity and do not fully fix gravitational gauge in thesense that we do not fully solve (gravitational) constraints of the theory, but instead we leavethe gravitational potentials as external stochastic sources (keeping in mind that they areeventually fixed by the linear Einstein equations).The condition of being in the classical stochastic regime of a quantum field theory ratherthan in the quantum regime can be formulated as follows: the (classical) correlators thatcontain anti-commutators (and no time ordering) are much larger than the quantum cor-relators defined in terms of anti-commutators with or without time ordering (example ofwhich include the causal or spectral two-point function h [ ˆ φ ( x ) , ˆ φ ( e x )] i , retarded and advancedpropagators, etc.). We therefore assume that our two-point functions obey,2 F ( x ; ˜ x ) ≡ h{ ˆ φ ( x ) , ˆ φ ( e x ) }i ≫ (cid:12)(cid:12) h [ ˆ φ ( x ) , ˆ φ ( e x )] i (cid:12)(cid:12) , (12)where { ˆ φ ( x ) , ˆ φ ( e x ) } and [ ˆ φ ( x ) , ˆ φ ( e x )] denote anti-commutator and commutator operation, re-spectively. Rigorously speaking, the classicality condition (12) is never satisfied for all space-time points. By assuming (12) we are saying that we restrict ourselves to those space-timepoints where (12) is amply satisfied. Rather than rigorously going through a procedure thatwould achieve that in practice, here we just sketch how such a procedure can be exacted. Inthe case of interest for dark matter, the condition (12) will be met for sufficiently large spa-tial separations. One can make use of a suitable window (smearing) function, which projectsout of the full two-point function its classical part. When the complementary (‘quantum’)part of the two-point function is integrated out, one will generate local geometric divergentcontributions (that can be renormalized by adding suitable local geometric counterterms).Apart from renormalizing the Newton and cosmological constant to its observable values, theremaining geometric terms will have a negligible effect on the evolution of late time two-pointfunctions, and we shall neglect them here. The remaining infrared parts of the two-pointfunctions will satisfy the classicality condition (12).7o get a better feeling on what classicality really means, it is helpful to assume adi-abaticity with respect to gradient expansion (discussed in more detail below), in whichcase one can perform a Wigner transform with respect to the relative spatial coordinate, x i − ˜ x i , resulting in the statistical two-point function, F ( X i , p i , t, t ′ ), X i ≡ ( x i + ˜ x i ) / F ( X i , p i , t, t ′ )[ ∂ t ∂ t ′ F ( X i , p i , t, t ′ )] t ′ = t ≫ ( ~ / is satisfied, then one is in the classicalregime. More concretely, in the case at study we expect the classicality condition (12) tobe satisfied for two-point functions that are smeared on distances larger than the co-movingdistance corresponding to the end of inflation, which is of the order ∼ k ~ ∂ ~X · ∂ ~p k ≪ , k ~ ∂ η ∂ E k ≪ , (13)where the norm is to be understood in the sense that one derivative acts on one test function(such as a two-point function) and the other on another object (such as a gravitational poten-tial). Assuming that two-point functions vary on scales of momentum (energy) given by themomentum (energy), i.e. ~ ∂ E ∼ ~ /E ∼ ~ / ( mc ) ∼ λ C /c , and ~ ∂ ~p ∼ ~ / k ~p k ∼ ~ / ( mv ) ∼ λ dB ,where λ C and λ dB denote the Compton and de Broglie wavelength, respectively, the condi-tions (13) can be recast as, L ≫ λ dB , T ≫ λ C c , (14)where L and T represent the characteristic length and time scales over which gravitationalpotentials or two-point functions vary. L can be as small as the smallest large-scale structureswe are interested in (which is of the order ∼ kpc) which implies that T must be much largerthan the time light crosses about one mega-parsec which is about a million years or so(this estimate follows from the observation that λ dB ∼ λ C as v ∼ (10 − − − ) c ).To get a feeling on how good that approximation is, note that the inequalities are amplysatisfied for a dark matter whose mass is of the electroweak scale ∼ GeV. However, whenone considers ultra-light scalar such as in references [3] [4], the scalar mass is of the order m ∼ − − − eV, the Compton and de Broglie wave lengths are λ C ∼ − − − kpc, An alternative (and related) criterion for classicality of a state is given by the von Neumann entropy ofthe Gaussian part of the density matrix being much larger than one [30] [31]. The validity of this expansion depends on the initial density matrix which ought to be classical enough.The initial density matrix can for example be taken to be Gaussian, containing initial one-point functionsand connected two-point functions. In particular, without any coarse-graining only the connected part of thetwo-point function can satisfy the conditions of gradient expansion. dB ∼ − −
10 kpc, the quantities in (14) can become comparable for smallest scales ofinterest, and hence one expects significant higher order gradient corrections. One is typicallyinterested in modeling dark matter at an accuracy better than 1% (as it will be testedby upcoming observations), which then defines the order in gradient expansion that oneought to keep. The corrections of the gradient expansion can be subsumed by the followingperturbation parameters ε ~ ∼ ( ε k ∼ ~ ∂ X ma , ε k/p ∼ ~ ∂ X p ∼ ~ ∂ X ∂p , ε H ∼ ~ H ma , ε ∂η ∼ ~ ∂ η ma ) . (15)Next, there are relativistic corrections due to the relativistic nature of dark matter. Thefact that the energy is not equal to the rest energy we can fully capture in our formalismas long as we keep on-shell energy roughly speaking equal to its quasi-particle value, E = q m + p ph , where p ph = g ij p i p j denotes the physical momentum squared (for simplicity inhere we do not include all of the metric corrections). To study these corrections one cansystematically include them order by order if, ε p ∼ p com ma ≪ p com denotes the comoving momentum today. These corrections occur e.g. as relativis-tic corrections to the energy-momentum tensor whose most important components are energydensity and pressure which source the generalized Poisson-like equations for the gravitationalpotentials.Furthermore, there are relativistic corrections induced by the relativistic nature of thescalar field Klein-Gordon equations, and these appear as higher order time derivatives inthe Vlasov (or collisionless Boltzmann) equation. These corrections are small if the secondcondition in equations (13) and (14) is met and if k ∂ η k ≪ H , where H denotes the Hubbleexpansion rate in conformal time.Next, there are relativistic corrections arising from general relativity being different fromNewton’s gravity. These corrections occur as higher time derivative corrections to gravita-tional potentials and as the corrections induced by the Universe expansion. The latter aresuppressed by the Hubble rate H and they are small if, T ≪ a H . (17)Furthermore, since general relativity has more degrees of freedom than Newton’s grav-ity, there are general relativistic corrections expressed as a non-vanishing gravitational slip.Finally, we expect that as a result of non-linear interactions between matter and gravity,gravitational vector and tensor perturbations will be (dynamically) generated (even if they9re not present at the initial time). In this paper we neglect these type of perturbations, butthey can be easily included in our formalism by including them in the Ansatz for the metrictensor.Of course, there are also higher order gravitational perturbations. However, since on thelarge scales we are interested in gravitational potentials do not grow much beyond their initialvalue, ε ∼ Φ G , Ψ G ∼ − ≪ , (18)we can safely neglect terms of the form Φ G and Ψ G ; higher order vector and tensor per-turbations can be also neglected since vectors and tensors remain smaller than gravitationalscalars throughout the evolution. We will also encounter the parameter ε H/k ∼ H ∂ − X , (19)that controls whether we are on sub- or superhorizon scales. The concept of the Wigner transformation (see for example [19] or [32]) was introduced to ex-tract phase-space distributions and its Boltzmann equation from particle wavefunctions, thengeneralized to field theory in Minkowski space-time and even later to field theory in curvedspace-time to yield a covariant Boltzmann or Vlasov equation. However, the generalizationof the Wigner transformation to arbitrary curved space-times must still be considered as anactive research field since there are as far as we know merely four major contributions to thisarea [21] [22] [23] [25] which agree only in the limit where ~ goes to zero. Apart from theproposal by [25], all approaches are based on perturbative expressions. On the other hand,all of these papers cover almost entirely off-shell phase-space distribution f ( X µ , p ν ) in thesense that the momentum conjugate to the time difference ∆ η in the two-point function, p ,is still an independent variable that needs to be put on shell by integrating it out since thestarting point for the Wigner transformation is a non-equal-time two-point function, ∆ η = 0.As it was to our knowledge first pointed out by [33] for electrodynamics in Minkowski space-times, going on-shell requires more than one moment in p space, i.e. R dp π ~ p n f ( X µ , p ν ).This approach of taking several moments and relating them has been applied to homoge-neous cosmological backgrounds by [24] to define a particle number density. Our goal is toobtain candidates for on-shell phase-space distributions f ( X µ , p i ) and their dynamics for thenon-homogeneous linearized longitudinal metric (4) we provided in the beginning. We seekthem by computing the dynamics of two-point functions on-shell or at equal-times in the op-erator formalism and then performing a ( D − D − F (cid:0) η, x i , e x i (cid:1) := D ˆ φ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1) E = Tr h ˆ ρ ini (cid:2) ˆ φ, ˆΠ , Φ G , Ψ G (cid:3) ˆ φ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1)i , (20) F (cid:0) η, x i , e x i (cid:1) := D ˆΠ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1) E = Tr h ˆ ρ ini (cid:2) ˆ φ, ˆΠ , Φ G , Ψ G (cid:3) ˆΠ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1)i , (21) F (cid:0) η, x i , e x i (cid:1) := D ˆ φ (cid:0) η, x i (cid:1) ˆΠ (cid:0) η, e x i (cid:1) E = Tr h ˆ ρ ini (cid:2) ˆ φ, ˆΠ , Φ G , Ψ G (cid:3) ˆ φ (cid:0) η, x i (cid:1) ˆΠ (cid:0) η, e x i (cid:1)i , (22) F (cid:0) η, x i , e x i (cid:1) := D ˆΠ (cid:0) η, x i (cid:1) ˆΠ (cid:0) η, e x i (cid:1) E = Tr h ˆ ρ ini (cid:2) ˆ φ, ˆΠ , Φ G , Ψ G (cid:3) ˆΠ (cid:0) η, x i (cid:1) ˆΠ (cid:0) η, e x i (cid:1)i , (23)where the expectation values are taken with respect to some initial density matrix ˆ ρ ini whichfunctionally depends on the operators ˆ φ and ˆΠ at some initial time η ini . Note that in thecontext of cosmology the initial density matrix ˆ ρ ini depends also functionally on the stochasticgravitational potentials Φ G , Ψ G at this initial time η ini , and in its general form it allows forimplementation of effects of coherent states, squeezing and state mixing. In this way, two-point functions can be stochastic quantities, F ( η, x i , e x i ) = h F i (Φ G , Ψ G ) ( η, x i , e x i ) := (cid:28) D ˆ φ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1) E ˆ ρ (cid:29) (Φ G , Ψ G ) := Z D Φ G D Ψ G P (cid:2) Φ G , Ψ G (cid:3) Tr h ˆ ρ ini (cid:2) ˆ φ, ˆΠ , Φ G , Ψ G (cid:3) ˆ φ (cid:0) η, x i (cid:1) ˆ φ (cid:0) η, e x i (cid:1)i , (24)where P is a probability distribution for the gravitational potentials. The reason to introducethis formalism lies in its application to cosmology in the sense that we want to bridge agap from semi-classical quantum field theory to cosmological perturbation theory. Thus,Φ G and Ψ G are stochastic, homogeneously distributed fields that evolve into non-Gaussianfields due to the evolution of large-scale structures. We want to think of this model moreas a conceptional test case on how to relate full quantum microscopic theories to modelsin cosmological perturbation theory as for example the cold dark matter model. Once wefind that this is a fruitful Ansatz , we will provide generalizations for arbitrary metrics andeven wave the semi-classical approach by integrating out the gravitational constraint fieldsΦ G , Ψ G , which at the moment act as approximate self-interactions of the scalar field theory We remark, that this might be closely related to the stochastic gravity framework proposed in [34]although we did not investigate this further. For us, the stochasticity of two-point functions is more anad-hoc
Ansatz that turns out to be very convenient in relation to cosmological perturbation theory. D ( η, x i ) := a D − ( η ) δ ij "h ∂ i Φ G i ( η, x i ) − ( D − h ∂ i Ψ G i ( η, x i ) ∂∂x j + a D − ( η ) h G ( η, x i ) − ( D − G ( η, x i ) i ∆ x − m ~ a D ( η ) h G ( η, x i ) − ( D − G ( η, x i ) i , (25)where we used the Laplace operator ∆ which on conformally flat cosmological spaces equals δ ij ∂ i ∂ j . We also define the following function as a shorthand h ( η, x i ) := a − ( D − ( η ) h G ( η, x i ) + ( D + 1)Ψ G ( η, x i ) i . (26)Then, based on the Hamilton’s equation for the canonical operators (10) and (11), we getthe following system of equations F ′ ( η, x i , e x i ) = h ( η, x i ) F ( η, x i , e x i ) + F ( η, x i , e x i ) h ( η, e x i ) , (27) F ′ ( η, x i , e x i ) = D ( η, x i ) F ( η, x i , e x i ) + F ( η, x i , e x i ) h ( η, e x i ) , (28) F ′ ( η, x i , e x i ) = D ( η, e x i ) F ( η, x i , e x i ) + h ( η, x i ) F ( η, x i , e x i ) , (29) F ′ ( η, x i , e x i ) = D ( η, x i ) F ( η, x i , e x i ) + D ( η, e x i ) F ( η, x i , e x i ) . (30)This is a system of four first-order differential equations with four independent initial con-ditions. However, we have to keep in mind that these two-point functions obey certainsymmetry properties and we realize by combining F and F that we can specify threesymmetric functions and one anti-symmetric function as initial conditions. We remark, thatthis system of equations closes in the sense that we need no information about higher n-pointfunctions. This is due to the following reasons: firstly, we neglected manifest self-interactionsof the scalar field (e.g. ∼ λφ ) that are not due to gravity, secondly, we approximate theself-interactions that are induced via gravity. These interactions are non-local in space butlocal in time via an inversion of the generalized Poisson equation. It means that the scalar12eld couples in this approximation only to its two-point functions since the gravitational po-tentials Φ G , Ψ G are - via the semi-classical Einstein equations - entirely expressible in termsof the scalar field two-point functions. We call this the hybrid approach. Thirdly, the scalarfield does not interact with dynamical part of gravity, the gravitons, since we put them tozero by hand as an assumed negligible effect.Let us continue to manipulate the system of equations (27) to (30) by switching to col-lective and difference coordinates for the spatial parts, X i := x i + e x i , r i := x i − e x i . (31)We define the Wigner transform with respect to covariant momenta p i and its zeroth momentdenoted by a bar as F ∗ ( η, X i , p i ) := Z d D − r e − i ~ p i r i F ∗ ( η, X i , r i ) , (32) F ∗ ( η, X i ) := Z d D − p (2 π ~ ) D − F ∗ ( η, X i , p i ) . (33)This definition is our equal-time, thus on-shell, version of the several curved space-time gen-eralizations of the Minkowski space-time Wigner transformation of course for the our specificchoice of a longitudinal linearized metric without gravitons. It is a direct generalization of the Ansatz in [24] from the homogeneous FLRW-space-time to its non-homogeneous perturbedform. This definition identifies the time coordinate η and spatial collective coordinates X i asa single point on the curved space-time manifold, whereas the spatial difference coordinates r i and the momenta p i belong to the tangent and cotangent space, respectively, that is associ-ated to that point. We neither make use of any geodesic expansion nor do we use Riemanniancoordinates. This implies also that our equations are exact apart from the linearization inthe gravitational potentials. We also would like to mention that the on-shell operator for-malism for curved space-times we are using resolves the problem of perturbatively solvingthe off-shell constraint equation which always accompanies the off-shell Vlasov equation byproviding a manifest closure for on-shell correlators [21] [22] [23] [25].It will turn out that upon Wigner transforming the equations (27) to (30), two othercorrelators are much more useful than F and F . Thus, we define the following combinationof equal-time two-point functions F + ( η, x i , e x i ) := 12 h F ( η, x i , e x i ) + F ( η, x i , e x i ) i (34)= 14 D n ˆΠ( η, x i ) , ˆ φ ( η, e x i ) o + n ˆ φ ( η, x i ) , ˆΠ( η, e x i ) o E ,F − ( η, x i , e x i ) := i h F ( η, x i , e x i ) − F ( η, x i , e x i ) i − ~ δ D − ( x i − e x i ) (35)= i D n ˆΠ( η, x i ) , ˆ φ ( η, e x i ) o − n ˆΠ( η, e x i ) , ˆ φ ( η, x i ) o E , (cid:8) . , . (cid:9) denotes the anti-commutator. The calculation of transforming the two-pointfunction dynamics into Wigner space is shown in appendix A.1. We remark that this cal-culation is exact up to the linearization in the gravitational potentials. Since we want toidentify phase-space distributions we have to treat the problem by utilizing the gradient ap-proximation . Therefore, we consider again the perturbation parameters in (15) to (18) andsolve perturbatively for F + and F which are determined in terms of F and F − , F + = ( h − Φ G − ( D − G i a D − F ′ − ~ ∂∂X i h Φ G + ( D − G i ∂∂p i F − ) × ( O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1)) , (36) F = a D − ( h a D − F ′ i ′ − " ~ ∂∂X i h Φ G + ( D − G i ∂∂p i F − ′ − ∆ X h a D − F i + p ~ h − D − G ih a D − F i + m ~ a h − D − G ih a D − F i) × ( O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1)) , (37)where p := δ ij p i p j . (38)The dynamics of F and F − is given by the following coupled equations, F ′− = ( ~ ∂∂X k h Φ G + Ψ G i ∂∂p k " p h a D − F i − p ~ · ∂ X h Φ G − ( D − G ih a D − F i − h G − ( D − G i p k ~ ∂∂X k h a D − F i + m a ~ ∂∂X i Φ G ∂∂p i h a D − F i) × ( O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1)) , (39) ( " a D − F ′ ′′ + ( D − H " a D − F ′ ′ + p ~ h − D − G i ′ h a D − F i + " m ~ a h − D − G i ′ h a D − F i + 2 m ~ a h − D − G ih a D − F i ′ − ∆ X h a D − F i ′ + 2 p ~ h − D − G ih a D − F i ′ − p ~ ∂∂X k h Φ G + Ψ G i ∂∂p k F − +2 h G − ( D − G i p ~ · ∂ X F − − m a ~ ∂∂X i Φ G ∂∂p i F − ) × ( O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1)) = 0 . (40)14e conclude that to this order in the gradient approximation, we still keep all degrees offreedom that were contained on the original first-order system (27) to (30). Dropping thethird-order time derivative would leave us with the degrees of freedom of a symmetric and ananti-symmetric function. Before we try to recover a Vlasov equation from these equations letus pause a bit and make it clear how this equation reflects the difference between products ofone-point functions and connected two-point functions. We will also realize that the higher-order time-derivatives correspond to oscillatory degrees of freedom. For simplicity, we set D = 4 and focus on the homogeneous part of equation (40) in the large mass limit ( p ≪ m ),where we denote the homogeneous approximation of F by F hom00 ,12 h a (cid:2) F hom00 (cid:3) ′ i ′′ + H h a (cid:2) F hom00 (cid:3) ′ i ′ + 2 H m ~ a h a F hom00 i + 2 m ~ a h a F hom00 i ′ ≈ . (41)Expanding this equation, we arrive at h a F hom00 i ′′′ − H h a F hom00 i ′′ + h m ~ a − H − H ′ ih a F hom00 i ′ + 3 h H + HH ′ − H ′′ ih a F hom00 i ≈ . (42)In order to make progress, we also have to provide the Einstein equations, which we derivein appendix A.2 in equations (194) to (195). The spatially homogeneous equations withneglected momenta p ≪ m read in D = 4 dimensions (all terms p m − are dropped), − H ′ − H ≈ ~ M P ( a − F hom11 − m a ~ F hom00 ) , (43)3 H ≈ ~ M P ( a − F hom11 + a m ~ F hom00 ) , (44)and from (37), we have F hom11 ≈ a h a (cid:2) F hom00 (cid:3) ′ i ′ + m ~ a F hom00 . (45)The equations (42) to (45) admit three independent solutions. We can guess them quicklyby noting once more that F hom00 is constructed out of one-point functions and a connectedpiece F hom00 = h ˆ φ i hom h ˆ φ i hom + h ˆ φ ˆ φ i homconnected . (46)In the limit H ≪ ~ − ma , the solutions for the one-point functions are through the Klein-Gordon equations approximately given by h ˆ φ i hom, sol 1 ≈ a − / cos (cid:16) Z dη ma (cid:17) , h ˆ φ i hom, sol 2 ≈ a − / sin (cid:16) Z dη ma (cid:17) . (47)15ad we used only one-point functions h ˆ φ i to construct F hom00 , our analysis would be completeat this stage since we can only impose two initial conditions for h ˆ φ i and they would completelydetermine F hom00 which in this case has always an oscillatory contribution. However, let usforget about the one-point functions and focus on the connected part of F hom00 . We see thatthe following functions are two independent solutions to (42), F hom, sol 100 ≈ a − cos (cid:16) Z dη ma (cid:17) , F hom, sol 200 ≈ a − sin (cid:16) Z dη ma (cid:17) , (48) F hom, sol 111 ≈ m ~ a sin (cid:16) Z dη ma (cid:17) , F hom, sol 211 ≈ m ~ a cos (cid:16) Z dη ma (cid:17) . (49)The, corresponding solutions for the Hubble rate are given by the following leading orderterms h H a i sol 1,2 ≈ const , (50) h H ′ + H i sol 1,2 ≈ ± H × cos (cid:16) Z dη ma (cid:17) , (51) h H ′′ + HH ′ i sol 1,2 ≈ ∓ ma H sin (cid:16) Z dη ma (cid:17) . (52)We than choose a linear combination of these solutions which is not oscillatory and can onlybe provided by means of the connected part of F hom00 . It is to leading order simply given by F hom, non-osc00 = 12 (cid:2) F hom, sol 100 + F hom, sol 200 (cid:3) ≈ a − , F hom, non-osc11 ≈ m ~ a . (53)The, corresponding solutions for the Hubble rate are given by the following leading orderterms h H a i non-osc ≈ const , (54) h H ′ + H i non-osc ≈ . (55)We of course get this solution by dropping all higher-order time derivatives on F hom00 in thelimit H ≪ ~ − ma . To summarize, we have shown that – by means of the connected part – thetwo-point function formalism allows to overcome the oscillatory behavior of time derivativesof the Hubble rate, and equivalently of pressure, without any averaging procedure. It is inthis respect significantly richer in comparison to the approach based on classical real scalarfields. Beeing ignorant about any p dependence for the moment, the initial density matrix for homogeneoustwo-point functions contains five initial conditions (see e.g. [26]): the one-point functions h ˆ φ i and h ˆΠ i , andthe connected parts of the two-point functions F , F and F + or equivalently F , F ′ and F ′′ . The fourthcondition for the connected part of F − is trivially satisfied in the homogeneous case. Generalized On-Shell Vlasov Equation
We now compare the real scalar field energy-momentum tensor expressed in terms of scalarfield two-point functions (see (202) to (206) in appendix A.2 ) with a general energy-momentum tensor in kinetic theory. This will allow us to identify phase-space distributionsbased on the scalar field. The phase-space distribution f cl of classical collisionless particlesin general relativity obeys the Vlasov equation [35] " ∂∂η + p icl p cl ∂∂X i + Γ αiβ p clα p βcl p cl ∂∂p icl f cl ( η, X j , p clk ) = 0 . (56)The energy-momentum tensor in kinetic theory is then given by T kin µν ( η, X i ) = Z d D − p cl h γ − / p clµ p clν E cl i ( η, X i , p cli ) f cl ( η, X i , p cli ) . (57)Here, the quantity γ is the determinant of the spatial metric. The particle energy E cl andthe temporal momentum p cl are related to the on-shell condition p clµ p µcl = − m cl , (58)which gives in longitudinal gauge p cl = g q m cl + g ij p cli p clj , g i = 0 , (59) E cl = −| g | / p cl . (60)Of course we want to identify similar quantities through the covariant Wigner momenta p i .By using a tilde from now on, we want to clearly distinguish between the covariant Wignermomentum p i , which is an integration variable and derived quantities that are related to itvia the metric, E ( η, X i , p i ) = h m + p a ( η ) i / h p m a ( η ) + p Ψ G ( η, X i ) i , (61) e p ( η, X i , p i ) = a − ( η ) (cid:2) − Φ G ( η, X i ) (cid:3) E ( η, X i , p i ) , (62) e p ( η, X i , p i ) = − a ( η ) (cid:2) G ( η, X i ) (cid:3)e p ( η, X i , p i ) , (63) e p k ( η, X i , p i ) = a − ( η ) (cid:2) G ( η, X i ) (cid:3) δ ki p i . (64)In particular, we emphasize that there is no independent integration variable p which isencountered in off-shell Wigner transformations, we have only the on-shell quantity e p ( p i ).17s we derive in appendix A.2, the 00- and 0 i -components of the real scalar field energy-momentum tensor are to leading order in ~ given by (see equations (202) and (203)) D ˆ T E ( η, X i ) = T ( η, X i ) = " m a ~ h G ( η, X i ) i F ( η, X i )+ h G ( η, X i ) + 2Ψ G ( η, X i ) i Z d D − p (2 π ~ ) D − p ~ F ( η, X i , p i ) × " O (cid:0) ε ~ (cid:1) , (65)and D ˆ T i E ( η, X i ) = T i ( η, X i ) = − a − ( D − h G ( η, X i ) + ( D − G ( η, X i ) i × Z d D − p (2 π ~ ) D − p i ~ F − ( η, X i , p i ) × " O (cid:0) ε ~ (cid:1) . (66)We realize that according to equation (65) a phase-space density candidate in the classicalparticle limit would be given by f evencl −→ f even φ := Eγ / (2 π ~ ) D − F ~ = ( m a + p ) / (2 π ~ ) D − h − ( D − G + p m a + p Ψ G i a D − F ~ . (67)However, when looking at the T i equation (66), we would rather come to the conclusion thatthe classical phase-space-density should be given by f oddcl −→ f odd φ := 1(2 π ~ ) D − F − ~ . (68)In equations (157) and (158) in appendix A.1 we summarize that according to their funda-mental definitions the two-point function F is of even parity in p i whereas the two-pointfunction F − is an odd parity function in p i . Thus, up to a rescaling and corrections in ourperturbation parameters, the quantity F seems to play the role of the phase-space-densityfor even moments and F − seems to play the role of the phase-space-density for odd momentsin p i . Based on the identifications in the previous paragraph, we rewrite (39) and (40) in terms ofthe definitions (67) and (68). We find "e p ∂∂η f odd φ + e p k ∂∂X k f even φ − e p i p i ∂∂X k h Φ G + Ψ G i ∂∂p k f even φ − m ∂∂X k Φ G ∂∂p k f even φ × " O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1) = 0 , (69)18 e p ∂∂η f even φ + e p k ∂∂X k f odd φ − e p i p i ∂∂X k h Φ G + Ψ G i ∂∂p k f odd φ − m ∂∂X k Φ G ∂∂p k f odd φ + ~ a − D − " a D − ∂∂η " a − ( D − ( m a + p ) − / f even φ − ~ a − ∆ X ∂∂η h ( m a + p ) − / f even φ i × " O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1) = 0 . (70)We emphasize again that e p is on-shell. We set f φ := f even φ + f odd φ = (2 π ~ ) − ( D − h Eγ / F ~ + F − ~ i , (72)and find "he p ∂∂η + e p k ∂∂X k − e p i p i ∂∂X k (cid:2) Φ G + Ψ G (cid:3) ∂∂p k − m ∂∂X k Φ G ∂∂p k i f φ + ~ a − D − " a D − ∂∂η " a − ( D − ( m a + p ) − / f even φ − ~ a − ∆ X ∂∂η h ( m a + p ) − / f even φ i × " O (cid:0) ε ~ · ε (cid:1) + O (cid:0) ε ~ (cid:1) = 0 . (73)Equation (73) is the main result of this paper. It tells us that we can obtain a corrected on-shell Vlasov equation from statistical two-point functions of a scalar field theory. It includesa third-order time derivative such that we keep all degrees of freedom from the initial first-order system (27) to (30) originating from the relativistic Klein-Gordon equation. However,dropping these third-order time derivatives as small corrections we are left with the degrees offreedom of a one-particle phase-space distribution. We note, that dropping these third-ordertime derivatives can be justified by either using certain initial conditions in the case that F is given only in terms of the connected two-point functions (concretely F + ∼ H F ). In thecase where F is only given by a product of oscillatory one-point functions, we can drop thosecontributions only after an averaging procedure. Apart from the third-order time derivativesthe generalized Vlasov equation (73) contains corrections in the gradient expansion. Droppingthe later, we find the same form as for the classical, collisionless on-shell Vlasov equation givenin (56). We note that since we used a spin zero field we expect corrections due to non-trivialspin in other quantum field theoretical settings. We also note that the degrees of freedom of Note, that we could also have considered f time-rev φ := f even φ − f odd φ , (71)which yields phase-space dynamics for reversed momenta or equally for reversed times. However, the equationfor the time-reversed density amounts only to a flip of the sign of the momentum p i and thus yields no newinformation. f φ and why we think that it can model a general fluid with gravitational inter-actions. First, we note that it is no surprise that the definition (72) yields a generalizationof the Vlasov equation since it is related to the Wigner transformation of the Schr¨odingeroperators ˆ ψ in the non-relativistic limit. We see this by writingˆ φ ( η, x i ) ∼ ˆ ψ ( η, x i ) exp h − i m ~ Z η d e η a ( e η ) i + ˆ ψ † ( η, x i ) exp h + i m ~ Z η d e η a ( e η ) i , (74)and after some manipulations we find f φ ( η, X i , p i ) ∼ a D − Z d D − r e − i ~ r i p i D ˆ ψ ( η, X k + r k /
2) ˆ ψ † ( η, X k − r k / E × " O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + oscillatory terms . (75)20e already remarked in section 2.3 that the oscillatory terms can be removed by an appro-priate initial density matrix. Thus, we rediscover in the large mass limit the definition of thenon-relativistic Wigner quasi-probability distribution based on an initial density matrix [20].This is another strong hint that our on-shell approach to construct a phase-space distribution f φ has a classical interpretation provided the classically condition and the gradient expansionwe discussed in 2.2 apply. This implies a suitable choice for the initial density matrix, inother words the state it represents has to be classical enough. We can also formulate theapproximate equivalence between a classical one-particle phase-space model and the classicallimit of the real scalar quantum field theory in the following way: since the dynamics for thescalar field phase-space distribution f φ and the classical distribution f cl agree on length andtime-scales that are associated to the classical limit, the difference between the two quanti-ties is encoded in the possibility to formulate arbitrary initial conditions. For any smoothclassical phase-space distribution we can write f cl ( η ini , X i , p i ) = Z d D − r e ir i p i f cl ( η ini , X k , r k )= Z d D − ( x − y ) e i ( x − y ) i p i e f cl ( η ini , x k , y k ) , (76)which in particular means that any moment can be written as f ( k ,...,k n ) cl ( η ini , X i ) = Z d D − p (2 π ~ ) D − p k ...p k n f cl ( η ini , X i , p i )= ∂ n ∂ ( x − y ) k ...∂ ( x − y ) k n e f cl ( η ini , x i , y i ) (cid:12)(cid:12)(cid:12) ( x − y ) i =0 . (77)From this we conclude that an arbitrary smooth, classical, one-particle phase-space distribu-tion is initially specified by an arbitrary function in two spatial coordinates. However, we canalways provide such a function based on a Gaussian initial density matrix for the quantumtheory which is encoded in the connected part of the two-point function, h ˆ ψ ( η ini , x k ) ˆ ψ ∗ ( η ini , y k ) i = h ˆ ψ ( η ini , x k ) ih ˆ ψ ∗ ( η ini , y k ) i + h ˆ ψ ( η ini , x k ) ˆ ψ ∗ ( η ini , y k ) i connected . (78)Note that the product of two one-point functions is not general enough to cover an arbitraryfunction of two arguments, so we really need the connected term. We can provide similararguments for the fully relativistic scalar field theory by splitting the classical distributioninto even and odd parts whose arbitrary initial conditions can always be specified by providingthe initial connected parts of the two-point functions F and F − as well as F + and F tofix oscillatory behavior. It is worth pointing out that the formalism in which one uses a single particle wave function resulting " ∂∂η + p i ma ( η ) ∂∂X i − ma ( η ) ∂∂X i Φ G ( η, X i ) ∂∂p i f φ ( η, X i , p i ) × " O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε (cid:17) = 0 . (79)The last equation is used for a collisionless gas in the context of cold dark matter as forexample in [2] or [38] and is simply the non-relativistic limit of (56), " ∂∂η + p cli m cl a ( η ) ∂∂X i − m cl a ( η ) ∂∂X i Φ G ( η, X i ) ∂∂p cli f cl ( η, X j , p clk ) = 0 . (80)We thus conclude that collisionless dark matter obeying a smooth phase-space distributioncan always be mimicked by a real scalar field theory based on scales where the mass dominates.Initial non-trivial moments of the phase-space distribution can be provided by non-trivialinitial density matrices for the connected parts of the two-point functions. Taking momentsof equation (79) shows that this generates in principle an infinite hierarchy of moments.Apart from the evolution of the phase-space distribution we also have to specify theEinstein equations that determine the gravitational potentials. The Einstein equations takea particularly convenient form if we write them in terms of hydrodynamic quantities. Toidentify those is the goal of the next section. from the evolution of a certain class of initial states may possess no classical limit in the sense that thephase-space distribution (Wigner) function can exhibit rapid (space and/or time) oscillations in the limitwhen ~ →
0, thus invalidating the gradient expansion. Since the Schr¨odinger equation is the non-relativisticlimit of the Klein-Gordon equation this argument could in principle also apply to our scalar field phase-spacedistribution f φ . However, the question whether a spatial gradient expansion does apply or not is tied to thespecification of the initial density matrix: it ought to be such that it yields two-point functions that satisfythe classicality criteria spelled out in section 2.2. Hydrodynamics Based on Two-Point Functions
A perfect fluid description is suitable for early (linear) evolution of cold dark matter. However,it stops being correct in the non-linear regime, in which gravitational slip, vorticity andanisotropic stresses (as well as the corresponding gravitational perturbations) get generatedeven if they were not present initially. The question we want to address in this subsectionis whether those non-perfect fluid components can generically be modeled with the scalarfield two-point function approach we present in this paper. All two-point functions usedto generate different components of the energy-momentum tensor discussed in this sectionare assumed to arise from suitably smeared (classical) two-point functions (a more detaileddiscussion on this important point can be found in section 2.2). In the above section weworked out phase-space distributions based on real scalar field two-point functions and arguedthat a non-trivial initial density matrix can give rise to a non-trivial hierarchy of moments.We now work out how this is reflected on the level of hydrodynamic quantities. In particularwe argue that although the energy-momentum tensor of the real scalar field theory has the apparent form of the energy-momentum tensor of a perfect fluid it does not correspond toone D ∂ µ ˆ φ∂ ν ˆ φ E − g µν h D ∂ α ˆ φ∂ α ˆ φ E + m ~ D ˆ φ E i = h T µν i 6 = T pf µν = ∂ µ φ cl ∂ ν φ cl − g µν h ∂ α φ cl ∂ α φ cl + m ~ φ i , (85)where φ cl := D ˆ φ E . (86)Thus, we will in general not speak of a perfect fluid in the two-point function approach. Inour scalar field model the fundamental reason is that the statistical two-point functions split It is well known (see for example [39]), that the energy-momentum tensor of a classical real scalar fieldtheory has the form of a perfect fluid T cl µν = ( e cl + P cl ) u cl µ u cl ν + g µν P cl , (81)with the following classical energy density e pf , pressure P pf and four-velocity u pf µ , e cl = − g µν ∂ µ φ cl ∂ ν φ cl + 12 m φ , (82) P cl = − g µν ∂ µ φ cl ∂ ν φ cl − m φ , (83) u µ cl = − h e cl + P cl i − / g µν ∂ ν φ cl . (84)This classical viewpoint, based on one-point functions, has been exploited linearly (see e.g. [40] or [41]) indifferent gauges and non-lineary for real and complex classical fields in the longitudinal gauge [3] [13]. D ˆ φ ( x ) ˆ φ ( y ) E = φ cl ( x ) φ cl ( y ) + D ˆ φ ( x ) ˆ φ ( y ) E connected , (87)where the connected piece can initially be an arbitrary function of x i and y i in position spaceor equivalently of X i and p i in Wigner space. The reducible piece is given by the productof one-point functions. The question whether the one-point function or the connected two-point function or both contribute to the whole two-point function depends on the dark matterproduction mechanism which is model dependent. Here we focus our attention mainly onthe connected piece when discussing statistical two-point functions of the scalar matter fieldbecause the scalar field in our model (2) does not couple linearly to external sources, suchthat one-point functions are generally absent (unless they are imposed as the initial conditionas it is e.g. done for the inflaton).In order to make contact to a hydrodynamic description, we define the following energydensity e and apparent pressure P analogously to the one-point function approach, e := − g µν D ∂ µ ˆ φ∂ ν ˆ φ E + 12 m D ˆ φ E , (88) P := − g µν D ∂ µ ˆ φ∂ ν ˆ φ E − m D ˆ φ E , (89)as well as a composite quantity containing a notion of four-velocity, h ( e + P ) u µ u ν i com := D ∂ µ ˆ φ∂ ν ˆ φ E = ∂ µ φ cl ∂ ν φ cl + D ∂ µ ˆ φ∂ ν ˆ φ E connected . (90)We then have T µν = h ( e + P ) u µ u ν i com + g µν P . (91)We stress that up to now, we do not have a definition of an irreducible four-velocity, we onlyhave a definition of a composite operator that will contain it . We now make the followingidentification 0-th moment ˆ= − h ( e + P ) u u i com , (92)1-st moment ˆ= h ( e + P ) u u i i com , (93)2-nd moment ˆ= h ( e + P ) u i u j i com , (94)and define a quantity that we will play the role of the stress tensor σ ij := δ ik h ( e + P ) u k u j i com − ( e + P ) − h ( e + P ) u u i i com h ( e + P ) u u j i com . (95) We remark that composite fluid quantities can also arise from genuine perfect fluids by introducing asmoothing scale [42]. However, this origin is conceptually different from the connected two-point functionapproach we are advertising here.
24e will see below that these identification are justified based on the continuity and Eulerequation. We find neglecting contributions of the metric h ( e + P ) u u i com ( X ) ∼ D ˆΠ ( X ) E , (96) h ( e + P ) u u i i com ( X ) ∼ h(cid:16) ∂ Xi − ∂ ri (cid:17) D ˆΠ( X + r/
2) ˆ φ ( X − r/ E i r =0 , (97) h ( e + P ) u i u j i com ( X ) ∼ h(cid:16) ∂ Xi ∂ Xj − ∂ ri ∂ rj (cid:17) D ˆ φ ( X + r/
2) ˆ φ ( X − r/ E i r =0 . (98)Since the various two-point functions appearing can be independently specified, we concludethat the composite four-velocity objects that we are referring to as moments are independent.One might object that the equations of motion enforce certain two-point functions to beproportional to other ones in the classical limit where higher-time derivatives or spatialderivatives are small. Indeed this is the case as we find in equation (37). So to lowest orderin the gradient expansion in the parameters (15) to (19) we find h ( e + P ) u u i com ( X ) ≈ m D ˆ φ ( X ) E , (99) h ( e + P ) u u i i com ( X ) ≈ ∂ ri h D ˆΠ( X + r/
2) ˆ φ ( X − r/ E − ( r → − r ) i r =0 , (100) h ( e + P ) u i u j i com ( X ) ≈ ∂ ri ∂ rj h D ˆ φ ( X + r/
2) ˆ φ ( X − r/ E i r =0 . (101)or in using the notion of section 2 h ( e + P ) u u i com ( X ) ≈ m h F ( X, r ) i r =0 , (102) h ( e + P ) u u i i com ( X ) ≈ ∂ ri h F − ( X, r i r =0 , (103) h ( e + P ) u i u j i com ( X ) ≈ ∂ ri ∂ rj h F ( X, r ) i r =0 . (104)Thus, in the classical limit even moments in Wigner momentum space will be related to evennumbers of r i -derivatives of the scalar field two-point function F ( X i , r j ) evaluated at zerowhereas for odd moments the same is true for odd numbers of r i -derivatives of the function F − ( X i , r i ), the two-point functions we already identified as phase-space densities on theprevious section. The zeroth and the second moment are the first two non-vanishing Taylorcoefficients of the arbitrary function F ( X i , r j ) and are thus independent. The function F ( X i , r j ) is arbitrary since its connected part can be freely specified by the initial densitymatrix. In other words, the stress tensor σ ij is completely generic and not only related tospatial X i -derivatives of the zeroth moments as it would be the case for one-point functionsor classical real scalar fields. In particular, it may contain an addition contribution to thepressure on top of the apparent pressure P we defined in (89). But not only this, we realizethat the first moment shows that we can specify a non-vanishing vorticity even at early times25n linear perturbation theory since we have ρǫ ijk ∂∂X j δv k ( X ) ≈ ǫ ijk ∂∂X j h ∂∂r k F − ( X, r ) i r =0 = 0 . (105)In (93) to (94) we labeled components of the composite energy-momentum tensor asmoments such that we are interpreting them as rest-mass-density, momentum-densities andstress tensor of particles in a certain non-relativistic limit. One might worry, that this fluidpicture does not hold since we do not have a conserved rest-mass density. However, as we showin appendix A.3, local energy- and momentum-conservation reduce to the familiar continuityand Euler equation in an expanding universe in the limit of small pressure, weak gravitationalfields and small velocity. This holds for every energy-momentum tensor of the form (91). Inparticular it holds for the real scalar field energy-momentum tensor. Identifying energydensity, apparent pressure and composite velocities in terms of the Wigner transformed two-point functions will allow us to reformulate the aforementioned non-relativistic limits clearlyas a large mass limit. We find e = m ~ F − a − ∂ X h Φ G − ( D − G i ∂ X F − a − h G i ∆ X F + 12 a − D h − Φ G + ( D − G i" a D − (cid:16) − Φ G − ( D − G (cid:17) F ′ ′ . (106)For the apparent pressure P we get then P = 12 a − D h − Φ G + ( D − G i" a D − (cid:16) − Φ G − ( D − G (cid:17) F ′ ′ − a − ∂ X h Φ G − ( D − G i ∂ X F − a − h G i ∆ X F , (107)and we realize that the choice of our perturbation parameters in (15) to (19) amounts to hav-ing a small apparent pressure. We note that the apparent pressure is due to the fundamentalfield theory we started with, i.e. it is build out of wave- rather than particle phenomena. Weidentify m F as the rest-mass density in the particle picture and remark that it is consistentwith the lowest-order expression we get in the phase-space language, ρ ( η, X i ) := m h m ~ F ( η, X i ) i = m ~ D ˆ φ ( η, X i ) ˆ φ ( η, X i ) E = m Z d D − pf even φ ( η, X i , p i ) γ − / ( η, X i ) " O (cid:0) ε (cid:1) . (108)With this definition we have e = ρ + P , (109)26s well as P = ~ m a − D h − Φ G + ( D − G i" a D − (cid:16) − Φ G − ( D − G (cid:17) ρ ′ ′ − ~ m a ∂∂X k h Φ G − ( D − G i ∂∂X k ρ − ~ m a h G i ∆ X ρ . (110)The composite fluid-four velocity quantities evaluate to h ( e + P ) u u i i com = − a − ( D − h G + ( D − G i Z d D − p (2 π ~ ) D − p i ~ F − + ~ m h G + ( D − G i ∂∂X i "h − Φ G − ( D − G i ρ ′ , (111)as well as h ( e + P ) u i u j i com = T ij − g ij P = ~ m ∂ ∂X i ∂X j ρ + Z d D − p (2 π ~ ) D − p i p j ~ F . (112)Note, that we consistently find h ( e + P ) u u i com = T − P = − " e + P + h ( e + P ) u i u i i com . (113)Late us now return to the scheme we presented (93) to (94) by applying the definition of thescalar field phase-space distribution we worked out in (72). We find in the large mass limit − h ( e + P ) u u i com ˆ= 0-th moment ≈ m Z d D − pf φ γ − / ≈ ρ , (114) h ( e + P ) u u i i com ˆ= 1-st moment ≈ m Z d D − p p i m f φ γ − / , (115) h ( e + P ) u i u j i com ˆ= 2-nd moment ≈ m Z d D − p p i p j m f φ γ − / , (116)and thus the stress tensor σ ij ≈ m Z d D − pδ ik ˜ p k p j m f φ γ − / − m ρ " Z d D − p p i m f φ γ − / d D − p p j m f φ γ − / , (117)is an arbitrary quantity since the phase-space distribution f φ can freely be specified via theinitial density matrix. This is just a different way of phrasing the independence of momentsin position space as we did in (99) to (101). We underpin again that this is even valid onscales where ma ≫ ∂ X where there is no quantum pressure term involved yet.27 .2 Einstein Equations Having identified hydrodynamic variables in the last section, we would like the express theEinstein equations in terms of these variables. We find G = 12 ( D − D − H + ( D − G − ( D − D − H Ψ ′ G = a ~ M P h G + 2Ψ G i" e + h ( e + P ) u i u i i com , (118) G i = ( D − ∂∂X i Ψ ′ G + ( D − H ∂∂X i Φ G = ~ M P h ( e + P ) u i u i com , (119) G ii = − ( D − H ′ −
12 ( D − D − H + ( D − ′′ G + D − D − h Φ G − ( D − G i + ( D − h H ′ + ( D − H i (Φ G + Ψ G ) + ( D − H h Φ ′ G + ( D − ′ G i = a ~ M P " P + 1 D − h ( e + P ) u i u i i com , (120)∆ X G kk D − δ ij − G ij = h ∆ D − δ ij − ∂ i ∂ j ih Φ G − ( D − G i = − ~ M P " h ( e + P ) u k u k i com D − δ ij − h ( e + P ) u i u j i com . (121)In particular the last equation shows that the gravitational slip is sourced by terms non-linear in the fluid velocities and should be taken into account for higher-order corrections.We also remind the reader that the spatial Einstein equations with neglected gravitons canbe obtained from the temporal equations via the Bianchi identity and the energy-momentumconservation. We combine this set of four redundant Einstein equations into two independentones that express the gravitational potentials in terms of the hydrodynamic fields. For theHubble rate we find12 ( D − D − H = a ~ M P D e + δ ij h ( e + P ) u i u j i com E Φ G , Ψ G ≈ a ~ M P h ρ i Φ G , Ψ G , (122)where we remark that the expectation value here is taken with respect to the stochastic vari-ables Φ G and Ψ G that are taken to be Gaussian. Together with the approximated dynamicalmatter equations we recover the usual behavior of the scalar factor of the cold dark matterscenario. The gravitational potentials themselves are determined by∆ X h ∆ X Ψ G − ( D − D − H Ψ G i = a D − ~ M P " ∆ X e + δ ij ∆ X h ( e + P ) u i u j i com + H ( D − ∂∂X i h ( e + P ) u i u i com , (123)28 X h Φ G − ( D − G i = a ~ M P " δ ij ∆ X D − h ( e + P ) u i u j i com − D − D − ∂ ∂X i ∂X j h ( e + P ) u i u j i com . (124)One has to be aware that the hydrodynamic fields, if again expressed in terms of two-pointfunctions, still contain the gravitational potentials but only with spatial derivatives. Theseterms can however be neglected for a leading order approximation in the large mass limit.We clearly see that the gravitational fields enter our calculation as non-dynamical constraintfields which was of course expected. Again, using our approximation we see that we recoverthe Poisson equation on subhorizon scales∆ X Ψ G ≈ a D − ~ M P h ρ − h ρ i Φ G , Ψ G i . (125)We also see that the gravitational slip is of higher order. In this section we want to derive non-linear equations for the real scalar field fluid based on ourtwo-point function approach. These equations are identical to local energy and momentumconservation and can be derived in a much more general setting as we show it in appendixA.3. However, as a consistency check, we want to explicitly use the dynamical equations forthe two-point functions of the first part of the paper. The computation may be found inappendix A.4. We obtain two differential equations for composite operators that are exactup to the linearization in the gravitational potentials and that correspond to the Euler andcontinuity equation ∂ η "h ( e + P ) u u i com + P + h − Φ G +( D − G i ∂ k "h G − ( D − G ih ( e + P ) u k u i com + ( D − h H − Ψ ′ G ih ( e + P ) u u i com − h H − Ψ ′ G ih ( e + P ) u k u k i com = 0 , (126) ∂ η h ( e + P ) u u i i com + ∂ k h ( e + P ) u k u i i com + ∂ i P + h D H + Φ ′ G − ( D − ′ G ih ( e + P ) u u i i com + h ∂ k Φ G − ( D − ∂ k Ψ G ih ( e + P ) u k u i i com − ∂ i Φ G h ( e + P ) u u i com + ∂ i Ψ G h ( e + P ) u k u k i com = 0 . (127)We want to see the non-relativistic limit of these equations. With the definition of therest-mass density in (108) which yields ( P ≪ e ) ρ = e − P ≈ e + P ≈ − h ( e + P ) u u i com − P . (128)29e also need to define the proper fluid velocity v i := − ρ − h G + Φ G ih ( e + P ) u i u i com " e + P ) − h ( e + P ) u k u k i com / , (129)and expand the composite term including spatial velocities in terms of the stress tensor aswe did in (95). We find approximately h ( e + P ) u i u k i com ≈ δ ij h σ jk + ρ · v j · v k + ~ m a ∂ j ∂ k ρ i . (130)We now have all ingredients to approximate equations (126) to (127) as the non-relativisticcontinuity equation in an FLRW-universe ∂ η ρ + ( D − H ρ + ∂ i h ρ · v i i ≈ , (131)and generalized Euler equation ∂ η h ρ · v i i + D H h ρ · v i i + ρ∂ i Φ G + ∂ k " δ ij δ km σ jm + ρ · v i · v k − ~ m a h δ ik ( D − H ∂ η ρ + δ ik ∂ η ρ i ≈ . (132)The higher time derivatives are the only terms remaining from the apparent pressure P. Weremark that equation (132) generizalizes the usual form of the classical non-linear scalar fielddark matter equations as stated for example in [3] or [4] by first, an stress tensor that canbe specified in accordance with inflationary predictions at initial times and which is not onlygiven in terms of a quantum pressure and second, a fluid velocity which generically allowsfor non-vanishing vorticity as was pointed out in section 4.1. As we pointed out earlier, theperfect fluid description is a suitable one at early times for linear evolution. However, non-linear evolution generates non-vanishing stresses and vorticity although they are negligiblesmall initially. Our model is capable of capturing these contributions that get more and moreimportant at late times, i.e. in the non-linear regime. We derived the correspondence between the approximated dynamics of the scalar field phase-space distribution f φ and a classical one-particle phase-space distribution in the cold darkmatter scenario. Before exploiting this correspondence in a more detail analysis, one wouldhave to specify initial conditions as well. A typical assumption within the non-relativistic The quantum pressure term for a pure one-point function approach may be recovered from the stress σ ij . f inicl ( X k , p i ) ≈ ρ inicl ( X k ) m γ / ( X k ) δ D − (cid:16) p i − δ ij ma ini (cid:2) − Ψ ini G ( X k ) (cid:3)(cid:2) v inicl (cid:3) j ( X k ) (cid:17) , (133)such that the initial physical rest-mass density is given by ρ inicl ( X k ) ≈ mγ − / ( X k ) Z d D − pf inicl ( X k , p l ) , (134)whereas the initial physical momentum density equals ρ inicl (cid:2) v inicl (cid:3) i ( X k ) ≈ δ ij a − (cid:2) ini G ( X k ) (cid:3) γ − / ( X k ) Z d D − p p j f inicl ( X k , p l ) . (135)Higher-order cumulants and thus the stress tensor are absent initially. We transfer thissetting to the scalar field phase-space density and make use of the large mass approximationused in the cold dark matter kinetic equation (80), f ini φ ( X k , p i ) ≈ ρ ini ( X k ) γ / ( X k ) m δ D − p i − m (cid:2) ( e + P ) u u i (cid:3) inicom ( X k ) ρ ini ( X k ) ! . (136)We rewrote the initial phase-space density such that we can relate it easier to the gravitationalpotentials and the scale factor, i.e. the metric constraint fields that contain initially theinformation from previous regimes in the cosmological evolution due to their relation toother matter that were dominant in those. Using the Einstein equations from section 4.2, wehave f ini φ ( X k , p i ) ≈ D − ma M P ~ γ / ( X k ) h D − H + ∆ X Ψ ini G ( X k ) i × δ D − p i − ma ini D − ∂ i Φ ini G ( X k ) H ini ! , (137)where we are sticking to the typical set-up in which the gravitational potentials are initiallyconstant and the decaying mode is neglected. The gravitational slip is initially also equal tozero within cosmological perturbation theory, however, we keep it here to illustrate that thepotential Ψ G is sourced by the density perturbation whereas the potential Φ G is to source toleading order by the velocity perturbation. We split (137) into even and odd parts in orderto be able to write down initial conditions for the scalar field two-point functions. We find F ini00 ( X k , p l ) ≈ ~ D − M P m a h D − H + ∆ X Ψ ini G ( X k ) i × Z d D − re − i ~ p i r i cos D − ma ini ~ r i ∂ i Φ ini G ( X k ) H ini ! , (138)31nd F ini − ( X k , p l ) ≈ imγ / ( X k ) D − M P m a h D − H + ∆ X Ψ ini G ( X k ) i × Z d D − re − i ~ p i r i sin D − ma ini ~ r i ∂ i Φ ini G ( X k ) H ini ! . (139)Carrying out the linearization in the gravitational potentials, these expressions reduce to F ini00 ( X k , p l ) ≈ ~ D − M P m a h D − H + ∆ X Ψ ini G ( X k ) i (2 π ~ ) D − δ D − (cid:0) p l (cid:1) , (140) F ini − ( X k , p l ) ≈ − ma D − D − H ini M P ma ini ∂∂X i Φ ini G ( X k )(2 π ~ ) D − ∂∂p i δ D − (cid:0) p l (cid:1) . (141)We perform a Wigner transformation to obtain the correlators in coordinate space F ini00 ( x k , y l ) ≈ D − M P ~ ~ m a h D − H + (cid:2) ∆Ψ ini G (cid:3)(cid:16) ( x + y ) k (cid:17)i , (142) F ini − ( x k , y l ) ≈ ia D − H ini D − M P ~ ( x − y ) i (cid:2) ∂ i Φ ini G (cid:3)(cid:16) ( x + y ) k (cid:17) . (143)As we found out earlier, we can express the remaining two-point functions F + and F interms of F and F − (see (36) to (37)), F ini+ ( x k , y l ) ≈ − a D − H ini D − M P ~ ~ m a h ( D − H + (cid:2) ( D − ini G + ∆Φ ini G (cid:3)(cid:16) ( x + y ) k (cid:17)i , (144) F ini11 ( x k , y l ) ≈ D − a D − M P ~ h D − H + (cid:2) ∆Ψ ini G (cid:3)(cid:16) ( x + y ) k (cid:17)i . (145)We have now specified all ingredients of a Gaussian initial density matrix for the real scalarfield theory we started with in the beginning. Note, that this is a very choice that mapson an initially perfect fluid. We could as well have taken into account higher cumulants inmomentum space. In this work we develop a formalism for the dynamics of dark matter in which we start witha tree-level relativistic action for a real scalar field and obtain an effective action descriptionthat includes leading order interactions mediated by gravity. Our formalism is relativistic,in that it allows for a systematic inclusion of both relativistic matter field effects as well asrelativistic gravitational effects. The non-relativistic limit of our dynamical equations couldbe obtained from a second quantized scalar field formalism akin to the one used in condensed32atter literature. However, since we are in particular interested in capturing relativisticcorrections (that can be important in the gravitational sector when one is interested in thescales comparable to the Hubble scale and in the matter sector e.g. when the scalar field isultralight), using such a formalism would restrict its validity too much.Furthermore, we identify a phase-space distribution f φ based on four on-shell, equal-timereal scalar field statistical two-point functions (72). The statistical two-point functions obeya system of first order differential equations that closes because we first neglected manifestself-interactions of the matter field and the dynamical gravitational fields and second, didnot integrate out the gravitational constraint fields. In the language of Feynman diagramsthis amounts to approximating loop contributions with external sources whose evolution isdetermined by the semi-classical Einstein equation. The evolution of the phase-space distri-bution f φ is determined by a generalized Vlasov equation including relativistic corrections,third-order time derivatives and corrections in a gradient expansion (73). Dropping the third-order time-derivatives as small corrections reduces the degrees of freedom to those of classicalone-particle phase-space distribution. The statistical two-point functions of the scalar matterfield entering the definition of f φ are evaluated with respect to an initial density matrix andthus have generically reducible and connected pieces, in other words they contain a part givenby one-point functions or classical fields. Focusing on the connected piece of the statisticalmatter two-point function makes the major distinction from previous approaches of model-ing real scalar field fluids that focused on one-point functions. The reason is that it allowsfor generic initial conditions in two arguments without coarse-graining, either in positionspace or in Wigner space which then translates into a hierarchy of non-related moments inmomentum space and in particular enables us to model a fluid that generically can includevorticity and anisotropy (95). At the level of hydrodynamics we realized this by facing whatwe called a composite term that written as fluid quantities reduces into products of velocitiesand an irreducible piece (87) comprising a stress tensor. We derived non-linear imperfect hy-drodynamic equations (131) (132) from the integrated system of statistical matter two-pointfunctions that are exact up the linearized scalar metric we worked with.We note that using statistical two-point functions from the beginning allows us to treatgravity on a semi-classical level where we introduced linearized stochastic gravitational po-tentials that couple to the statistical two-point functions and thereby make the phase-spacedistribution f φ stochastic. This is what we call hybrid approach and it bridges the gap tocosmological perturbation theory since we now can calculate in a second step two-point func-tion with respect to the gravitational potentials. The Einstein equations relating them werederived 4.2. We also provided initial conditions for the statistical matter two-point functions33nd thus provided all ingredients to treat this problem with non-equilibrium quantum fieldtheory techniques like the Schwinger-Keldysh formalism which we intent to do in the future.We once more remark that - to model dark matter on non-linear scales at late times - itis necessary to go beyond the perfect fluid approximation. The perfect fluid description isvalid only in the linear regime which is reflected in the initial conditions and breaks down onscales k > k nl ∼ . − primarily due to shell-crossing and generation of other types ofperturbations that in the non-linear regime get dynamically generated. These perturbationsinclude gravitational slip, vector and tensor metric perturbations as well as vorticity andanisotropic stresses at the matter side. Except for vector and tensor metric perturbations,all of these can be consistently treated in our formalism, which models dark matter byutilizing statistical (Hadamard) two-point functions. In future work we intent to go beyondthis hybrid approach, including a full non-linear treatment of gravity that is not restrictedto the Newtonian gauge. Acknowledgments.
We are grateful for conversation on this subject with C. Uhlemann.This work is part of the research programme of the Foundation for Fundamental Research onMatter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).This work is in part supported by the D-ITP consortium, a program of the Netherlands Or-ganization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education,Culture and Science (OCW).
A Appendix
A.1 Wigner Tranformation of 2-Point Function Dynamics
In this appendix we drop the ubiquitous η dependence to save some space. We define thefollowing operator h f ( X i , p i ) i ←− ∂ X · −→ ∂ p h g ( X i , p i ) i := " ∂f∂X k ( X i , p i ) " ∂g∂p k ( X i , p i ) . (146)By using partial integration, one can show that the following relation holds up to boundaryterms Z d D − ( x − e x ) e − ip i ( x i − e x i ) Z d D − zA ( x i , z i ) B ( z i , e x i )= A ( X i , p i ) e i ~ (cid:0) ←− ∂ X ·−→ ∂ p −←− ∂ p ·−→ ∂ X (cid:1) B ( X i , p i ) + boundary terms . (147)Using the definition p := δ ij p i p j (148)34e rewrite the equations (27) - (30) in Wigner space in the following way a D − F ′ ( X i , p i ) = h G ( X i ) + ( D − G ( X i ) i e i ~ ←− ∂ X ·−→ ∂ p F ( X i , p i )+ F ( X i , p i ) e − i ~ ←− ∂ p ·−→ ∂ X h G ( X i ) + ( D − G ( X i ) i , (149) F ′ ( X i , p i ) = ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i e i ~ ←− ∂ X ·−→ ∂ p "h ∂∂X k + i p k ~ ih a D − F i ( X i , p i ) + " G ( X i ) − ( D − G ( X i ) e i ~ ←− ∂ X ·−→ ∂ p (h ∆ X i p ~ · ∂ X − p ~ ih a D − F i ( X i , p i ) ) − m ~ a h G ( X i ) − ( D − G ( X i ) i e i ~ ←− ∂ X ·−→ ∂ p h a D − F i ( X i , p i )+ h a − ( D − F i ( X i , p i ) e − i ~ ←− ∂ p ·−→ ∂ X h G ( X i ) + ( D − G ( X i ) i , (150) F ′ ( X i , p i ) = (h ∂∂X k − i p k ~ ih a D − F i ( X i , p i ) ) · e − i ~ ←− ∂ p ·−→ ∂ X ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i + (h ∆ X − i p ~ · ∂ X − p ~ ih a D − F i ( X i , p i ) ) e − i ~ ←− ∂ p ·−→ ∂ X ( G ( X i ) − ( D − G ( X i ) ) − m ~ a h a D − F i ( X i , p i ) e − i ~ ←− ∂ p ·−→ ∂ X h G ( X i ) − ( D − G ( X i ) i + h G ( X i ) + ( D − G ( X i ) i e i ~ ←− ∂ X ·−→ ∂ p h a − ( D − F i ( X i , p i ) . (151) a − ( D − F ′ ( X i , p i ) =+ ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i · e i ~ ←− ∂ X ·−→ ∂ p "h ∂∂X k + i p k ~ i F ( X i , p i ) + ( G ( X i ) − ( D − G ( X i ) ) e i ~ ←− ∂ X ·−→ ∂ p (h ∆ X i p ~ · ∂ X − p ~ i F ( X i , p i ) ) − m ~ a h G ( X i ) − ( D − G ( X i ) i e i ~ ←− ∂ X ·−→ ∂ p F ( X i , p i )+ (h ∂∂X k − i p k ~ i F ( X i , p i ) ) · e − i ~ ←− ∂ p ·−→ ∂ X ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i + (h ∆ X − i p ~ · ∂ X − p ~ i F ( X i , p i ) ) e − i ~ ←− ∂ p ·−→ ∂ X ( G ( X i ) − ( D − G ( X i ) ) − m ~ a F ( X i , p i ) e − i ~ ←− ∂ p ·−→ ∂ X h G ( X i ) − ( D − G ( X i ) i . (152)35y using the definitions (34) and (35), we get the following system of equations a D − F ′ ( X i , p i ) = 2 h G ( X i ) + ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i F + ( X i , p i )+ 2 h Φ G ( X i ) + ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i F − ( X i , p i ) , (153) F ′ + ( X i , p i ) = ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i ∂∂X k h a D − F i ( X i , p i )+ h G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i(h ∆ X − p ~ ih a D − F i ( X i , p i ) ) − h Φ G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i( p ~ · ∂ X h a D − F i ( X i , p i ) ) − ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i · ( p k ~ h a D − F i ( X i , p i ) ) − m ~ a h G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p ih a D − F i ( X i , p i )+ h G ( X i ) + ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p ih a − ( D − F i ( X i , p i ) . (154) F ′− ( X i , p i ) = − ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i ∂∂X k h a D − F i ( X i , p i ) − h Φ G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i(h ∆ X − p ~ ih a D − F i ( X i , p i ) ) − h G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i( p ~ · ∂ X h a D − F i ( X i , p i ) ) − ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i · " p k ~ h a D − F i ( X i , p i ) + m ~ a h Φ G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p ih a D − F i ( X i , p i )+ h Φ G ( X i ) + ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p ih a − ( D − F i ( X i , p i ) . (155)36 − ( D − F ′ ( X i , p i ) = − m ~ a h G ( X i ) − ( D − G ( X i ) i cos h ~ ←− ∂ X · −→ ∂ p i F + ( X i , p i )+ 2 m ~ a h G ( X i ) − ( D − G ( X i ) i sin h ~ ←− ∂ X · −→ ∂ p i F − ( X i , p i )+ 2 ( G ( X i ) − ( D − G ( X i ) ) cos h ~ ←− ∂ X · −→ ∂ p i(h ∆ X − p ~ i F + ( X i , p i ) ) − ( Φ G ( X i ) − ( D − G ( X i ) ) sin h ~ ←− ∂ X · −→ ∂ p i(h ∆ X − p ~ i F − ( X i , p i ) ) − ( Φ G ( X i ) − ( D − G ( X i ) ) sin h ~ ←− ∂ X · −→ ∂ p i( p ~ · ∂ X F + ( X i , p i ) ) − ( G ( X i ) − ( D − G ( X i ) ) cos h ~ ←− ∂ X · −→ ∂ p i( p ~ · ∂ X F − ( X i , p i ) ) + ( ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i) cos h ~ ←− ∂ X · −→ ∂ p i · ( ∂∂X k F + ( X i , p i ) ) − ( ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i) sin h ~ ←− ∂ X · −→ ∂ p i · ( ∂∂X k F − ( X i , p i ) ) − ( ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i) sin h ~ ←− ∂ X · −→ ∂ p i · ( p k ~ F + ( X i , p i ) ) − ( ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i) cos h ~ ←− ∂ X · −→ ∂ p i · ( p k ~ F − ( X i , p i ) ) . (156)For later discussion, we remark the following important equal time properties F ( X i , p i ) = 12 Z d D − r e − i ~ p i r i (cid:28) n ˆ φ (cid:16) X i + r i (cid:17) , ˆ φ (cid:16) X i − r i (cid:17)o (cid:29) = F ( X i , − p i ) , (157) F − ( X i , p i ) = i Z d D − re − i ~ p i r i " (cid:28) n ˆΠ (cid:16) X i + r i (cid:17) , ˆ φ (cid:16) X i − r i (cid:17)o (cid:29) − (cid:28) n ˆ φ (cid:16) X i + r i (cid:17) , ˆΠ (cid:16) X i − r i (cid:17)o (cid:29) = − F − ( X i , − p i ) , (158) Z d D − p (2 π ~ ) D − h p k ...p k n +1 i F ( X i , p i ) = 0 , (159) Z d D − p (2 π ~ ) D − h p k ...p k n i F − ( X i , p i ) = 0 . (160)These equations tell us that F is even in p i and that F − is odd in p i . After an appropriaterescaling that also accounts for the right dimensions, these two quantities will play the roleof even and odd phase-space densities. 37 hase-space dynamics including ε ~ and ε ~ · ε g corrections. In this section we want tomanipulate the unintegrated dynamical equations for these two-point functions to see whetherwe can reproduce equations that mimic the Vlasov-equation. First, write down the phase-space dynamics perturbatively including ε ~ that correspond to the next order in the gradientexpansion we described in subsection 2.2. We also include for illustration ε ~ · ε g correctionsthat result from multiplying terms of the gradient expansion with the gravitational potentials. a D − F ′ = 2 h G + ( D − G i F + − ~ ∂ ∂X i ∂X j h Φ G + ( D − G i ∂ ∂p i ∂p j F + + ~ ∂∂X i h Φ G + ( D − G i ∂∂p i F − − ~ ∂ ∂X i ∂X j ∂X k h Φ G + ( D − G i ∂ ∂p i ∂p j ∂p k F − , (161) F ′ + = 12 ∂∂X k h Φ G − ( D − G i ∂∂X k h a D − F i + h G − ( D − G ih ∆ X − p ~ ih a D − F i + 18 ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j h p h a D − F ii − ∂∂X k h Φ G − ( D − G i ∂∂p k h p i ∂∂X i h a D − F ii − ∂ ∂X k ∂X i h Φ G − ( D − G i ∂∂p i h p k h a D − F ii − m ~ a h G − ( D − G ih a D − F i + m a ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j h a D − F i + h G +( D − G ih a − ( D − F i − ~ ∂ ∂X i ∂X j h Φ G +( D − G i ∂ ∂p i ∂p j h a − ( D − F i , (162)38 ′− = − ~ ∂ ∂X k ∂X i h Φ G − ( D − G i ∂ ∂p i ∂X k h a D − F i − ~ ∂∂X k h Φ G − ( D − G i ∂∂p k "h ∆ X − p ~ ih a D − F i − ~ ∂ ∂X k ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p k ∂p i ∂p j " p h a D − F i − h G − ( D − G i p k ~ ∂∂X k h a D − F i + ~ ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j h p k ∂∂X k h a D − F ii + 12 m ~ a ∂∂X i h Φ G − ( D − G i ∂∂p i h a D − F i − m a ~ ∂ ∂X i ∂X j ∂X k h Φ G − ( D − G i ∂ ∂p i ∂p j ∂p k h a D − F i + ~ ∂∂X i h Φ G + ( D − G i ∂∂p i h a − ( D − F i − ~ ∂ ∂X i ∂X j ∂X k h Φ G + ( D − G i ∂ ∂p i ∂p j ∂p k h a − ( D − F i , (163) a − ( D − F ′ = − m ~ a h G − ( D − G i F + + 14 m a ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j F + + 2 h G − ( D − G ih ∆ X − p ~ i F + + 14 ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j h p F + i − ~ ∂∂X k h Φ G − ( D − G i ∂∂p k "h ∆ X − p ~ i F − − ~ ∂ ∂X i ∂X j ∂X k h Φ G − ( D − G i ∂ ∂p i ∂p j ∂p k h p F − i − ∂∂X k h Φ G − ( D − G i ∂∂p k h p · ∂ X F + i − h G − ( D − G i p ~ · ∂ X F − + ~ ∂ ∂X i ∂X j h Φ G − ( D − G i ∂ ∂p i ∂p j " p · ∂ X F − + ∂∂X k h Φ G − ( D − G i ∂∂X k F + − ~ ∂ ∂X k ∂X i h Φ G − ( D − G i ∂ ∂X k ∂p i F − − ∂ ∂X k ∂X i h Φ G − ( D − G i ∂∂p i h p k F + i − p ~ · ∂ X h Φ G − ( D − G i F − + ~ ∂ ∂X i ∂X j ∂X k h Φ G − ( D − G i ∂ ∂p i ∂p j " p k F − + m a ~ ∂∂X i h Φ G − ( D − G i ∂∂p i F − − ~ m a ∂ ∂X i ∂X j ∂X k h Φ G − ( D − G i ∂ ∂p i ∂p j ∂p k F − . (164)39 hase-space dynamics including ε ~ but dropping ε ~ · ε g corrections. Let us furtherdrop the ε ~ · ε g corrections which is at least naively consistent with our linearization ingravity, i.e. consistent with not keeping ε g terms as we did from the very beginning of thispaper. We do not expect the corrections ε ~ from the gradient expansion to become importantunless we have a very light scalar field ( m ≈ − eV) as pointed out in section 2.2 which isconsidered an extreme case. Thus, for typical masses at scales & eV, the following equationsare perfectly accurate and we will even drop the ε ~ corrections when discussing them furtherin the main text. a D − F ′ = 2 h G + ( D − G i F + + ~ ∂∂X i h Φ G + ( D − G i ∂∂p i F − , (165) F ′ + = ∆ X h a D − F i − p ~ h G − ( D − G ih a D − F i − m ~ a h G − ( D − G ih a D − F i + h G + ( D − G ih a − ( D − F i , (166) F ′− = 12 ~ ∂∂X k h Φ G − ( D − G i ∂∂p k " p h a D − F i − p ~ · ∂ X h Φ G − ( D − G ih a D − F i − h G − ( D − G i p k ~ ∂∂X k h a D − F i + 12 m ~ a ∂∂X i h Φ G − ( D − G i ∂∂p i h a D − F i + ~ ∂∂X i h Φ G + ( D − G i ∂∂p i h a − ( D − F i , (167) a − ( D − F ′ = − m ~ a h G − ( D − G i F + + ∆ X F + − p ~ h G − ( D − G i F + + 1 ~ ∂∂X k h Φ G − ( D − G i ∂∂p k " p F − − h G − ( D − G i p ~ · ∂ X F − − p ~ · ∂ X h Φ G − ( D − G i F − + m a ~ ∂∂X i h Φ G − ( D − G i ∂∂p i F − . (168)40 .2 Einstein Equations in Longitudinal Gauge with Scalar Pertur-bations The dynamical equations of the previous section are supplemented by the Einstein equa-tions. Since we neglected gravitons with the choice of our metric, the Einstein equations areconstraint equations that will determine the gravitational potentials in terms of two-pointfunctions of the scalar field and thereby induce non-linear interactions. We write down themetric in this gauge g ( η, x i ) = − a ( η ) (cid:2) G ( η, x i ) (cid:3) , g ij ( η, x i ) = a ( η ) δ ij (cid:2) − G ( η, x i ) (cid:3) , (169) g ( η, x i ) = − a − ( η ) (cid:2) − G ( η, x i ) (cid:3) , g ij ( η, x i ) = a − ( η ) δ ij (cid:2) G ( η, x i ) (cid:3) , (170) √− g ( η, x i ) = a D ( η ) (cid:2) G ( η, x i ) − ( D − G ( η, x i ) (cid:3) . (171)Let us collect the linearized connection coefficients in longitudinal gaugeΓ = H + Φ ′ G , (172)Γ i = ∂ i Φ G , (173)Γ i = δ ij ∂ j Φ G , (174)Γ ij = H δ ij − h H (Φ G + Ψ G ) + Ψ ′ G i δ ij , (175)Γ ij = H δ ij − Ψ ′ G δ ij , (176)Γ ijk = − ∂ j Ψ G δ ik − ∂ k Ψ G δ ij + ∂ l Ψ G δ il δ jk . (177)We have the temporal Ricci tensor components R = − ( D − H ′ + ∆Φ G + ( D − ′′ G + ( D − H h Φ ′ G + Ψ ′ G i , (178) R i = ( D − ∂ i Ψ ′ G + ( D − H ∂ i Φ G . (179)as well as the purely spatial part R ij = h H ′ + ( D − H i δ ij + ( D − ∂ i ∂ j Ψ G − ∂ i ∂ j Φ G + ∆Ψ G δ ij − h Ψ ′′ G + 2( D − H (Φ G + Ψ G ) + 2 H ′ (Φ G + Ψ G ) + H Φ ′ G + (2 D − H Ψ ′ G i δ ij . (180)This leaves us with a R = ( D − h H ′ + ( D − H i − G + (2 D − G − D − ′′ G − D − G h H ′ + ( D − H i − D − H Φ ′ G − D − H Ψ ′ G . (181)The 00- and 0 i -components of the linearized Einstein tensor then read G = 12 ( D − D − H + ( D − G − ( D − D − H Ψ ′ G , (182) G i = ( D − ∂ i Ψ ′ G + ( D − H ∂ i Φ G . (183)41he ij -components of the linearized Einstein tensor read G ij = − h ( D − H ′ + 12 ( D − D − H i δ ij + ( D − ′′ G δ ij + ∆ h Φ G − ( D − G i δ ij − ∂ i ∂ j h Φ G − ( D − G i + " ( D − h H ′ + ( D − H i (Φ G + Ψ G ) + ( D − H h Φ ′ G + ( D − ′ G i δ ij . (184)The energy-momentum tensor operator of the scalar field theory is given byˆ T µν = ∂ µ ˆ φ∂ ν ˆ φ − g µν h g αβ ∂ α ˆ φ∂ β ˆ φ + m ~ ˆ φ i . (185)The composite operator (185) needs to be renormalized by introducing on the gravity sidehigher order geometrical counterterms ( R , square of the Weyl tensor, Gauss-Bonnet term)as well as lower order geometrical counterterms containing a bare Newton constant and a barecosmological constant such that after substraction of UV-divergencies we are left with theobservable values of the renormalized Newton constant and the renormalized cosmologicalconstant [43]. Contributions of the renormalized higher order geometrical terms completelyirrelevant for the studies of large scale structures. Regularizing the two-point functionscontained in the energy-momentum tensor (185) related to the conditions we spelled outin (12). In order to regularize the two-point functions we will split them into infrared andultraviolet parts by introducing a cut-off in such a way that the conditions (12) can besatisfied up to that cut-off. However, for the scope of this paper we will not have to worrymore about renormalization issues.The semi-classical Einstein equation relates the gravitational potentials to two-point func-tions of the scalar field in the coincidence limit G µν = ~ M P T µν = ~ M P D ˆ T µν E . (186)Lets see how this works by looking at the purely temporal equation. We find (suppressingthe time dependence in the argument) G ( X i ) = 12 ( D − D − H +( D − G ( X i ) − ( D − D − H Ψ ′ G ( X i ) = ~ M P D ˆ T ( X i ) E , (187)where in our scalar field modelˆ T ( X i ) = ∂ ˆ φ ( X i ) ∂ ˆ φ ( X i ) − g h ∂ ˆ φ ( X i ) ∂ ˆ φ ( X i ) + ∂ k ˆ φ ( X i ) ∂ k ˆ φ ( X i ) + m ~ ˆ φ ( X i ) ˆ φ ( X i ) i , = 12 ˆ φ ′ ( X i ) ˆ φ ′ ( X i ) + 12 h G ( X i ) + 2Ψ G ( X i ) i ∂ k ˆ φ ( X i ) ∂ k ˆ φ ( X i )+ a h G ( X i ) i m ~ ˆ φ ( X i ) ˆ φ ( X i ) , (188)42nd thus taking expectation values yields D ˆ T ( X i ) E = 12 a − D − F ( X i ) h G ( X i ) + 2( D − G ( X i ) i + 12 h G ( X i ) + 2Ψ G ( X i ) i( ∂ X − ∂ r ! F ( X i , r i ) )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r i =0 + 12 m a ~ h G ( X i ) i F ( X i ) , where F ( X i ) = F ( X i , r i = 0) . (189)For the 0 i -equations we get( D − ∂ i Ψ ′ G + ( D − H ∂ i Φ G = ~ M P D ˆ φ ′ ∂ i ˆ φ E . (190)For the ij -components of the energy-momentum tensor we getˆ T ij = ∂ i ˆ φ∂ j ˆ φ + 12 h − G + Ψ G ) i ( ˆ φ ′ ) δ ij − ∂ k ˆ φ∂ k ˆ φδ ij − m a h − G i ˆ φ δ ij . (191)Projecting on the trace-free part we get " ∆ D − δ ij − ∂ i ∂ j Φ G − ( D − G i = ~ M P * ∂ i ˆ φ∂ j ˆ φ − ∂ k ˆ φ∂ k ˆ φD − δ ij + . (192)The equation for the trace reads G ii D − − ( D − H ′ −
12 ( D − D − H + ( D − ′′ G + D − D − h Φ G − ( D − G i + ( D − h H ′ + ( D − H i (Φ G + Ψ G ) + ( D − H h Φ ′ G + ( D − ′ G i = ~ M P (h − G + Ψ G ) i D ( ˆ φ ′ ) E − D − D − D ∂ k ˆ φ∂ k ˆ φ E − m a h − G i D ˆ φ E ) . (193)We rewrite all Einstein equations in terms of the two-point functions F , F + , F − and F : − ( D − H ′ −
12 ( D − D − H + ( D − ′′ G + D − D − h Φ G − ( D − G i + ( D − h H ′ + ( D − H i (Φ G + Ψ G ) + ( D − H h Φ ′ G + ( D − ′ G i = ~ M P (h D − G ) ih a − D − F i − D − D − " ∆ X − ∆ r ! F r =0 − m a ~ h − G i F ) , (194)432 ( D − D − H + ( D − G − ( D − D − H Ψ ′ G = ~ M P ( a − D − F h G + 2( D − G i + h G ( X ) + 2Ψ G ( X ) i" ∆ X − ∆ r ! F r =0 + a m ~ h G i F ) , (195)( D − ∂ X Ψ ′ G + ( D − H ∂ X Φ G = ~ a ( D − M P h G + ( D − G i( ∂ X F + + 2 i∂ r F − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =0 ) , (196) " ∆ X D − δ ij − ∂ Xi ∂ Xj Φ G − ( D − G i = ~ M P " ∂ Xi ∂ Xj ∂ Xi ∂ rj − ∂ Xj ∂ ri − ∂ ri ∂ rj − δ ij D −
1) ∆ X + δ ij D − r F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =0 . (197) Eliminating the two-point functions F + and F from energy-momentum tensor. Let us set T µν := D ˆ T µν E . (198)By integrating the dynamical matter equations over the momenta we can rewrite the energy-momentum tensor in such a way that it only depends on the two-point functions F and F − . We first get a D − F ′ ( X i ) = 2 h G ( X i ) + ( D − G ( X i ) i F + ( X i ) , (199) F ′ + ( X i ) = ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i ∂∂X k h a D − F i ( X i )+ h G ( X i ) − ( D − G ( X i ) i Z d D − p (2 π ~ ) D − (h ∆ X − p ~ ih a D − F i ( X i , p i ) ) − m ~ a h G ( X i ) − ( D − G ( X i ) ih a D − F i ( X i )+ h G ( X i ) + ( D − G ( X i ) ih a − ( D − F i ( X i ) , (200)44nd thus F ( X i ) = m ~ a D − h − D − G ( X i ) i F ( X i )+ 12 a D − h − Φ G ( X i ) − ( D − G ( X i ) i" a D − h − Φ G ( X i ) − ( D − G ( X i ) i F ′ ( X i ) ′ − a D − ∂∂X k h Φ G ( X i ) − ( D − G ( X i ) i ∂∂X k F ( X i ) − a D − h − D − G ( X i ) i Z d D − p (2 π ~ ) D − (h ∆ X − p ~ i F ( X i , p i ) ) . (201)We plug this expression into the energy-momentum tensor and get T ( X i ) = m a ~ h G ( X i ) i F ( X i )+ a − ( D − h G ( X i ) + ( D − G ( X i ) i" a D − h − Φ G ( X i ) − ( D − G ( X i ) i F ′ ( X i ) ′ − ∂ X h Φ G ( X i ) − ( D − G ( X i ) i ∂ X F ( X i )+ " G ( X i ) + 2Ψ G ( X i ) d D − p (2 π ~ ) D − p ~ F ( X i , p i ) . (202) T i ( X i ) = a − ( D − h G ( X i ) + ( D − G ( X i ) i × ( ∂∂X i " a D − h − Φ G ( X i ) − ( D − G ( X i ) i F ′ ( X i ) − Z d D − p (2 π ~ ) D − p i ~ F − ( X i , p i ) ) . (203) T ii ( X i ) D − a − ( D − " − h Φ G ( X i ) − ( D − G ( X i ) i × " a D − h − Φ G ( X i ) − ( D − G ( X i ) i F ′ ( X i ) i ′ − ∂ X h Φ G ( X i ) − ( D − G ( X i ) i ∂ X F ( X i ) − D − D − X F ( X i )+ 1 D − Z d D − p (2 π ~ ) D − p ~ F ( X i , p i ) . (204) T ij ( X i ) − δ ij D − T kk ( X i )= Z d D − p (2 π ~ ) D − " ∂ Xi ∂ Xj p i p j ~ − δ ij D − (cid:16) ∆ X p ~ (cid:17) F ( X i , p i ) . (205)45 ij ( X i ) = δ ij ( a − ( D − " − h Φ G ( X i ) − ( D − G ( X i ) i × " a D − h − Φ G ( X i ) − ( D − G ( X i ) i F ′ ( X i ) ′ − ∂ X h Φ G ( X i ) − ( D − G ( X i ) i ∂ X F ( X i ) − ∆ X F ( X i ) ) + Z d D − p (2 π ~ ) D − " ∂ Xi ∂ Xj p i p j ~ F ( X i , p i ) . (206)In the equation for the traceless spatial energy-momentum-tensor we used the fact that F is even in p i to eliminate two terms. By using the Bianchi identities, one can show thatthe purely spatial constraint equations follow from the constrained equations involving time-components. We thus have two independent constraint equations that relate the gravitationalpotentials Φ G and Ψ G to momentum integrals over the two-point functions F and F − . A.3 Energy-Momentum Conservation for Composite Fluids
In this section we want to derive non-relativistic fluid equations by assuming an energy-momentum tensor in a composite form that is for example due to taking expectation valuesof two-point functions. We also assume a linearized scalar metric in longitudinal gauge. Wehave e T npf µν := h ( e e + e P ) e u µ e u ν i com + g µν e P , (207)where the composite term is basically a placeholder for any symmetric two-tensor whoseindices can be raised and lowered with the metric. Thus, the quantity e P is only an apparentpressure and the real pressure might get contributions from the composite term. We use atilde to make a distinction between a general case and the real scalar field case we discussthroughout the paper. The superscript NPF refers to the idea that although the energy-momentum tensor (207) has the apparent form of a perfect fluid, it does not need to be theenergy-momentum tensor of such due to its composite nature. Since the composite termought to be a generalization of the non-composite perfect fluid term we impose h ( e e + e P ) e u µ e u µ i com =: − ( e e + e P ) , (208)which acts also as a definition for the energy e e . Local energy conservation then gives0 = ∇ µ h e T npf i µ = ∂ η "h ( e e + e P ) e u e u i com + e P + ∂ i h ( e e + e P ) e u i e u i com +Γ ii h ( e e + e P ) e u e u i com − Γ ik h ( e e + e P ) e u k e u i i com +Γ iik h ( e e + e P ) e u k e u i com − Γ i h ( e e + e P ) e u e u i i com . (209)46e raise and lower indices and commute u u i , ∂ η "h ( e e + e P ) e u e u i com + e P + ∂ i h ( e e + e P ) e u i e u i com + Γ ii h ( e e + e P ) u u i com − Γ ik h ( e e + e P ) e u k e u i i com + h Γ iik − g Γ i g ik ih ( e e + e P ) e u k e u i com = 0 . (210)Next we plug in the perturbed metric in longitudinal gauge and get ∂ η "h ( e e + e P ) e u e u i com + e P + h − Φ G +( D − G i ∂ k "h G − ( D − G ih ( e e + e P ) e u k e u i com + ( D − h H − Ψ ′ G ih ( e e + e P ) e u e u i com − h H − Ψ ′ G ih ( e e + e P ) e u k e u k i com = 0 . (211)To find the non-relativistic limit of this composite equation we have to define a quantity thatshould capture the rest-mass in the non-relativistic limit e ρ := e e − e P , (212)as well as a proper fluid velocity e v i := − e ρ − h G + Φ G ih ( e e + e P ) e u i e u i com " e e + e P ) − h ( e e + e P ) e u k e u k i com / . (213)We then have the non-relativistic continuity equation in an FLRW-universe " ∂ η e ρ + ( D − H e ρ + ∂ i he ρ · e v i i O (cid:16) Φ G , Ψ G (cid:17) + O (cid:16)e v , e P (cid:17) = 0 . (214)The local momentum conservation reads0 = ∇ µ h e T npf i µi = ∂ h ( e e + e P ) e u e u i i com + ∂ k h ( e e + e P ) e u k e u i i com + ∂ i P + h D H + Φ ′ G − ( D − ′ G ih ( e e + e P ) e u e u i i com + h ∂ k Φ G − ( D − ∂ k Ψ G ih ( e e + e P ) e u k e u i i com − ∂ i Φ G h ( e e + e P ) e u e u i com + ∂ i Ψ G h ( e e + e P ) e u k e u k i com . (215)Defining the proper composite velocity term he v i e v k i com := e ρ − h G − Φ G ih ( e e + e P ) e u i e u k i com , (216)we find the Euler equation in an FLRW-space-time0 = " ∂ η he ρ · e v i i + D H he ρ · e v i i + ∂ k (cid:16)e ρ · he v i e v k i com (cid:17) + e ρ δ ij ∂ j Φ G + δ ij ∂ j e P × " O (cid:16) Φ G , Ψ G (cid:17) + O (cid:16)e v , e P (cid:17) . (217)47e can now discuss two options.Option one is simply given by declaring the composite term of the energy-momentum tensor(207) and similarly its non-relativistic descendant (216) to be non-composite as it would bethe case for a perfect scalar field fluid based classical field theory including non-linear termsin the fluid quantities but keeping gravitational potentials linear. Equations (214) and (217)were for example derived in the scalar field context by by [3] and [13] for real and complexfields, respectively.The second option is of course to take into account the composite nature of the term in(216) which happens for a real scalar field fluid based on non-vanishing connected two-point functions as we discuss it in section 4.1. However, we stress once more that theabove derivation made no reference to the real scalar field theory and we only assumed localconservation of a fluid energy-momentum tensor of the form (207). We will now give anindependent derivation of the the fluid equations (211) and (215) in the next section as across-check and will explicitly use the scalar field matter equations. A.4 Continuity and Euler Equation for Real Scalar Field Fluid
Local energy conservation.
Let us start with the local energy conservation by integratingthe dynamical equations (153) and (154) as well as (156). We find F + = ~ m a D − h − Φ G − ( D − G i ρ ′ , (218) F = − a D − h − D − G ih ( e + P ) u u i com , (219) Z d D − p (2 π ~ ) D − p ~ F + − ∂∂X k h Φ G + ( D − G i Z d D − p (2 π ~ ) D − p k ~ F − = 12 a D − h − Φ G − ( D − G i" a h − G ih ( e + P ) u i u i i com ′ − ~ m a D − h − Φ G − ( D − G i ∆ X ρ ′ − ~ m a D − ρ ′ ∆ X h Φ G + ( D − G i . (220)We also have a − ( D − F ′ + 2 m a ~ h G − ( D − G i F + = ∆ X "h G − ( D − G i F + − F + ∆ X h Φ G − ( D − G i − h G − ( D − G i Z d D − p (2 π ~ ) D − p ~ F + − ∂∂X k "h G − ( D − G i Z d D − p (2 π ~ ) D − p k ~ F − . (221)48e also write down the following relation for the fluid-four velocity for convenience h − Φ G − ( D − G i ∂∂X k "h G + 2Ψ G i Z d D − p (2 π ~ ) D − p k ~ F − = − a D h − Φ G − ( D − G i ∂∂X k "h G − ( D − G ih ( e + P ) u k u i com + a ( D − ~ m h − Φ G − ( D − G i ∂∂X k "h G +2Ψ G i ∂∂X k "h − Φ G − ( D − G i ρ ′ . (222)We plug everything in and indeed end up after a series of straightforward manipulations with, "h ( e + P ) u u i com + P ′ + ( D − h H − Ψ ′ G ih ( e + P ) u u i com + h − Φ G + ( D − G i ∂∂X k "h G − ( D − G ih ( e + P ) u k u i com − h H − Ψ ′ G ih ( e + P ) u k u k i com = 0 , (223)which agrees with the result (214) of the more general derivation in appendix A.3. Thedefinition of the rest-mass in (108) yields ρ = e − P , (224)such that h ( e + P ) u µ u µ i com = − ( ρ + 2 P ) . (225)In order to find the non-relativistic limit of this equation we have to define proper irreduciblefluid velocity v i := − ρ − h G + Φ G ih ( e + P ) u i u i com " e + P ) − h ( e + P ) u k u k i com / . (226)We then have the non-relativistic continuity equation in an FLRW-universe " ∂ η ρ + ( D − H ρ + ∂ i h ρ · v i i O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε p (cid:17) = 0 . (227)49 ocal momentum conservation. Now, we continue with the Euler equation by integrat-ing equation (155), " Z d D − p (2 π ~ ) D − p i ~ F − ′ = 14 ∂ ∂X i ∂X k h Φ G − ( D − G i ∂∂X k h a D − F i + 12 ∂∂X i h Φ G − ( D − G i Z d D − p (2 π ~ ) D − h ∆ X − p ~ ih a D − F i − h G − ( D − G i Z d D − p (2 π ~ ) D − p i p k ~ ∂∂X k h a D − F i − ∂∂X k h Φ G − ( D − G i Z d D − p (2 π ~ ) D − p i p k ~ h a D − F i − m ~ a ∂∂X i h Φ G − ( D − G ih a D − F i − ∂∂X i h Φ G + ( D − G ih a − ( D − F i . (228)We plug in the expressions for the integrated two-point functions in terms of hydrodynamicalvariables " ~ m a D − ∂∂X i "h − Φ G − ( D − G i ρ ′ + h G − ( D − G i a D h ( e + P ) u u i i com ′ = ~ m a D − ∂ ∂X i ∂X k h Φ G − ( D − G i ∂∂X k ρ + ~ m a D − ∂∂X i h Φ G − ( D − G i ∆ X ρ − ∂∂X i h Φ G − ( D − G i" a D h − G ih ( e + P ) u k u k i com − a D − ~ m ∆ X ρ − ∂∂X k "h G − ( D − G i" a D h − G ih ( e + P ) u k u i i com − a D − ~ m ∂ ∂X i ∂X k ρ − a D ∂∂X i h Φ G − ( D − G i ρ + 12 a D ∂∂X i h Φ G + ( D − G ih ( e + P ) u u i com . (229)Manipulating these expressions finally leads to h ( e + P ) u u i i ′ com + h D H + Φ G − ( D − G i ′ h ( e + P ) u u i i com + ∂∂X i P + ∂∂X i Ψ G h ( e + P ) u k u k i com − ∂∂X i Φ G h ( e + P ) u u i com + h − Φ G + ( D − G i ∂∂X k "h G − ( D − G ih ( e + P ) u k u i i com = 0 , . (230)which agrees with the result (215) of the more general derivation in appendix A.3. Definingthe proper composite velocity term h v i v k i com := ρ − h G − Φ G ih ( e + P ) u i u k i com , (231)50e find0 = " ∂ η h ρ · v i i + D H h ρ · v i i + ∂ k h ρ · h v i v k i com i + ρ δ ij ∂ j Φ G + δ ij ∂ j P × " O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε p (cid:17) . (232)We evaluate the apparent pressure term,0 = ( ∂ η h ρ · v i i + D H h ρ · v i i + ρ δ ik ∂ k Φ G + ∂ k h ρ · h v i v k i com i − ~ m a h ∂ ∂X i ∂X k ρ + δ ik ( D − H ∂ η ρ + δ ik ∂ η ρ i × ( O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε p (cid:17)) . (233)As already mentioned above, the composite operators contains information about anisotropyor in other words a non-vanishing second cumulant. Using the definition in (95) togetherwith the definition (226) we find the stress tensor in terms of proper fluid velocities ρ · h v i v k i com = " δ ij δ km σ jm + ρ · v i · v k + δ ij δ kl ~ m a ∂ j ∂ l ρ × " O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε p (cid:17) , (234)and thus0 = ( ∂ η h ρ · v i i + D H h ρ · v i i + ρ δ ik ∂ k Φ G + ∂∂X k " δ ij δ km σ jm + ρ · v i · v k − ~ m a h δ ik ( D − H ∂ η ρ + δ ik ∂ η ρ i × ( O (cid:16) ε g (cid:17) + O (cid:16) ε ~ (cid:17) + O (cid:16) ε p (cid:17)) = 0 . (235)The equations we find here are identical to the equations (214) and (217), except that weidentify the perturbation parameters on a more fundamental level and plugged in a concreteexpression for the composite velocity term and the apparent pressure.Let us once more point out that a local rest-mass-density conservation due to some internalsymmetry is not necessary to derive typical fluid equations in an FLRW-universe once weimpose a large mass limit. This is the reason why a minimally-coupled real scalar field iswell-suited to model the fluid dynamics of cold dark matter. It is of course true that the realscalar field model has no number current that is locally conserved on all scales (in contrastto a complex scalar field for example) and thus the probability associated to a single-particle51ave function of a real scalar field in Minkowski space is a priori not conserved. However,the energy-momentum tensor is another locally conserved quantity in the minimally-coupledreal scalar field theory. Imposing that the mass m dominates all other scales, the energydensity current coincides with the rest-mass density current with corrections given by termsthat go like momentum over mass, Hubble rate over mass and so on which can be tuned to besmall by imposing a certain initial momentum distribution for instance. 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