May 6, 2019

Cosmic Structure Formation with Kinetic Field Theory

1, 1 1,2 1 1 Matthias Bartelmann ∗, Elena Kozlikin , Robert Lilow , Carsten Littek , Felix Fabis , Ivan Kostyuk1, Celia Viermann1,3, Lavinia Heisenberg4, Sara Konrad1, and Daniel Geiss1,5

regions, so-called voids, surrounded by filaments which Kinetic Field Theory (KFT) is a statistical field theory for intersect in knots. The typical size of the voids is of order an ensemble of point-like classical particles in or out of 1 1 10h− Mpc . The knots form the so-called galaxy clus- equilibrium. We review its application to cosmological ters. In addition to this final, present-day state of cosmic structure formation. Beginning with the construction of structures, we can also observe seed structures in the very the generating functional of the theory, we describe in early universe. According to the well-established standard detail how the theory needs to be adapted to reflect the model of cosmology (see [1] for a review), the universe expanding spatial background and the homogeneous originated in a hot, dense state called the Big Bang. The electromagnetic heat radiation left over from this event and isotropic, correlated initial conditions for cosmic was predicted in [2], observed in [3] and confirmed to have structures. Based on the generating functional, we de- a near-perfect black-body spectrum with a temperature velop three main approaches to non-linear, late-time of 2.7K in [4]. It forms the so-called cosmic microwave cosmic structures, which rest either on the Taylor expan- background (CMB), which was released when the cosmic sion of an interaction operator, suitable averaging proce- plasma recombined, approximately 400,000 years after dures for the interaction term, or a resummation of per- the Big Bang. turbation terms. We show how an analytic, parameter- The CMB is almost perfectly isotropic. At closer in- free equation for the non-linear cosmic power spectrum spection, it reveals temperature fluctuations with a rel- 5 ative amplitude of 10− . We have good reasons to be- can be derived. ≈ lieve that these temperature fluctuations trace the cosmic We explain how the theory can be used to derive the structures present very shortly after the Big Bang. Notwith- density profile of gravitationally bound structures and standing the important question as to how these struc- use it to derive power spectra of cosmic velocity densi- tures originated, we consider them as reflecting the initial ties. We further clarify how KFT relates to the BBGKY state of the evolved cosmic structures we see today. Fig- hierarchy. We then proceed to apply kinetic field theory ure1 shows both, the initial state as reflected by the CMB to fluids, introduce a reformulation of KFT in terms of temperature fluctuations as observed by the Planck satel- macroscopic quantities which leads to a resummation lite [5], and the final state as shown by a galaxy survey conducted at 2µm wavelength (the 2MASS survey, [6]). scheme, and use this to describe mixtures of gas and dark matter. We discuss how KFT can be applied to study cosmic structure formation with modified theories

of gravity. As an example for an application to a non- ∗ corresponding author cosmological particle ensemble, we show results on 1 Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidel- arXiv:1905.01179v1 [astro-ph.CO] 3 May 2019 the spatial correlation function of cold Rydberg atoms derived from KFT. berg, Germany 2 Department of Physics, Technion, Haifa 3200003, Israel 3 Universität Heidelberg, Kirchhoff-Institut für Physik, Im Neuen- heimer Feld 227, 69120 Heidelberg, Germany 1 Introduction 4 Institut für Theoretische Physik, ETH Zürich, Wolfgang-Pauli- Strasse 27, 8093 Zürich, Switzerland In our cosmic neighbourhood, we see ourselves sur- 5 Institut für Theoretische Physik, Universität Leipzig, Brüderstr. rounded by rich and pronounced, large-scale structures. 16, 04103 Leipzig, Germany Considering individual galaxies as the smallest con- 1 Astronomical units of length: 1Mpc 3.1 1024 cm; h is the =1 · 1 stituents of these structures, we see large, almost empty Hubble constant in units of 100kms− Mpc−

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tions of ideal hydrodynamics and evaluates them on an expanding background to show that the density contrast grows in place proportional to the so-called linear growth factor D , +

δ(a) δ0 D (a) , (1) = + with a being the cosmic scale factor describing the expan- sion of the universe (see Sect.3 below), and δ0 the density contrast at an arbitrary reference time when D is set to + unity. As long as the growth remains linear, structures do not change their size against the expanding background. A Fourier analysis in wave numbers co-moving with the mean cosmic expansion shows that the Fourier modes of the density-contrast field neither couple to each other nor change their wave number with time. Large-scale density- fluctuation modes are still linear today. Despite formidable efforts and successes, it has turned out to be quite hard to follow cosmic structure formation into the non-linear regime with analytic means. See [11] for an excellent review on standard perturbation theory, Figure 1 These two full-sky maps illustrate the problem of cos- [12–21] as examples for several innovative approaches, mic structure formation. The top panel shows the temperature [22–30] as examples for Lagrangian perturbation theory fluctuations in the Cosmic Microwave Background as observed and [31–33] for recent developments towards an effec- by the Planck satellite [5]. They reflect the initial conditions for tive field theory of cosmic structure formation. One of cosmic structures at approximately 400,000 years after the Big the main reasons for this is that the fluid description of Bang. The bottom panel shows the distribution of galaxies in the predominantly dark matter must ultimately fail where our cosmic neighbourhood as observed by the 2-MASS survey and when the matter flow converges. When streams meet [6]. in a fluid, a discontinuity or a shock forms, keeping the velocity field of the flow unique. This is a consequence of the central assumption of (ideal) hydrodynamics that the mean-free path of the fluid particles is negligibly It is for several reasons most important for cosmology small. When dark-matter streams meet, however, multiple to understand the evolution, the amplitude and the mor- streams will form that simply cross each other, leading to phology of these structures. In the cosmological standard a multi-valued velocity field. This notorious shell-crossing model, reconciling the amplitudes of the initial and the problem of cosmic structure formation is most likely the final state is possible only if we assume that by far most of most important reason hampering progress in analytic the matter in the universe does not partake in the electro- theories of cosmic structure formation. magnetic interaction, and is therefore called dark matter Of course, one can resort to fully numerical simula- [7]. If this was not the case, the amplitude of the CMB tions. Corresponding algorithms have reached an impres- temperature fluctuations would have to be two orders of sive level of sophistication and have delivered overwhelm- magnitude larger. Within the limits of even the most pre- ingly detailed results that, by and large, agree very well cise measurements, the initial state can be considered as with observations. Yet, good reasons remain for trying to a Gaussian random field [8]. The formation of filamentary understand the formation of late-time, non-linear cos- structures from a Gaussian random field has been shown mic structures on an analytical basis. Conceptually the to be a natural consequence of gravitational collapse of most important of these reasons is that numerical simu- overdense regions against the cosmic expansion [9,10]. lations strictly speaking do not explain the properties of It is quite straightforward to analyze the early stages cosmic structures, even though they succeed in reproduc- of cosmic structure formation analytically as long as the ing them in astounding detail. Of particular importance relative density fluctuations are small, more precisely as are universal features of cosmic structures, such as the long as the so-called density contrast δ (cf. Eq.45) is less density profiles of gravitationally-bound objects. Analytic than unity. The standard procedure begins with the equa- theories, on the other hand, could be expected to trace

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physical properties of non-linear structures back to their how KFT can be used to explain the density profiles of foundations in fundamental physics. dark-matter haloes. Section6 presents the derivation of A second convincing reason in favour of an analytic velocity power spectra. In Sect.7, we establish the con- theory is its flexibility against changing assumptions. Run- nection between KFT and the BBGKY hierarchy of kinetic ning sufficiently detailed and well-resolved numerical theory, offering a closure condition for the hierarchy. We simulations is computationally expensive, and scanning discuss in Sect.8 how fluids can be incorporated into KFT. wider parameter ranges is thus often forbiddingly time In Sect.9, we give an exact re-formulation of KFT purely consuming. A third good reason is that shot noise is in- in terms of macroscopic fields and show how this leads evitable in numerical simulations, caused by the finite to an efficient resummation of perturbation terms. The number of particles in any simulation volume. An analytic fluid description and the macroscopic re-formulation are theory could be expected to allow taking the thermody- then combined in Sect. 10 to describe mixtures of dark namic limit. Higher-order measures for the statistics of the matter and gas with KFT. We outline in Sect. 11 how KFT cosmic structures, such as three- or four-point correlation can be extended to be applied within modified theories functions which are indispensable to quantify non-linear of gravity, and illustrate this outline with results for one and in particular non-Gaussian structures, could then be specific example. Finally, in Sect.12, we give an outlook calculated without uncertainties due to shot noise and on how to apply KFT to ensembles of cold Rydberg atoms the finite number of samples that can be drawn from any as one example for a completely different physical system. numerical simulation. We summarize this review in Sect. 13. Kinetic field theory is based on field-theoretical de- scriptions of classical particles [34,35] and was developed mainly for studying glasses, fluctuation-dissipation the- 2 Kinetic Field Theory orems and the ergodic-non-ergodic transition [36–39]. We have adapted it to cosmology and applied it to var- 2.1 Generating Functional ious aspects of cosmological structure formation [40–44], but also to completely different kinds of physical systems Kinetic field theory is a statistical field theory for classical whose elementary constituents can be described as classi- particle ensembles in or out of equilibrium. Its central cal degrees of freedom. For cosmology, the KFT approach mathematical object is a generating functional Z . In close has several decisive advantages, the most important of analogy to the partition sum of equilibrium thermody- which is that the motion of the classical particles is de- namics, this generating functional integrates the proba- scribed in phase space by Hamilton’s equations. Since bility distribution P(ϕ) for system states ϕ over the state trajectories in phase space do not cross, this approach space, avoids the shell-crossing problem by construction. KFT Z incorporates the dynamics of the particles into a gener- Z DϕP(ϕ) . (2) ating functional. Structurally, this generating functional = resembles the generating functionals of statistical quan- It is generally a path integral if the states space is a func- tum field theories. One important simplification of the tion space of ïˇn˛Aeld conïˇn˛Agurations. application to classical particles is due to the symplectic For classical (canonical) ensembles consisting of N nature of Hamilton’s equations, which gives rise to Liou- point particles, we can immediately specify the generating ville’s theorem. Thus, by construction, KFT is built on a functional further. The state space is then the phase space diffeomorphic map with unit functional determinant of Γ of the N particles whose trajectories (q j ,p j ) : x j with = an initial phase-space configuration to any later time. 1 j N are tuples of position q j and momentum p j . ≤ ≤ We begin this review in Sect.2 with the foundations To allow a compact notation, we bundle the phase-space of the theory and the construction of its central object, trajectories x j into a tensorial object i.e. its generating functional. We specialize it to cosmol- x x j e j , (3) ogy in Sect.3 and summarize several further develop- = ⊗ ments in Sect.4, mainly offering different ways of tak- where summation over j is implied and the vector e j has ing particle interactions into account in approximate or components (e ) δ , 1 i N. For such tensors, we j i = i j ≤ ≤ averaged ways. Here, we derive a closed, analytic, non- define the scalar product perturbative, parameter-free equation for the non-linear x, y ¡x y ¢¡e e ¢ x y . (4) density-fluctuation power spectrum which agrees very 〈 〉 = i · j i · j = j · j well with the results from numerical simulations up to Point particles are described by Dirac delta distributions 1 wave numbers of k 10h− Mpc. In Sect.5, we review instead of smooth fields. The path integral in (2) then ≈

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turns into an ordinary integral over the N-particle phase time t space, Z Z dx P(x) . (5) =

We introduce a generator field J conjugate to the phase- space trajectories x(t) into the generating functional Z , trajectories (p, q)(t) Z ½ Z ¾ ∞ ­ ® Z Z [J] dx P(x) exp i dt 0 J(t 0),x(t 0) , (6) → = 0 momentum p such that the functional derivative of Z [J] with respect to the component J j (t) of the generator field returns the average position x j (t) of particle j at time t, 〈 〉 position q ¯ δ ¯ x (t) i Z [J]¯ . (7) j ¯ 〈 〉 = − δJ j (t) J 0 = Figure 2 Illustration of the main idea of KFT. An initial proba- Like x, the generator field has components conjugate to bility distribution on the phase space is mapped to any later position q and momentum p, time by means of the Hamiltonian flow. Since phase-space trajectories do not cross, this approach avoids the notorious à ! Jq shell-crossing problem by construction. J J e j e . (8) = j ⊗ j = ⊗ j Jp j

We further split the probability P(x) for the state x with a functional delta distribution singling out the deter- to be occupied into a probability P(x(i)) for the particle ministic trajectories solving the equations of motion. ensemble to occupy an initial state x(i) at time t 0, and For classical point particles, the equation of motion = the conditional probability P(x x(i)) for the ensemble to for a single particle is the Hamiltonian equation, | move from there to the time-evolved state x, x˙ J x H 0 , (13) Z − ∇ = P(x) dx(i) P ¡x x(i)¢P ¡x(i)¢ . (9) = | where H is the Hamiltonian function on the phase space and For particles on classical trajectories, the transition prob- à ! ability P(x x(i)) must be a functional delta distribution of 0 I3 | J : (14) the classical equation of motion, = I 0 − 3 ¡ (i)¢ £ ¡ (i)¢¤ P x x δD x Φcl x , (10) is the usual symplectic matrix, with In denoting the unit | = − matrix in n dimensions. Let us first assume the particles to (i) where Φcl(x ) denotes the classical (Hamiltonian) flow be non-interacting, in which case the Hamiltonian equa- on the phase space, beginning at the initial phase-space tions are linear and admit the construction of a Green’s (i) points x of the particle ensemble. Writing the equation function G(t,t 0). To prepare for the inclusion of interac- of motion in the form E(x) 0, the Hamiltonian flow tions, we augment this free equation of motion by an in- = consists of all solutions of this equation for initial points homogeneity or source term K , within a certain domain of phase space, as illustrated in x˙ J H K , (15) Figure2. − ∇x = Denoting the solution of the classical equations of mo- and write its solution x¯(t) beginning at x(i) in terms of the tion beginning at x(i) as Green’s function as ¡ (i)¢ Z t E x,x 0 , (11) (i) = x¯(t) G(t,0)x dt 0G(t,t 0)K (t 0) . (16) = + 0 we can write the transition probability as We shall henceforth assume that the inhomogeneity is P ¡x x(i)¢ δ £E ¡x,x(i)¢¤ (12) caused by an interaction with the potential V . Then, we | = D

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can write where we have introduced the conventional short-hand notation (k ,t ) :(j). We shall abbreviate the integral à ! j j = 0 expressions in (23) as K (t 0) . (17) = V (t 0) −∇ Z Z Z Z d3k : d3q , : . (25) The Green’s function G itself is a 6 6 matrix which we = = 3 × q k (2π) write as à ! Replacing the particle position q j (t1) by a functional gqq (t,t 0)I3 gqp (t,t 0)I3 derivative with respect to the generator-field component G(t,t 0): . (18) = 0 gpp (t,t 0)I3 Jq j (t1), we find the density operator

N Its component functions gqq (t,t 0), gqp (t,t 0), and gpp (t,t 0) X will be specified later in a cosmological context. For sim- ρˆ(1) ρˆ j (1) , (26) = j 1 plicity, we abbreviate g (t,0) : g (t). = qp = qp For the entire particle ensemble, we introduce the N- composed of the one-particle density operators particle Green’s function G(t,t 0): G(t,t 0) IN and the N- = ⊗ à ! component source field K : K j e j . Then, the solutions δ = ⊗ ˆ (1) : exp k . (27) of the equations of motion for the particle ensemble are ρ j 1 = − δJq j (t1) E ¡x,x(i)¢ x(t) x¯(t) 0 . (19) = − = For indistinguishable particles, which we shall henceforth assume, we can set the particle index j to an arbitrary Inserting this expression into (12), the result into (9) and value without loss of generality. Since the operator (27) (i) the probability distribution P(x ) into the generating is an exponential of a derivative with respect to the gen- functional Z [J] from (6) gives erator field, it corresponds to a finite translation of the Z ½ Z ¾ generator field. After applying r 1 of these operators, ∞ ­ ® ≥ Z [J,K ] dΓ exp i dt 0 J(t 0),x¯(t 0) , (20) the generator field is translated by = 0 r à ! where we have introduced the initial phase-space mea- X ¡ ¢ 1 J J δD t 0 t j k j e j . (28) sure → − j 1 − · 0 ⊗ = (i) ¡ (i)¢ dΓ : dx P x . (21) Setting the generator field J to zero after application of = the density operator turns the generating functional into Z i Lq ,q i Lp ,p iS 2.2 Density Operator Z [L] dΓe + + I (29) = 〈 〉 〈 〉

The standard application of KFT in this review will con- with the components cern the calculation of density power spectra. We shall r thus proceed to define a density operator and study its X L k e , action on the generating functional (20). The number den- q j j = − j 1 ⊗ = sity of the particles at time t1 is r X N Lp k j gqp (t j ) e j , (30) X ¡ ¢ = − j 1 ⊗ ρ(q,t1) δD q q j (t1) . (22) = = j 1 − = of the translations and the interaction term

In a Fourier representation, defined by the convention r X Z t j SI k j dt 0 gqp (t j ,t 0) j V (t 0) . (31) Z Z 3 = · 0 ∇ 3 ikq d k ikq j 1 f˜(k) d q f (q)e− , f (q) f˜(k)e , (23) = = = (2π)3 In (29), q and p in the exponent are the initial particle the density mode with the wave number k1 is positions and momenta,

N Ã ! X ik1q j (t1) q (i) ρ˜(k1,t1) : ρ˜(1) e− , (24) x . (32) = = j 1 p = =

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The generating functional (20) and the result (29) after acting on the free generating functional, applying r one-particle density operators and setting the ˆ Z [L] eiSI Z [L] , (40) source field J to zero are fully general in the sense that = 0 neither the initial particle distribution in phase space, nor with the Green’s function, nor the interaction potential have Z i Lq ,q i Lp ,p been specified yet. So far, we have just assumed that a Z0[L] dΓe + . (41) = 〈 〉 〈 〉 Green’s function and an interaction potential exist. One approach to perturbation theory can now begin with a Taylor expansion of exponential factor in terms of the in- 2.3 Interaction Operator teraction operator. Anticipating the specialisation of KFT to cosmology detailed in Sect.3, Fig.3 shows cosmologi- We suppose that the potential V is a linear superposition cal results obtained in first-order perturbation theory [42]. of contributions by individual particles, A summary of this calculation is presented in Appendix A.

N N Z X ¡ ¢ X ¡ ¢ V (q,t) v q q j (t) v(q y)δD y q j (t) = j 1 − = j 1 y − − 3 Specialisation to Cosmology = = Z v(q y)ρ(y,t) , (33) We shall now specify the theory to its cosmological ap- = − y plication. Where not stated otherwise, we use the cos- where we have identified the density ρ from (22). The mological parameters Ωm 0.3, Ω 0.7, Ω 0.04 for = Λ = b = potential gradient at position qi is the density parameters of matter, the cosmological con- Z stant, and baryonic matter, respectively; further h 0.7 ¡ ¢ = V (q,t) δ q q (t) V (q,t) (34) for the dimension-less Hubble constant and σ8 0.8 for i D i = ∇ = q − ∇ the normalisation of the power spectrum. Moreover, we Z Z choose the form of the cold-dark matter transfer function ρi (q,t) V (q,t) : Bi (q,t)V (q,t), = q ∇ = − q provided by [10] to specify the initial power spectrum. where we have introduced the response field

B (q,t): δ ¡q q (t)¢ (35) 3.1 Equation of Motion for Point Particles i = −∇ D − i after partial integration. With (33), we can write this result The cosmological standard model is built upon the theory as of general relativity and two symmetry assumptions. They Z Z assert that there exists a mean flow in the Universe with i V (q,t) Bi (q,t)v(q y)ρ(y,t) . (36) respect to which all sufficiently averaged observable prop- ∇ = q y − erties appear spatially isotropic and homogeneous. This Applying the convolution theorem, this expression turns mean flow can then only either expand or shrink isotrop- into ically and must thus be characterised by a scale factor Z a(t), depending only on cosmic time t. The dynamics V (q,t 0 ) B˜ ( 10)v˜(10)ρ˜(10) (37) ∇i 1 = i − of a(t) is determined by Einstein’s field equations which, k10 under the symmetry assumptions made, simplify to Fried- where ( 10) :( k0 ,t 0 ). The Fourier transform of the den- mann’s equations. The relative cosmic expansion rate is − = − 1 1 sity ρ is expressed by the operator ρˆ from (26), while we the Hubble function H(a) a˙/a. Usually, a is normalised = can assign the operator to unity today, but it is more convenient for our purposes to normalise a to unity at the initial time t 0 such that ˆ ik10 qi (t10 ) = Bi ( 10) ik10 e ik10 ρˆi ( 10) (38) a(0) : a 1. − = = − = i = to the response field. The existence of the mean flow suggests to introduce 3 3 coordinates q comoving with the flow, defined in terms With the definition d10 : dt 0 d k0 /(2π) , this allows us = 1 1 of the physical coordinates r via r aq. The Lagrange to write the interaction term SI as the interaction operator = function of point particles of mass m, expressed in terms of the comoving coordinates, is r X Z t j Sˆ k d1 g (t ,t )Bˆ ( 1 )v˜(1 )ρˆ(1 ) (39) m I j 0 qp j 10 j 0 0 0 L(q,q˙,t) a2q˙2 mΦ (42) = j 1 · 0 − = = 2 −

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105 time early in the matter-dominated epoch, as we intend linearly evolved to do throughout, we can safely set Ω 1 and thus bring KFT, 1st-order term mi = 104 KFT, non-linear the Poisson equation (43) into the form Cosmic Emulator ¯ 3 2 3 2 δ ρ ρ

] 10 Φ H , δ : − , (45) 3 ∇ = 2 i a = ρ¯

Mpc 2 3 10 with the density fluctuations now being expressed by the − h [

δ dimension-less density contrast δ. P 101 Linear perturbation theory shows that the density fluctuations δ grow proportional to the so-called linear 100 growth factor D . It is convenient to replace the cosmic + time t by the growth factor, 1 10− t D (t) D (0) , (46) 1 0.4 → + − + − 0.2 where the time t 0 is now set to some initial time which emu

P = / 0.0 is late enough for matter to dominate over radiation, but

KFT early enough for density fluctuations to be very small. A

P 0.2 − 0.001 0.01 0.1 1 10 suitable choice is the time when the cosmic microwave 1 wave number k [h Mpc− ] background was released. In this new time coordinate, the Lagrange function reads g L¯(q,q˙,t) q˙2 Φ¯ (q) (47) Figure 3 Example of a perturbative result in a cosmological = 2 − application. (Upper panel) The dark-blue curve shows the ini- ¯ tial CAMB density-fluctuation power spectrum [45](z 1100), (see e.g. [40]), with the potential Φ satisfying the Poisson i = linearly evolved to the present time. The light-blue curve is equation the first-order perturbation term with improved Zel’dovich tra- 3 a 2Φ¯ δ . (48) jectories, with dashed parts illustrating negative values. The ∇ = 2 g curve thus illustrates how power is removed from intermediate and transported to smaller scales. The red curve is the sum Here, g(t) is defined to be the function of the linearly evolved power spectrum and the first-order per- 2 dlnD turbation term. The black curve finally shows the non-linearly g(t): a D f H , f : + , (49) = + = dlna evolved power spectrum obtained from numerical simulations. Already at first order, the analytic result comes close to the fully normalised to unity at t 0 or a ai. Notice that the ¯ = = numerical result. (Lower panel) The relative deviation of the potential Φ defined in (48) now has the dimension of a KFT result as compared to the numerical simulation result of squared length. The equation of motion reads Cosmic Emulator [46–49] is shown for the k-range provided by g˙ Φ¯ Cosmic Emulator. q¨(t) q˙(t) ϕ 0 , ϕ : , (50) + g + ∇ = = g

with the potential ϕ now obeying the Poisson equation (see, e.g. [50]), where Φ is the gravitational potential 3 a sourced by the fluctuations of the matter density ρ around 2 ϕ 2 δ . (51) its time-dependent mean ρ¯ via the Poisson equation ∇ = 2 g The usual Legendre transform of the Lagrange func- 2Φ 4πGa2 ¡ρ ρ¯¢ . (43) ∇ = − tion (47) leads to the Hamiltonian The mean cosmic density after the radiation-dominated p2 epoch is H¯ (q,p) Φ¯ (52) = 2g + 2 3Hi 3 ρ¯(a) a− Ωmi , (44) and thus to the Newtonian equation of motion = 8πG à ! p/g where Hi and Ωmi are the Hubble function and the cosmic x˙ , (53) matter-density parameter at the initial time. If we set this = Φ¯ −∇

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which is solved by the Green’s function Another choice for the propagator gqp begins with defining the effective force term such that only its velocity- Ã ! Z t I3 g¯qp (t,t 0) dt¯ dependent contribution is initially absent, G¯(t,t 0) , g¯qp (t,t 0) . (54) = 0 = t g(t¯) I3 0 g˙ 1 f (t) hq˙ ϕ , h : 1 . (59) = g − ∇ = g − It is important to note that g¯qp (t,t 0) is limited from above because of the cosmic expansion. Consequently, the iner- The remaining initial force then causes the particles to tial trajectories described by this Green’s function deviate lag behind the inertial motion. This choice has been sug- strongly from the true, fully interacting trajectories. It will gested to avoid part of the re-expansion in the Zel’dovich thus be advantageous to find a replacement for g¯qp that approximation of cosmic structures after their formation already captures part of the gravitational interaction. An [40]. Relative to this effective force, the equation of motion example for such a replacement is given by the Zel’dovich (50) reads approximation [51], but we prefer a slightly more general approach here. q¨(t) h˙(t)q˙(t) f (t) , (60) − = which implies the homogeneous solution q˙(t) q˙(0)eh = 3.2 Particle Trajectories and Effective Force and thus the propagator Z t h(t¯) h(t ) We wish to solve the equation of motion (50) with an ex- g (t,t 0) dt¯e − 0 . (61) qp = pression of the form t 0 At late times, h 1, and the propagator (61) approaches Z t → − e 1 times the Zel’dovich propagator. q(t) q0 gqp (t)q˙0 dt 0 gqp (t,t 0) f (t 0) (55) − = + + 0 Since the peculiar velocity q˙ itself depends on the time- integrated force f (t), (58) and (59) are integral equations such that the propagator g (t,t ) can play the role of the qp 0 for the force. After transforming them to differential equa- q-p component of the Green’s function G(t,t ). Taking the 0 tions, they can be solved by variation of constants. The first two time derivatives of (55) gives solution of (59), which we will use here together with the propagator (61), turns out to be Z t q¨(t) g¨qp (t)q˙0 g˙qp (t,t)f (t) dt 0 g¨qp (t,t 0)f (t 0). gh˙ Z t = + + 0 f (t) ϕ 2 g ϕdt 0 . (62) (56) = −∇ − g 0 ∇ We shall use this expression later as a starting point for an For (56) to agree with the equation of motion (50), the averaging and an approximation scheme. effective force f (t) has to be appropriately adapted once gqp has been chosen. A frequent and convenient choice in cosmology is the Zel’dovich approximation [51], which 3.3 Initial Conditions describes particle trajectories as inertial in the time coor- dinate t D , = + Supported by observations of the temperature fluctua- tions in the cosmic microwave background, we can as- gqp (t,t 0) t t 0 . (57) = − sume that the density fluctuations in the early universe can be characterised as a Gaussian random field (for re- Inserting this particular choice into (56) and comparing cent empirical support of this assumption, see [8]). By to (50) immediately results in the Helmholtz theorem, the peculiar velocity field can be g˙ decomposed into a curl and a gradient. By the cosmic ex- f (t) q˙ ϕ . (58) = − g − ∇ pansion and angular-momentum conservation, the curl component will quickly decay, so we can model the initial This is thus the effective force term in the Zel’dovich ap- peculiar velocity as the gradient of a velocity potential Ψ. proximation. It is chosen such that the total initial force, Then, by the continuity equation, the density contrast δ i.e. the sum of the potential gradient and the velocity- has to satisfy the Poisson equation δ 2Ψ. Thus, given = −∇ dependent contribution, vanishes initially, corresponding the velocity potential, both density contrast and peculiar to the initial inertial motion of the particles. velocity will be determined. The velocity potential also

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has to be a Gaussian random field. If the density contrast 4 Further Developments is statistically characterised by its power spectrum Pδ(k), the velocity potential has the power spectrum After these preparations, we return to some further devel- 4 opments simplifying the evaluation of the free generating P (k) k− P (k) (63) Ψ = δ functional (41) and the interaction term (31). according to the Poisson equation. With the peculiar ve- locity and the density contrast both being Gaussian ran- dom fields derived from the velocity potential, either of 4.1 Factorisation of the Free Generating Functional the power spectra PΨ(k) or Pδ(k) completely determines their statistical properties. Note that we assume irrota- With the Gaussian initial distribution (64) of particle po- tionality of the velocity field and thus absence of vorticity sitions and momenta, the momenta can immediately be at the initial time only. During the subsequent nonlinear integrated out in the free generating functional (41), lead- evolution of the particles vorticity can and will neverthe- ing to less be created. Z µ 1 ¶ To specify the generating functional of KFT, we need N i Lq ,q Z0[L] V − dq exp L>p Cpp Lp e 〈 〉 . (67) the probability distribution P(x(i)) for the initial phase- = −2 space coordinates of the point particles of our ensemble. With the momentum-correlation matrix depending only Drawing particle positions randomly by a Poisson pro- on the relative particle separations, this remaining expres- cess with a probability proportional to the density con- sion can be fully factorised. In the simplest case of a two- trast δ 2Ψ, and assigning momenta to these par- = −∇ point function, leaving out an irrelevant delta distribution, ticles proportional to Ψ, the probability distribution ∇ the result is P(x(i)) P(q,p) turns out to be = 3 2 Q0 Z0[L] (2π) δD (k1 k2) V − e− P (k1,t) , (68) N µ ¶ = + V − 1 P(q,p) (p) exp p C 1 p , q C > pp− where the non-linearly evolved power spectrum P (k1,t) = 3N −2 (2π) detCpp is given by (64) Z ¡ Q ¢ ik1 q P (k1,t): e 1 e · (69) where Cpp is the momentum-correlation matrix depend- = q − ing on the particle positions [42]. For late-time cosmologi- with the expressions cal applications using the unbound Zel’dovich propagator (57) or its improvement (61), the factor C (p), which is 2 σ1 2 2 2 2 a polynomial in the momenta, can safely be set to unity. Q0 : gqp (t)k1 , Q : gqp (t)k1 a (q) . (70) = 3 = − ∥ We shall see in Sect.9 how C (p) can be fully taken into account. appearing in the exponentials. The quadratic form Q0 The momentum-correlation matrix is given by independent of q appearing in the exponential damping term derives from the momentum dispersion. 2 σ1 The function a (q) appearing in this expression is the Cpp I3 IN Cp j pk E jk , (65) ∥ = 3 ⊗ + ⊗ correlation function of those momentum components where j k and E e e . The variance σ2 is defined parallel to the line connecting the positions of the two 6= jk = j ⊗ k 1 by momenta being correlated. In terms of the power spec- trum P (k) of the density fluctuations, it is given by Z Z P (k) δ σ2 k2 P (k) δ . (66) 1 Ψ 2 2 = k = k k a (q) a1(q) µ a2(q) (71) ∥ = + The first term on the right-hand side of (65) is the mo- mentum dispersion caused by the momenta being drawn with Z from a Gaussian random field. The second term on the 1 ∞ j1(kq) a1(q) 2 dk Pδ(k) , right-hand side is the correlation between the momenta = −2π 0 kq of different particles, separated by q jk : q j qk . Since Z = | − | 1 ∞ we assume that the initial density fluctuations are a sta- a2(q) dk Pδ(k) j2(kq) . (72) = 2π2 tistically isotropic and homogeneous random field, the 0 momentum correlations can only depend on the absolute Here, the spherical Bessel functions j1,2(kq) appear, and value of the relative particle separation. µ is the cosine of the angle enclosed by k1 and q. Notice

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that, since j (x) x/3 and j (x) x2/5 for x 1, the A seemingly radical approximation of the interaction 1 ≈ 2 ≈ ¿ functions a1,2(q) and a (q) have the limits term (77) consists in replacing the potential gradient 1ϕ2 ∥ ∇ between the particles 1 and 2 by a suitable average. Simply σ2 σ2 1 1 averaging 1ϕ2 over all particle pairs would return zero a1(q) , a2(q) 0 , a (q) (73) ∇ → − 3 → ∥ → − 3 because of the statistical isotropy of the particle distribu- for q 0, and thus Q Q in the same limit. tion. It is thus important to realise that the interaction → → 0 If the exponential in (69) can be approximated by its term (77) contains the projection of the potential gradient Taylor expansion to first order, the power spectrum P on the wave vector k1 of the density mode considered. We thus need to calculate the projection k ϕ in a evolves linearly, 1 · ∇1 2 suitable average which does not vanish for particles corre- 2 P (k ,t) g (t)P (k ) . (74) lated with the density mode k1. 1 ≈ qp δ 1 The probability P12 for finding particles 1 and 2 at a separation q : q q has a spatially independent 12 = | 1 − 2| 4.2 Averaged Interaction Term mean contribution ρ¯ and a conditional contribution due to the particle correlations, expressed by the spatial corre- We now return to the interaction term (31) and evaluate it lation function ξ(q12) of the particles, for a two-point function at equal times, t1 t2 t, taking ¡ ¢ = = P12 ρ¯ 1 ξ12(q12) . (80) the constraint k1 k2 0 into account which is enforced = + + = by the delta distribution in (68). We then have For uncorrelated particles in a homogeneous random Z t field, the direction of 1ϕ2 is random with respect to k1, ¡ ¢ ∇ S (t) k dt 0 g (t,t 0) f (t 0) f (t 0) , (75) thus its contribution to the average must vanish, while I = − 1 · qp 1 − 2 0 the correlated part remains. We thus weigh the potential where fi is the force (62) acting on particle i. Dropping gradient ϕ with the conditional probability ξ(q ) and ∇1 2 12 the time argument for brevity, we bring the force terms take the Fourier transform of the result to obtain the aver- into the form age Fourier component of the potential gradient at wave

N number k1, X ¡ ¢ f1 f2 2f12 f1j f2j , (76) Z − = + − ik1 (q1 q2) j 3 ϕ (k ) ξ(q ) ϕ e− · − . (81) = 1 2 1 12 1 2 〈∇ 〉 = q12 ∇ where f is the force on particle i due to particle j, and i j By means of the Fourier convolution theorem, we can we have used f f 2f by Newton’s third law. If we 12 − 21 = 12 then write the Fourier transform of the averaged potential can neglect three-point correlations for now, the second gradient as a convolution of the Fourier transforms of term on the right-hand side of (76) can be neglected in an the potential gradient itself with that of the correlation isotropic random field since the forces exerted by particles function, i.e. with the power spectrum. For the latter, we j with j 3 on particles 1 and 2 will vanish on average ≥ take the linearly time-evolved initial density-fluctuation because there is no preferred direction they could point power spectrum P (k), damped on the free-streaming to. We can then simplify the interaction term to δ scale. Thus, we insert t Z 2 2 2 2 2 σ gqp k /3 SI(t) 2k1 dt 0 gqp (t,t 0) f12(t 0) (77) D P¯δ(k): D e− 1 Pδ(k) (82) = − · 0 + = + for the Fourier transform of the correlation function and containing the projection of the force f12 between parti- obtain cles 1 and 2 on the wave vector k1. The Fourier transform of the potential v of a unit point Z k κ ­ ® 2 1 − ¯ 1ϕ2 (k1,t) iA(t)D 2 Pδ(κ) . (83) mass satisfying the Poisson equation (51) is ∇ = − + κ (k κ) 1 − A(t) 3a The angular integral remaining in (83) can be carried v˜(t) , A(t) , (78) = − ρ¯ k2 = 2g 2 out and results in Z k κ Z 1 y2dy thus the gradient of the potential of particle 2 at the posi- 1 − ¯ 2 ˆ ¯ 2 Pδ(κ) k1k1 2 Pδ(k1 y) J(y) , (84) tion of particle 1 is κ (k κ) = 0 (2π) 1 − A(t) Z k where kˆ is the unit vector in direction of k , and y : κ/k . ik(q1(t) q2(t)) 1 1 1 1ϕ2(t) i e − . (79) = ∇ = − ρ¯ k2 We truncate the integration at y 1 (κ k1) to suppress k = =

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modes on scales smaller than the density fluctuation con- 105 sidered. The function J(y) is 104 Z 1 1 µy 1 y2 1 y − − + J(y): dµ 2 1 ln . (85) = 1 1 y 2µy = + 2y 1 y 3

] 10

− + + | − | 3 Inserting the averaged potential gradient (83) into the Mpc 2 3 10 force term (62), we obtain a scale-dependent, mean force − h [ term f12 (k1,t) weighed by the correlation function be- δ

〈 〉 P 1 tween correlated particle pairs, which defines the mean 10 interaction term via (77), 100 linearly evolved Z t ­ ® numerically simulated (Smith et al. 2003) SI(t) 2k1 dt 0 gqp (t,t 0) f12 (k1,t 0) , (86) KFT, mean interaction term 〈 〉 = − · 0 1 10− 1

which by construction does not depend on the particle − 0.1 positions or momenta any more. 0.0

Replacing SI by this average SI , we can thus pull the numerical

〈 〉 P

/ 0.1 interaction term in front of the integral (29) and use the − form (68) of the free generating functional to obtain the KFT 0.2 P − 0.001 0.01 0.1 1 10 closed expression 1 wave number k [h Mpc− ]

Q0 i S P¯ (k,t) e− + 〈 I〉P (k,t) (87) = for the non-linearly evolved, density-fluctuation power Figure 4 Upper panel: The analytic result (87), assuming an spectrum. Note that this expression is non-perturbative initial power spectrum as provided by [10] at z 1100, is i = and parameter-free. Comparing the perturbative first- shown here for the present epoch (red curve) and compared to order result shown in Figure3 with the power spectrum the linearly evolved power spectrum (blue curve) and the non- obtained from (87) shown in Figure4 clearly demonstrates linear power spectrum obtained from numerical simulations the significant improvement. [52] (black curve). Lower panel: The relative deviation of the The non-linear power spectrum in the mean-field ap- KFT result from the numerical simulation result of [52] is shown. 1 proximation shown in this Figure falls below the numer- The agreement up to wave number k 10h Mpc− is very 1 ≈ ically simulated spectrum for k & 10h Mpc− . This does good. not indicate a conceptual breakdown of KFT, but rather a limitation of the mean-field approximation of the force 1 term. The scale of 10h Mpc− at z 0 is thus not of fun- Exchanging the order of integration over k and t, we find ≈ = damental importance, but depends on the normalisation i Z k · gh˙ Z t ¸ of the power spectrum. ik qi j iK pi j iK pi j fi j (t) 2 e · Ae · 2 dt 0 g Ae · = ρ¯ k k + g 0 (89)

4.3 Interaction in the Born Approximation The second term in brackets in (89), gh˙ Z t Let us now return to the force term (58) with the potential iK pi j B(κ,t): 2 dt 0 g Ae · (90) gradient expressed by its Fourier transform (79). If we wish = g 0 to avoid averaging over particle positions, as we did before, with κ : k p , is easily evaluated by numerically integrat- = · i j the essential difficulty is that the actual particle positions ing its real and imaginary parts. The potential of the force q1,2(t) appear in the Fourier phase. This difficulty can be term fi j from (89) thus has the effective, k-dependent by-passed by approximating the particle trajectories by amplitude their unperturbed or inertial trajectories, iK p A (κ,t): Ae · i j B(κ,t) , (91) eff = + q j (t) q j gqp (t)p j , j 1,2 , (88) → + = whose real and imaginary parts we show in Figs.5 and where q j and p j without a time argument are meant to in- 6. Figure5 shows the dependence on the scale factor for dicate initial particle positions and momenta. We abbrevi- three different choices of κ, while Fig.6 displays the de- ate q q q , p p p and introduce K g (t)k. pendence on κ for a 1. i j = i − j i j = i − j = qp =

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101 101 κ = 0.1 κ = 0.1 κ = 1 κ = 1 1 κ = 10 1 κ = 10 10− 10−

3 3 10− 10− eff eff A A Re 5 Im 5 10− 10−

7 7 10− 10−

9 9 10− 10− 0.001 0.01 0.1 1 0.001 0.01 0.1 1 scale factor a scale factor a

Figure 5 Real (left) and imaginary (right) parts of the effective amplitude Aeff(κ,t) of the interaction potential defined in (91). The curves are shown in dependence of the scale factor a and are parameterised by the scalar product κ : k p between = · i j wave vector k and initial momentum difference pi j . The momentum difference is scaled by σ1 here. Dashed parts of the curves indicate negative values.

5 10− but particles oscillate within them. Alternatively, these os- cillations can be interpreted as the effect of an oscillating effective propagator gqp , which mimicks the motion of particles in gravitationally-bound structures. 6 eff 10− Second, the imaginary part of the potential amplitude A is typically small at early times and for small values of κ,

Im Re, a = 1.0 , Im, a = 1.0 but becomes large at late times for large κ. This indicates eff

A damping by the transport of particles and the associated 7 Re 10− mixing of phases of particle trajectories. This damping justifies the exponential cut-off in the power spectrum (82) which we have used for calculating the mean force term before. 8 10− 0.01 0.1 1 10 It remains to be seen in comparison with numerical κ simulations how useful this variant of the Born approxi- mation will be for cosmological structure formation. The agreement between numerical simulations and the non- Figure 6 Real and imaginary parts of the effective potential linear power spectrum (87) including the averaged inter- V (κ,t) for a 1 in dependence of κ. As in Fig.5, dashed action, however, suggests that the Born-approximated in- eff = sections indicate negative values. teraction term may return similarly or even more accurate results.

These Figures show several interesting properties of the effective potential amplitude. First, for large values of 5 Halo profiles in KFT κ, i.e. for small scales or large initial momenta, the real part of Veff begins oscillating at late times. This indicates The formation of large-scale, dark-matter structures in that the displacement of the particles becomes compa- our KFT formulation as well as in numerical simulations is rable or larger than the structures they belong to, thus governed by a few physical properties only: the interaction reducing the effective potential amplitude at late times: potential between dark matter particles, the equations of structures on such scales are then not built up any more, motion governing particle trajectories, the underlying cos-

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mological background model, i.e. the expansion history result of particle dynamics including particle interactions of the model universe, and the initial density and momen- and correlations. tum correlations set, as is most commonly believed, by To account for the fact that a general power-law po- quantum fluctuations during inflation. tential leads to a linear growth factor that depends on the It would appear that in numerically simulated as well wave number k, the Poisson equation has to be modified as in observed, gravitationally-bound dark-matter struc- accordingly. This is achieved through an algebraic, ad-hoc tures, these physical properties or their interplay lead to a modification of Poisson’s equation in Fourier space, particular shape of the radial density profile [53–55] that f (k)Φ˜ 4πGρ¯a2δ˜ , (92) is valid for halos on scales ranging from dwarf galaxies to − = massive galaxy clusters. Since the KFT formalism allows to with Φ˜ the gravitational potential, G the gravitational con- easily change all the ingredients for large-scale structure stant and ρ¯ the mean matter density. The modifying func- formation, it is a natural playground for investigating the tion f (k) is the inverse of the Fourier-transformed particle origin of the dark-matter, halo-density profile. interaction potential and is, in that sense, fixed. For the We found in [43] that taking into account initial mo- Newtonian gravitational interaction, it is simply f (k) k2. = mentum correlations leads to a characteristic deforma- A first analysis of the small-scale behaviour of the non- tion of the non-linear power spectrum on scales of the linear contributions to the power spectrum under differ- 1 order of k 0.3h Mpc− but not at wave numbers large ent gravitational laws using the perturbative approach ≈ enough for dark-matter halos to appear. Furthermore, we described in Sect. 2.3 yields the results shown in Fig.7 for have shown that the full initial density correlations play a interaction potentials of the form role for the shape of the density-fluctuation power spec- ¡ 2 2¢ n/2 trum at early times, but are washed out at late times when v(r ) A r ε − , (93) = + a 1, and therefore are unlikely responsible for the shape → of density profiles of highly non-linear objects [44]. where the parameter ε is introduced for technical rea- sons in order to perform the Fourier transforms of the In fact, it is the Newtonian gravitational potential that interaction potential for all n. It can be interpreted as a is often charged with being responsible for the shape and 4 smoothing scale. We set ε 10− , which corresponds to universality property of dark matter halos. We have there- = scales much smaller than the scales we are currently inter- fore used the KFT formalism to investigate whether the ested in. We have verified that a small enough ε does not shape of the interaction potential is responsible for the influence the power spectrum on the much larger scales shape of the non-linear power spectrum on small scales, 1 that we are considering. k 1h Mpc− , where contributions to the non-linear ≥ Particle interactions are included up to first order in power spectrum from inner structures of dark-matter ha- the interaction operator (39) in the same way as for the re- los begin to dominate. sults shown in Fig.3 since the agreement with non-linear According to the widely used halo model, the non- power spectra from Cosmic Emulator [46–49] that uses linear power spectrum is a convolution of Fourier-trans- state-of-the-art N-body simulations is already very good. formed Navarro-Frenk-White (NFW) profiles [53], weigh- The initial correlations appear in a quadratic form in ed with the mass function (for an extensive review see an exponential in the KFT formalism (cf. 64) and must [56]). Since the non-linear power spectrum is reproduced therefore be Taylor-expanded before the integration over 1 with KFT at least up to k 10h Mpc− , un-weighing with ≈ initial particle positions and momenta can be performed the mass function leads to the density profile of dark- in (40). Initial correlations are included up to second order matter halos. here. Including the full hierarchy of initial momentum cor- The non-linear contributions to the power spectrum relations, as shown in Sect. 4.1, proves to be much more obtained from KFT approximately correspond to the one- cumbersome in the perturbative approach compared to halo term of the halo model. Thus, leaving the relative the non-perturbative schemes of Sects. 3.2 and 3.3. Includ- abundances of haloes with different mass unchanged, we ing the full hierarchy of initial momentum correlations can study with KFT how different choices for the gravita- in this context would not alter the results shown here, tional potential affect the density-profile shape. as they only affect the shape of the power spectrum on 1 We point out that, while the halo model must be pro- intermediate scales (k 0.3h Mpc− ) as shown in [43]. ≈ vided with the explicit form of the halo density profile, the The amplitudes of the interaction potential (93) and of KFT approach does not need any input of this kind. The the initial density-fluctuation power spectrum are chosen non-linear density power spectrum and the halo density such that the amplitudes of the non-linear contributions profile that is produced in the KFT approach are thus the to the power spectrum for any potential match that of the

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Newtonian case, and that the amplitude of linear growth ] 1 3 linear on large scales, i.e. k 1h Mpc− , also matches the one v(r) 1 ¿ 104 ∝ r

Mpc 0.26 observed today. v(r) 1.5 3 ∝ r − 2 10− h v(r) 2

The linear power spectrum can be obtained, as usual, [ ∝ r

δ 6 (i) 10− 3 v(r) 3 by multiplying the initial power spectrum Pδ with the 10 ∝ r appropriate linear growth factor corresponding to the in- teraction potential being used, 102 2 (i) Pδ(k) D (a,k)Pδ (k) . (94) = + It is clear from this expression that the linear growth for 101 non-Newtonian potentials will be different from the New- tonian case due to the wave-number dependence of the non-lin. contributions to P growth factor. Therefore, the linear growth together with 100 the non-linear growth for non-Newtonian gravity will pro- 0.01 0.1 1 10 wave number k [h Mpc 1] duce power spectra that look quite different from the New- − tonian case. However, this is irrelevant for this analysis, be- cause only those contributions to the density-fluctuation power spectrum which are due to the inner structure, i.e. Figure 7 The non-linear contributions to the power spectrum today (z 0) are shown together with the linear power spec- density profile, of dark matter halos are of interest here. = Switching to the picture of the halo model, the argu- trum for Newtonian gravity which serves here as a reference ment becomes clearer: While the linear power spectrum, only, using an initial power spectrum as obtained with CAMB corresponding to the two-halo term, describes the cor- [45]. The power spectrum in the gray-shaded area, to a large relations between two different dark matter halos, the degree, does not depend on the inner correlations on dark one-halo term describes the contributions from the inner matter halos and is therefore neglected in our analysis. Starting 1 from k 1h Mpc− , it is assumed that inner halo structures structures of halos, which are highly non-linear. There- = fore, also in KFT the information on the density profile begin to affect the shape of the non-linear power spectrum. 1 On scales k 1h Mpc− , there is very good agreement of of dark matter halos will be contained exclusively in the > non-linear contributions. the non-linear corrections independent of the shape of the Figure7 clearly shows that, for smaller scales, i.e. interaction potential. 1 k 1h Mpc− , the non-linear contributions are almost > insensitive to the slope of the interaction potential. In the gray-shaded area in Fig.7, there is some sensitivity to the small scales, and higher-order interaction terms should potential slope, however these scales are too large to be become more dominant. Hence, the perturbative ansatz relevant for the density profiles of even the largest dark is expected to perform worse and may eventually break matter halos. down. To test whether this will indeed be the case, higher- Since the slopes of the non-linear contributions re- order terms and a check of the convergence of the pertur- main the same even for strongly varying interactions laws, bation series are needed. it appears that different choices for the gravitational po- We emphasise that the above results were interpreted tential would not affect the density-profile shape of dark assuming that the relative abundances of haloes with dif- matter halos. ferent mass remains unchanged. This assumption was This result is surprising and should still be perceived made because we also assumed that only non-linear con- with some caution. Although the non-linear power spec- tributions corresponding to the one-halo term were con- trum for Newtonian gravity obtained with the first-order sidered. In actuality, this does not need to be the case. We perturbation term in KFT agrees well with predictions could and should as well have contributions from the two- from numerical simulations as shown in Fig.3, it is still halo term in the non-linear part of the power spectrum. uncertain whether the perturbation series converges re- This poses a considerable problem for the analysis using gardless of the interaction potential. If the perturbation the perturbative approach as well as for the same analysis series does converge, for which there are good reasons, employing the non-perturbative approaches presented in then higher-order terms should yield ever smaller correc- Sects. 3.2 and 3.3. To consider the fully non-linear density- tions to the non-linear power spectrum. fluctuation power spectrum which corresponds to the However, for short-ranged interaction potentials, most one-halo and two-halo terms combined, the mass func- of the structure is expected to be accumulated on very tion in the halo model would have had to be adjusted

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accordingly, i.e. more low-mass halos and less high-mass specifying the operator to halos for short ranged interaction potentials. In short, the δ halo mass function has to be known for each particle inter- Πˆ α(1) : ρˆ(1) , (99) j = iδJ α (t ) action potential that is considered. It is only then possible p j 1 to make conclusive statements about the halo density where α (1,2,3) enumerates the Cartesian vector com- profiles. = ponents. The application of r 1 of the operators (98) to the ≥ generating functional translates the generator field, as 6 Momentum-Density Correlations discussed in Sect. 2.2, and pulls down the momentum trajectories from the phase factor in (6). Having applied 6.1 Operator Expression these operators and setting the generator field J to zero afterwards, we arrive at the expression Another important and conceptually straightforward ap- r · t ¸ plication of KFT is the calculation of momentum power Z Y Z j Z 0[L] dΓ gpp (t j )p j dt 0 gpp (t j ,t 0) j V (t 0) spectra [57]. These are hardly accessible with the standard = j 1 − 0 ∇ analytic approaches to cosmic structure formation. The = i Lq ,q i Lp ,p iS e + + I , (100) momentum field could naively be constructed as · 〈 〉 〈 〉 N N X 1 X where we have used the definitions (30) and (31). p(t1) p j (t1) u j (t1) , (95) In order to facilitate results obtained for density power = j 1 = m j 1 = = spectra from Sect. 4.1, we employ the Fourier transform with the velocities u j (t1) of the particles j, assuming all of the potential gradient (37) and replace the initial mo- particle masses to be equal. This momentum field lacks mentum of particle j by a partial derivative with respect any spatial dependence, since the microscopic degrees of to the shift vector, freedom are the phase-space coordinates of all particles. ∂ In order to retain any spatial information, we have to im- p j i . (101) = − ∂Lp pose that each particle can contribute to the momentum j at a position q if and only if it is at this position, In this way, we can write the momentum operator as

N X ¡ ¢ ∂ Π(q,t1) p j (t1)δD q q j (t1) pˆj (t j ) igpp (t j ) (102) = j 1 − = − ∂Lp j = N Z t j Z X ˆ p (t )ρ (q,t) . (96) dt 0 gpp (t j ,t 0) B j ( 10)v˜(10)ρˆ j (10), j 1 j − 0 k − = j 1 10 = such that the previous expression may be written as In the last step, we have identified the Dirac delta distribu- tion with the density of the j -th particle at position q and r Y Z [L] £pˆ (t )¤ Z [L] , (103) time t1. Thus, the new field Π is a momentum density. 0 j j = j 1 Fourier-transforming Π(q,t1) and replacing the phase- = space coordinates of particle j by functional derivatives with Z [L] given by (40). Thus, the calculation of r -point with respect to the corresponding source fields, we find correlations of the momentum-density differs from the the momentum-density operator calculation of density correlations only by the application

N of the product of pˆj -operators. In this way, the factorisa- X Πˆ (1) Πˆ j (1) , (97) tion of the free generating functional can be applied in = j 1 this context as well. = composed of the one-particle operators

δ 6.2 Perturbation Theory and Diagrams Πˆ j (1) : ρˆ(1) (98) = iδJp j (t1) In order to include particle interactions, a perturbative with ρˆ(1) from (27). At this point, we note that compo- approach is necessary. One such approach begins with nents of the momentum-density field are available by the Taylor expansion of the exponential factor in Z [L] in

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terms of the interaction operator SˆI. In this Section, we c. At a dashed wave vector, either no or one solid circle formulate rules allowing the systematic calculation of all segment must end, but arbitrarily many dashed circle terms in the perturbation series by employing diagrams segments may end at the same wave vector, internal representing these terms. These rules have been stated or external. for pure density correlations in [43]. We extend them here (v) Each diagram is assigned a multiplicity counting the to also include momentum-density correlations. We note equivalent diagrams. that this is effectively a perturbative expansion in the inter- To illustrate the construction of the diagrams and action potential, and thus corrections to the momentum the physical meaning of the terms, we consider the trajectories need to be accounted for at the appropriate linear-order correction to an equal-time 2-point cross- order. Thus, the linear correction to a momentum-density correlation function of a density ρ and a momentum- correlation is given by the acceleration of one particle in density field Π. The respective diagrams are shown in Fig. (102), which is of zero-th order in the Taylor expansion 8. of the exponential, plus the linear term of the perturba- tive expansion. At the end of this Section, we will give a k1 k2 k1 k2 k1 k2 simple example of the linear correction to a ­ρΠ® power spectrum. Formulating the rules below, we assume that an n- point correlation function is to be calculated at m-th per- Z Z Z turbation order. The diagrammatic representation of the 0 0 0 terms correcting for forces between the particles can then be constructed following these rules:

v v v (i) a. Attach s n 2m wave vectors pointing outward = + −k0 k0 −k0 k0 −k0 k0 (represented by arrows) to a circle marking the free 1 1 1 1 1 1 generating functional Z0, where s is the total number of density, momentum-density, and response field operators. Figure 8 At linear order, three terms contribute to the correc- b. Of these, distinguish r by dashed arrows, represent- tion of the 2-point cumulant of a density and momentum-density ˆ field. The first two come from the Taylor expansion representing ing the momentum-density operators Πjl , from the (s r ) solid arrows representing density fields associ- a deviation from the final freely-evolved position due to interac- − tions and the last diagram accounts for the deviation from the ated with either ρ jl or B jl . (ii) The time ordering is counter-clockwise along the cir- free momentum trajectory. cle, with the latest time being at the top. Interactions are represented by two lines attached to Z0 at the same point in time. If the correlator is simultaneous, the ex- By rules (i) and (ii), we attach one solid density line and ternal wave vectors are also attached to the same point. one dashed momentum-density line at the same point to (iii) The interaction potential is represented by a circled the circle representing the free generating functional for v connected to a pair of internal wave vectors, which an equal-time correlation. Rule (iii) connects the internal are marked with a prime. If the potential is translation- wave vectors by the interaction potential. Following rule invariant, the connected internal wave vectors point in (iv), we construct all possible particle identifications. The opposite directions and have the same magnitude. first two diagrams represent the linear term of the Taylor (iv) a. Each response field is marked by a circle segment expansion starting at a negative internal wave vector and con- 2 Z t (lin) X necting two different wave vectors. The circle seg- D δj,4 k j d10 v˜(10)gqp (t,t 0) (104) SI ments always end at a later time. = j 1 · 0 = b. Distinguish dashed circle segments, representing a ∂ deviation of the actual from the freely evolved parti- k10 gpp (t) Z0[L]. · ∂Lp2 cle position and given by the application of SˆI, from solid circle segments corresponding to a change of Without loss of generality, we here enumerate the particles the particle momentum by the second term in (102). in clockwise order starting at the top left. This term ac- These lines can only connect a response field (solid counts for the displacement of a particle from its position arrow) with a momentum-density (dashed arrow). according to inertial motion at the time of evaluation. The

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third diagram represents the deviation of the particle’s In Fig.9, we compare our results with those obtained momentum due to the interaction with other particles. by [60] who used Eulerian standard perturbation the- The term is ory. The authors of [60] calculate the power spectrum Z t of the momentum-density by separating it into uncon- (lin) D δ d10 v˜(10)k0 g (t,t 0) Z [L] . (105) nected and connected terms using Wick’s theorem and p = 2,4 1 pp 0 0 evaluating these at one-loop order. This captures the in- The translation tensor is given by (30), and the Kronecker teractions only partially, while the KFT approach pre- symbols identify an internal with an external particle. dicts a higher amplitude when using the averaged inter- At quadratic order in the interaction potential, con- action term (86). In addition, our results suggest that the siderably more diagrams contribute because two inter- spectrum increases proportional to k2 at wave numbers 2 actions of particles occur in the evolution of the system. . 10− h/Mpc, while the SPT results fall below this slope 3 Three scenarios can be distinguished: (1) one external approximately at wavenumbers above 10− h/Mpc. particle undergoes two interactions, (2) each external par- ticle undergoes one interaction, (3) one external particle scatters from a previously scattered internal particle. 1 10− KFT Park et al. (2016) 6.3 Momentum-Density Power Spectra 2

] 10− 3 In a similar fashion as for the density power spectra (69) Mpc

and (87), we can write down a closed expression for the 3 −

h 3 momentum-density power spectrum. We focus here on [ 10− i one scalar function that can be constructed from the ⊥ Π Π Π correlation tensor. More specifically, we consider ⊥ Π

〈 × 〉 h 4 the projection of Π perpendicular to the mode k. This 10− power spectrum is needed in order to calculate the ampli- tude of secondary temperature fluctuations in the cosmic 5 microwave background caused by Thomson scattering off 10− 0.001 0.01 0.1 1 10 free electrons moving with the bulk of structures. This is 1 wave number k [h Mpc− ] called the kinetic Sunyaev-Zel’dovich effect [58]. The result is

Z Figure 9 Power spectrum for the projection of the momentum- 2 Q0 i SI(t) Q ik q Π Π (k,t) 2gpp (t)e− + 〈 〉 a (q)e − · , density Π perpendicular to the mode k at present time. The red 〈 ⊥ ⊥〉 = − q ⊥ curve shows the KFT result (106), the blue curve the results (106) obtained by [60] using Eulerian standard perturbation theory at one-loop order. The cosmological parameters are chosen to be with the definitions for Q0, Q and SI (t) of equations (70) 〈 〉 Ω 0.279,Ω 0.721,h 0.701,Ω 0.0462,σ 0.8, and (86). m = Λ = = b = 8 = and we use an initial BBKS spectrum [10] in our calculations. The function a (q) is the correlation function of the ⊥ initial momentum components perpendicular to the line connecting the positions of the two momenta being cor- related. It can also be expressed by the initial power spec- trum Pδ(k) of density fluctuations and is given by

2 1 µ 7 The BBGKY hierarchy in KFT a (q) a1(q) − a2(q) (107) ⊥ = + 2 using the definitions for a1(q) and a2(q) in (72). To relate KFT to conventional kinetic theory, we now want Taking the limit of expression (106) for large scales to extract evolution equations for the phase-space den- or small wave numbers, k 0, we recover the standard sity and its higher-order correlators. We will first define a → result of [59] for the slope of the power spectrum, phase-space density operator and generalise the generat- ing functional as well as the interaction term from Sect.2 lim Π Π (k,t) k2. (108) k 0 〈 ⊥ ⊥〉 ∝ to cover all of phase space. →

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In analogy to the number density (22) and its operator with its Fourier-space operator (27), we introduce the phase-space density f u(s,t) (118) X c f (x,t) δ ¡x x (t)¢ (109) = = D − j à ! j X δ δ 1 δ exp ik i` . − · − · and its operator expression in a six-dimensional Fourier j iδJq j (t) iδJp j (t) m iδJp j (t) space Applying a phase-space density-current operator f u à ! c X δ δ together with r 1 phase-space density operators f to the fˆ(s,t) exp ik i` , (110) − = j − · iδJq j (t) − · iδJp j (t) free generating functional and setting the source fields to zero results in where x (q,p) is again the six-dimensional phase-space = Z ¯ p position with its Fourier conjugate s (k,`). i Lq ,q i Lp ,p r = Z0[L] dΓe 〈 〉+ 〈 〉 . (119) In this Section, we will always consider equal-time, = m r -point phase-space density cumulants with a Taylor ex- The time evolution of the phase space correlator (111) can pansion of the interaction up to N-th order, now be found by direct computation, N ¡ ¢n (N) iS¯I(t) N n ­ ® X (N) (iS¯ (t)) f˜(1) f˜(2)... f˜(r ) Z0[L] , (111) ­ ® X I ∂t f˜(1)... f˜(r ) ∂t Z0[L] (120) = n 1 n! = = n 1 n! = ˜ ˜ ˜ ˜ where f (1) f (t,s1), f (2) f (t,s2) and so on, and Z0[L] N (iS¯ (t))n 1 = = X I − £ ¯ ¤ denotes the free generating functional with r phase-space ∂t SI Z0[L]. + n 1 (n 1)! density operators having been applied and the source = − fields set to zero. Similar to (20), the free generating func- In the first term, the time derivative acts on the gener- tional is then ating functional Z i Lq ,q i L¯p ,p Z Z0[L] dΓe + , (112) i Lq ,q i L¯ p ,p ­ ® 〈 〉 〈 〉 ∂ Z [L] dΓe + i ∂ L¯ ,p (121) = t 0 = 〈 〉 〈 〉 t p with the shift operators r Z p j X i Lq ,q i L¯ p ,p ik j dΓe 〈 〉+ 〈 〉 , r r = − j 1 m X X ¡ ¢ = Lq k j e j , L¯ p k j gqp (t j ) `j e j . = − j 1 ⊗ = − j 1 + ⊗ while it acts on the time-integral boundary in the interac- = = (113) tion in the second term, r Z d6s The interaction (39) can then be extended to cover all ¯ X 10 ˆ ˆ ∂t SI(t) 6 `j D j ( 10)v0(10)f (10) , (122) of phase space, = j 1 (2π) − =

r Z t where we used g (t,t) 0 and set t 0 t. ¯ X ¡ ¢ ˆ ˆ qp = 1 = SI(t) d10 k j gqp (t,t 0) `j D j ( 10)v0(10)f (10). Inserting the derivatives back into (120), we can use = j 1 0 + − = (119) in the first term, and (116) in the second term. Iden- (114) tifying applied operators in the form of the free generating functional and collecting them into operators finally re- Here, the potential v˜(10) from (39) is extended to read sults in 3 ¡ ¢ v0(10) v˜(k0 )(2π) δ `0 , (115) 1 D 1 r D E(N) = ­ ˜ ˜ ®(N) X ˜ ˜ 6 6 ∂t f (1)... f (r ) ik j f (1)... f u(j)... f (r ) the integration measure is d10 dt 0 d s0 /(2π) , and the = − = 1 1 j 1 phase-space response-field operator is defined by = r Z 6 X d s10 ˆ ˆ i` ik v0(10) D j ( 10) ik10 f j ( 10) . (116) j j 6 − = − + j 1 (2π) = Before computing the evolution equations of phase- ­ ®(N 1) f˜(1)... f˜(j 10)... f˜(r )f˜(10) − , (123) space density cumulants, we need to introduce the phase- · − space density current where f˜(j 10): f˜(t,s s0 ). − = j − 1 X ¡ ¢ p j This shows that the time evolution of the r -point f u(x,t) δD x x j (t) (117) = j − m phase-space correlator is divided into two effects: The first

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is the convection of the phase-space density described namics: We introduce mesoscopic particles similar to con- by the first term. The second term contains all possible ventional fluid elements and demand a hydrodynamical interactions with an additional point in phase space. Here, scale hierarchy. Then, on the scale of the mesoscopic par- r 1-point phase-space correlators appear. Consequently, ticles, local thermal equilibrium can be assumed to have + evolution equations for r 1-point phase space density been established. The properties of the mesoscopic parti- + correlators are needed in order to solve the r -point evo- cles are then described by state variables such as pressure lution equation. These again contain convection terms and energy density. At the same time, we demand that the as well as r 2-point phase space density correlators. For mesoscopic particles are much smaller than any scale we + the latter, evolution equations are needed that contain are interested in and can be modelled as point-like. r 3-point correlators, and so on. An (almost) infinite hi- For simplicity, we assume an isotropic fluid which is + erarchy of coupled differential equations unfolds, known fully described by a spatially dependent velocity field u(q), as the BBGKY hierarchy. A Fourier transform back into a pressure field P(q) and an energy-density field ²(q). In configuration space shows that (123) are the exact terms addition, we assume that the pressure is only caused by of the conventional BBGKY hierarchy [41]. the random velocities of the microscopic particles. Then, A truncation criterion of the BBGKY hierarchy emerges both the pressure and the internal energy-density are pro- here from the order of the Taylor expansion of the interac- portional to the enthalpy-density h, tion: In the evolution equation for the r -point correlator, 5 5 the r 1-point correlators appear only in the (N 1)-st h ² P ² P . (124) + − = + = 3 = 2 order in the interaction, reduced by one. This is repeated in the time evolution of the two r 1-point correlators: In order to sample the fluid, the mesoscopic particles + the r 2 point correlator is again reduced by one order need to capture these properties: they need to contain + in the interaction and so on for each further step up the information about their position, the momentum, and the hierarchy. Once the 0-th order is reached, only convec- enthalpy at that position. Hence, we endow each particle tion terms and no higher-order correlators appear in the with these three degrees of freedom, evolution equation, and the hierarchy ends. ¡ ¢ ϕ q ,p ,H for the j-th particle , (125) j = j j j and describe their non-interacting dynamics by ballis- 8 Fluids in KFT tic motion, conserving momentum and enthalpy. These dynamics are expressed by the Green’s function 8.1 Introduction  t ti  I3 m− I3 0 G˜R (t,ti)  0 I 0 θ (t ti) . (126) Cosmic structure formation is governed by both dark and =  3  − baryonic matter. Therefore, it is crucial to extend KFT to 0 0 1 capture the physics of both particle types. Unlike dark matter, baryonic interactions are not limited to gravity From here on, a free generating functional for KFT can only. This causes the cosmic matter power spectrum to be constructed in analogy to (20) attain additional features such as baryon acoustic oscil- Z µ Z ¶ ˜ ∞ lations (BAO) as well as disturbances in the spectrum at Z0 [J,K ] dΓi exp i dt 0 J(t 0),ϕ¯ (t 0) , (127) = 0 〈 〉 small scales due to e.g. pressure and baryonic cooling. We describe here one way towards including mixtures of dark where the tilde marks the free generating functional for and baryonic matter into KFT. the fluid model, J (J , J , J ) and j = q j p j H j µ Z t ¶ ˜ (i) ˜ ϕ¯ (t) GR(t,0)ϕj dt 0 GR(t,t 0)K j (t 0) e j , 8.2 Extension of the generating functional = − 0 ⊗ K (t 0) (K ,K ,K ) . (128) = q p H In principle, fluid dynamics should be captured by KFT if microscopic gas-particle interactions were taken into account. However, it is a regime hard to reach in any ex- 8.3 Interaction operator pansion of the interaction. To acquire access to fluid dy- namics nonetheless, we implement a fluid model based In addition, acceleration due to pressure gradients and on an idea close to the conventional approach to hydrody- pressure-volume work must be taken into account. For

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viscous hydrodynamics, diffusion of energy and momen- isotropic fluid can then be constructed, tum must be added. Within KFT, these effects are included via interactions between the mesoscopic particles. The 2i Z ˆid X ˆ δ exact form of the interaction operator is found via an ap- SI d1k1 ρˆ j ( 1)Hi ρˆi (1) (133) = 5ρ¯ (i,j) − iδKp j (1) proach similar to smooth-particle hydrodynamics (SPH) Z [61]. For the purposes of this review, we only sketch the 2i X £ ¤ δ d1k1 ρˆ j ( 1) uˆi uˆ j Hˆi ρˆi (1) , derivation of the interaction operator for pressure-volume + 3ρ¯ (i,j) − − iδKH j (1) work. A thorough derivation including all terms of ideal and viscous fluid dynamics can be found in [62]. 3 with d1 dtd k1 and the one-particle operators The pressure field split into particle contributions = reads à ! δ ρˆ j (1) exp ik1 , (134) 2 2 X ¡ ¢ = − iδJ P(q) H ρ(q) Hi δD q qi , (129) q j = 5 = 5 i − δ 1 δ Hˆ j , uˆ j . where Hi is the enthalpy and qi the position of the i-th = iδJH j = m iδJp j particle. Following the Euler equation, the time evolution of the For an ensemble characterized by the free generat- momentum field contains the term ing functional (127) and the interaction operator (133), macroscopic evolution equations for density, momentum ∂ P(q ) q1 1 2 X ¡ ¢ density, and energy density can be extracted. As shown in p˙(q1) Hi ∂q1 δD q1 qi , = − ρ(q1) = −5ρ(q1) i − [62], these are indeed the continuity, Euler, and energy- (130) conservation equations of ideal hydrodynamics. If an in- teraction operator for diffusive effects is added, the Navier- where we inserted the discretised pressure field from Stokes and energy conservation of viscous hydrodynam- (129). ics are recovered. The momentum change at position q1 is associated to particles according to their (spatial) contribution at that position. To this end, we weigh the momentum change at 9 Macroscopic Reformulation of KFT q1 with a Dirac delta distribution around the j-th parti- cle’s position and integrate over the entire space, From the perspective of most quantum and statistical field Z theories, the generating functional (5) of KFT is rather p˙ d3q p˙(q )ρ (q ) (131) j = 1 1 j 1 unusual in the sense that the path integral is expressed Z in terms of microscopic degrees of freedom even though 2 X 3 1 ¡ ¢ ¡ ¢ d q1 δD q1 q j Hi ∂q δD q1 qi . we are actually interested in macroscopic fields like the = −5 ρ(q ) − 1 − i 1 density. As described before, the decisive advantage of this approach is the simplicity of the equations of motion For the interaction operator, this object must be ex- for the microscopic degrees of freedom. pressed by functional derivatives acting on the free gen- erating functional. However, this is not possible for the However, to facilitate the application of established inverse density. We handle this complication by approx- perturbative as well as non-perturbative field-theoretical imating the inverse densities in a Taylor series around techniques, we would like to reformulate the KFT parti- the mean density ρ¯ of the ensemble. As a first step, we tion function as a path integral over macroscopic fields. It truncate the approximation already at 0-th order, turns out that this will lead to a resummation of an infinite subset of terms appearing in the microscopic perturba- 1 1 tive expansion in the interaction operator. This will allow . (132) ρ j (q1) ≈ ρ¯ us to treat dark matter particles in terms of their funda- mental Newtonian dynamics rather than the improved For the pressure-volume work, a similar term for the Zel’dovich dynamics. The presentation here is based on enthalpy change of the j-th particle, H˙ j , can be derived the more detailed and general derivation in [63] which in an analogous calculation. From the momentum and uses the full phase-space density rather than the spatial enthalpy changes, the interaction operator for an ideal, density to preserve momentum information.

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9.1 Macroscopic action (141)

We begin by using (9) and (12) to write the partition func- Pulling all ψ-independent parts to the front, we find that tion as the remaining microscopic part of the path integral as- Z Z sumes the form of the free generating functional of ρ- and £ (i) ¤ Z dΓ Dx δ E(x,x ) F -correlators, = D

Z Z Z (i) Z Z Z Z iχ E(x,x ) iφρ φ iSψ,0 iφ ρ iφρ F dΓ Dx Dχe · , (135) Z Dφ Dφ e− · β dΓ Dψe + β· + · , = = ρ β | {z } ρ,F £ ¤ where we represented the delta distribution in terms of :Z φ ,φ = 0 β ρ a functional Fourier integral with respect to an auxiliary (142) field χ with components χ ¡χ ,χ ¢. We further intro- j = q j p j duce the combined microscopic field ψ : (x,χ) and its with φ and φ playing the roles of the source fields for = β ρ action the collective fields ρ and F , respectively.

(i) ! Finally, defining the combined macroscopic field φ : Sψ : χ E(x,x ) Sψ,0 Sψ,I , (136) = = · = + (φρ,φβ) and the free generating functional of collective- ρ,F ρ,F which splits into a part S describing the free motion of field cumulants W : ln Z , we arrive at the result ψ,0 0 = 0 the particles and a part Sψ,I describing their interactions. Using (15) and (17), the latter can be expressed as Z Z DφeiSφ (143) N = Z X S dt χ (t) V (t) . (137) ψ,I p j j with the macroscopic action = − j 1 ∇ = ρ, S : φ φ iW F £φ ,φ ¤ . (144) Inserting (37) and defining the dressed response field F φ = − ρ · β − 0 β ρ via its Fourier transform, We emphasise that this reformulation is exact and N X hence (143) still contains the complete information on F˜ (1) : χp (t1)B˜j (1)v˜(1) , (138) j the microscopic dynamics, even though Sφ does not de- = − j 1 = pend on ψ any more. The microscopic information is now allows us to write the interaction term as ρ,F £ ¤ encoded in the free generating functional W0 φβ,φρ Z and thus, by means of a functional Taylor expansion, in S d1F˜ ( 1)ρ˜(1) : F ρ , (139) ψ,I = − = · the free collective-field cumulants (0) where we also introduced the dot product as a short-hand Gρ ρF F (1,...,nρ,10,...,nF0 ) (145) ··· ··· notation for integrating over field arguments. ­ ® : ρ˜(1) ρ˜(n )F˜ (10) F˜ (n0 ) We now replace the explicitly ψ-dependent field ρ with = ··· ρ ··· F 0,c ¯ a new formally ψ-independent field φρ, using a functional nρ à ! nF à ! Y δ Y δ ρ,F £ ¤¯ delta distribution, W φβ,φρ ¯ . = iδφ˜ (u) iδφ˜ (r ) 0 ¯ u 1 β r 1 ρ 0 ¯φ 0 Z Z Z = = = £ ¤ iS ,0 iF φ Z Dφ dΓ Dψδ φ ρ e ψ + · ρ . (140) = ρ D ρ −

This way, the new field φρ effectively still carries all the 9.2 Including density-density and density-momentum information contained in ρ. Most importantly, φρ- and correlations ρ-correlation functions are identical. However, to empha- sise their different origin we will deliberately call φρ the Since we will use this macroscopic reformulation of KFT macroscopic and ρ the collective density field in the fol- in Sect. 9.4 to treat Newtonian dynamics, whose Green’s lowing. function (54) is limited from above, we have to take the Similar to (92), we now express the delta distribution complete expression for the initial probability distribu- as a functional Fourier transform with respect to a new tion (64) into account, which includes all correlations macroscopic auxiliary field φβ, between the initial phase-space coordinates of particles. This means that the lengthy polynomial C (p) introduced Z Z Z Z in (64) must be included. This makes the explicit calcula- iφ (φ ρ) iS ,0 iF φ Z Dφ Dφ dΓ Dψe− β· ρ − + ψ + · ρ . = ρ β tion of the above cumulants anything but straightforward.

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By adopting the essential ideas of the so-called Mayer where we introduced the generalisation of the damping cluster expansion [64], we have however been able to con- amplitude Q0 from (70) as dense this process into a small set of rules for Feynman- 2 ` σ1 X like diagrams. Here, we will restrict ourselves to reviewing Q(I ,...,I ): L 2 (t ) . (150) 1 ` p,Ij i = − 6 j 1 the recipe for cumulants involving the particle number = density ρ, while the complete technical derivation for cu- The term Σ˜ (`) (I ,...,I ) represents the sum over all mulants involving the phase-space density f can be found CDiag 1 ` possible connected `-particle correlation diagrams. The in [44]. possible diagram line types are The general form of the diagrams in this approach is Z ³ ´ very simple: particles are represented by dots, and correla- ¡ (i)¢ iθq θp P p p K : e− e− 1 , (151) j k = j k jk = (i) − tions between them by different types of connecting lines. q jk Since we are only interested in connected contributions Z to correlations, we need to consider connected diagrams ¡ (i)¢ : C e θ , Pδj δk K jk δj δk − j k = = q(i) only, i.e. such through which we can trace a continuous jk path. A general free cumulant as in (145) is then ordered Z ³ ´ ¡ (i)¢ : iC L e θ , Pδj pk K jk δj pk p,Ik − in terms of the number of particles ` being correlated in j k = = q(i) − · such a connected way, jk ¡ (i)¢ (0) P(δp)2 K jk Gρ ρF F (1,...,nρ,10,...,nF0 ) j k = jk ··· ··· = nρ Z ³ ´³ ´ X (0,`) θ G (1,...,nρ,10,...,n0 ) . (146) : iCδj pk Lp,Ik iCδk p j Lp,Ij e− , ρ ρF F F = q(i) − · − · ` 1 ··· ··· jk = Note that, as a consequence of statistical homogeneity, with the phase θ : iθ θ and = q + p the sum over particle numbers truncates at the number (i) (i) θq : K q , θp : L> Cp p Lp,I . (152) of density fields in the cumulant. Combined with the di- = jk · jk = p,Ij i j k agram rules, this ensures that all cumulants have a fi- In addition to the momentum correlations introduced nite number of terms, i.e. KFT can exactly describe the in Sect. (3.3), Cδj δk represents density autocorrelations, highly non-linear effects of initial correlations on the free- and Cδj pk density-momentum correlations between the streaming evolution with a finite number of explicit ex- initial positions of particles j and k. Due to statistical pressions. homogeneity, the above Fourier integrals only need to be We begin with the calculation of pure density cumu- (i) (i) performed over the relative initial coordinates q jk q j lants since mixed cumulants involving the dressed re- = − q (i) of particles, where j k in all cases by convention. All sponse field F are derived from the former by the inser- k < translation vectors are evaluated at the initial time. One tion of simple response factors. We first need the crucial may draw these lines between particle dots according to concept of a field-label grouping. For any fixed number of the following rules: particles `, we group the field labels (1,...,nρ,10,...,nF0 ) of the cumulant into j 1,...,` non-empty sets I . Any – No self-correlations in the form of subdiagrams may = j such collection of sets is called a field-label grouping. We occur for any kind of correlation line. define a variant of the phase-translation vectors (30) in – Any pair of particles can be connected directly by at terms of these groupings as most one line, i.e. no subdiagrams may occur X X for any combination of correlation lines. Lq,Ij (t) Lq,r (t): kr gqq (tr ,t) , (147) = r I = r I – No particle may have more than one solid δ-line at- ∈ j ∈ j X X tached to it, i.e. no subdiagrams my occur which in- L (t) L (t): k g (t ,t) . (148) p,Ij p,r r qp r clude . = r I = r I ∈ j ∈ j By convention, there is a flow of a general Fourier A general `-particle density cumulant is then given as a (i) momentum K jk along each line from smaller to larger sum over all possible field-label groupings, labels. The algorithm of Feynman rules for evaluating à n ! (`) ρ Σ˜ (I1,...,I`) for a given grouping of field labels is then (0,`) ` 3 X CDiag Gρ ρ(1,...,nρ) ρ¯ (2π) δD Lq,r (ti) as follows. ··· = r 1 = (1) Choose a fixed graphical arrangement of ` dots la- X Q(I1,...,I`) ˜ (`) e− ΣCDiag(I1,...,I`) , (149) beled j 1,...,`, representing the particles carrying {I ,...,I } = 1 ` the label sets Ij . These are the vertices of the diagrams.

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(2) Given this fixed set of vertices, draw all possible dia- 9.3 Macroscopic perturbation theory grams using the lines in (151), subject to the above rules. Repeat the next three steps for all these dia- The reformulated path integral (143) allows us to set up a grams. new perturbative approach to KFT following the standard (3) For any closed loop in a diagram, assign one of the procedure familiar from quantum and statistical field the- lines in the loop connecting vertices labeled j k a ory, i.e. in terms of propagators and vertices. For this pur- (i) R< loop momentum k and introduce an integral (i) . pose, we first split the action (144) into a propagator part jk k jk (4) Pick a vertex j which has only one line left with unde- S∆, collecting all terms quadratic in φ, and a vertex part termined momentum and use the conservation law SV , containing the remaining terms, ! S S S (156) φ = ∆ + V j 1 ` X− X with the definitions L K (i) K (i) 0 (153) q,Ij i j ji 1 Z Z + i 1 − i j 1 = 1 iS∆ : d1 d2φ˜( 1)∆− (1,2)φ˜( 2) , (157) = = + = −2 − − n to fix the momentum of this line. Incoming momenta 1 β µZ ¶ nρ µZ ¶ X∞ Y ˜ Y ˜ from vertices with smaller labels are counted positive iSV : du φβ( u) dr 0φρ( r 0) n= ,n 0 nβ!nρ! u 1 − r 1 − β ρ = = = while momenta outgoing to vertices with larger la- n n 2 β+ ρ 6= bels are counted negative. Momenta associated with Vβ βρ ρ(1,...,nβ,10,...,nρ0 ) , (158) × ··· ··· vertices not connected to vertex j are zero. 1 (5) Consecutively go through all other vertices to fix the introducing the inverse macroscopic propagator ∆− and the macroscopic (nβ nρ)-point vertices Vβ βρ ρ. remaining undetermined momenta by repeatedly ap- + ··· ··· plying the previous step. We further define the macroscopic generating func- φ ¡ ¢ tional Z by introducing a source field M Mρ,M for Once a pure density n-point cumulant is known any = β the combined macroscopic field φ into the partition func- mixed (n n n0 )-point cumulant is simply obtained = ρ + F tion, by first inserting n response factors and interaction po- F0 Z φ iS iM φ tentials as Z [M]: Dφe φ+ · . (159) = (0,`) Then, the vertex part of the action can be pulled in front Gρ ρF F (1,...,nρ,10,...,nF0 ) ··· ··· = of the path integral by replacing its φ-dependence with µ n ¶ ¯ X functional derivatives with respect to M, Sˆ : S δ , ¯` 3 ˜ ˜ V V ¯φ ρ (2π) δD Lq,r (ti) v(10)...v(nF0 ) = → iδM r 1 acting on the remaining path integral, = ³ ´ X Q(I1,...,I`) ˜ (`) ˆ Z 1 1 e− bI(1 ) ...b Σ (I1,...,I`). φ iSV φ ∆− φ iM φ 0 I(nB0 ) CDiag Z [M] e Dφe− 2 · · + · {I1,...,I`} = (154) iSˆ 1 (iM) ∆ (iM) e V e 2 · · . (160) = The response factors are defined simply as Expanding the first exponential in (160) in powers of the vertices now gives rise to a new perturbative approach bI(r ) iLI(r )(tr ) kr , (155) that we will refer to as the macroscopic perturbation the- = · ory. where I(r ) is the set of field labels containing r . The re- Within this approach, the interacting macroscopic- tarded nature of the particle propagators contained in field cumulants are obtained by taking appropriate func- these response factors also leads to the general property tional derivatives of the macroscopic cumulant-generat- ing functional W φ[M]: ln Z φ[M]. In particular, the low- that only such field-label groupings {I1,...,I`} give non- = vanishing contributions to mixed cumulants (154) which est perturbative order of the 2-point density cumulant Gρρ is given by the ρρ-component of the macroscopic have the following property: For all Ij , j 1,...,`, there = propagator, must be at least one ρ-label r Ij such that tr tu for ∈ ≥ 0 ¯ all F -labels u0 Ij . This can be understood as the causal δ δ ¯ ∈ G (1,2) W φ[M]¯ flow of responses having to terminate at some density ρρ ¯ = iδMρ(1) iδMρ(2) M 0 field label r at later time since there is no instant response = ∆ (1,2) terms involving vertices . (161) of the spatial density to forces acting on particles. = ρρ +

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Adopting the usual field-theoretical language, we will re- (167) formally in terms of a Neumann series by expanding (0) fer to this as the tree-level expression for Gρρ. For fur- the functional inverse in orders of iGρF , ther detail on the general properties of the macroscopic perturbation theory, we refer the reader to [63], where n X∞ ³ (0) ´ we introduce a Feynman-diagram language to compute ∆R(1,2) ∆A(2,1) iGρF (1,2) = = n 0 the higher-order perturbative contributions – i.e. the so- = called loop-contributions – systematically. (0) I (1,2) iGρ (1,2) 1 = + F To find expressions for the inverse propagator ∆− and Z (168) (0) (0) the vertex term Vβ βρ ρ, we insert the functional Taylor d10 iGρF (1, 10)iGρF (10,2) . ρ,F··· ··· + − + ··· expansion of W0 into the macroscopic action (144) and identify terms with (157) and (158), respectively, (0) From the definition of F in (138), it follows that GρF v. Ã (0) (0) ! ∝ G iI G Therefore, ∆R, ∆A and ∆ρρ contain terms of arbitrarily 1 FF + F ρ ∆− (1,2) (0) (0) (1,2) , (162) high order in the interaction potential v. This demon- = iI G Gρρ + ρF strates that the lowest perturbative order within the

Vβ βρ ρ(1,...,nβ,10,...,nρ0 ) macroscopic approach already captures effects which ··· ··· could only be described at infinitely high order within nβ nρ (0) i + Gρ ρF F (1,...,nβ,10,...,nρ0 ) . (163) the microscopic perturbation theory. = ··· ··· Here, I denotes the identity 2-point function, A closer analysis reveals that, in a statistically homo- geneous system, the transition from the micro- to the 3 ¡ ¢ I (1,2) : (2π) δ k k δ (t t ) . (164) macroscopic formulation entails a resummation of all = D 1 + 2 D 1 − 2 contributions which do not lead to any mode-coupling The propagator ∆ is then obtained by a combined ma- beyond the one already introduced by the free evolution. trix and functional inversion of (162), defined via the fol- Due to this property, we will refer to the macroscopic re- lowing matrix integral equation, formulation of KFT as resummed KFT (RKFT). Z 1 ! d10∆(1, 10)∆− (10,2) I (1,2)I . (165) − = 2 The matrix part of this inversion can be performed imme- 9.4 Resumming Newtonian dynamics diately and yields Using RKFT enables us to treat dark-matter particles in à (0) ! ∆R G ∆A i∆R terms of their fundamental Newtonian dynamics rather ∆(1,2) · ρρ · − (1,2) , (166) = i∆ 0 than the improved Zel’dovich dynamics. Since the inertial − A motion of particles on Zel’dovich-type trajectories already where we defined the retarded and advanced (macro- contains part of the gravitational interaction, even a first- scopic) propagators order calculation in the microscopic perturbation theory ³ ´ 1 captures the nonlinear evolution of the power spectrum ∆ (1,2) ∆ (2,1) : I iG(0) − (1,2) , (167) R = A = − ρF over a wide range of scales remarkably well, as shown in Fig.3 and discussed in detail in [42]. containing the remaining functional inverses. They de- scribe the linear response of the density at time t1 to a This is not the case for Newtonian dynamics, and any perturbation of the interacting system at time t2. expansion to finite order in the Newtonian gravitational Consequently, ∆ ∆ G(0) ∆ is the 2-point den- potential does not even reproduce the linear growth of ρρ = R · ρρ · A sity cumulant resulting from the linear response of the structures correctly. Working with an averaged interaction interacting system to the correlated motion of the free- term or the Born approximation, as described in Sects. 4.2 and 4.3, does not improve the situation either. This is streaming particles. Generally, ∆ρρ will nevertheless be nonlinear in the initial density correlation because the because both approaches require deviations from the in- (0) ertial trajectories caused by the (effective) forces acting free-streaming of particles in Gρρ itself builds up those nonlinearities as long as the initial particle momenta are between them to be small, which is not the case for the correlated [44]. truly non-interacting inertial trajectories in Newtonian dynamics. In [63], we describe how ∆R, ∆A and ∆ρρ can be com- puted explicitly for a given physical system. But even with- Using RKFT, however, allows to overcome this limita- out specifying the system, it is always possible to express tion. In Fig.10, we show the tree-level RKFT result for the

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density contrast power spectrum, weaker. At the same time, the resummation does not com- pensate the small-scale damping effect caused by the par- Z 3 ¯ (∆) 1 d k2 ¯ 2 P (1) : ∆ρρ(1,2)¯ , (169) ticles’ initial momentum variance σ1. This behaviour is δ = ρ¯2 (2π)3 ¯ t2 t1 reminiscent of the small-scale decay of the power spec- = evolved from the time of CMB decoupling until today trum found in Renormalized Perturbation Theory (RPT) assuming Newtonian particle dynamics, and compare it [16,17]. to the power spectrum resulting from Newtonian free- However, we can also infer that RKFT does not en- (∆) streaming as well as the power spectrum obtained from hance this damping effect either, as Pδ never drops linear Eulerian perturbation theory (EPT). below the free-streaming spectrum. This differs from the power spectra found in RPT or Zel’dovich dynamics, which overpredict the dissolution of small-scale struc- tures [11,16,17,43]. 104 Altogether, the tree-level RKFT result for the density 101 contrast power spectrum is thus found to capture the linear effects introduced by gravitational interactions in ]

3 2 10− a way that is fully consistent with the underlying non-

Mpc interacting Newtonian particle dynamics. Higher-order 3 5 − 10− h

[ perturbative corrections within RKFT are currently inves- δ

P 8 tigated. 10− tree-level RKFT 11 free-streaming 10− linear EPT initial 10 Mixtures of gas and dark matter in 10 14 − 4 2 0 2 4 10− 10− 10 10 10 KFT 1 wave number k [h Mpc− ] Baryonic matter with its additional non-gravitational in- teractions has a profoundly different effect on large-scale Figure 10 Comparison of different density contrast power cosmic structures than dark matter. To gain a complete spectra evolved from the time of CMB decoupling to redshift understanding of structure growth, it is therefore neces- zero, using cosmological parameters Ω 0.3, Ω 0.7, m = Λ = sary to investigate how a mixture of dark and baryonic Ω 0, h 0.7, σ 0.8: the tree-level Newtonian RKFT b = = 8 = matter co-evolves through the cosmic history. spectrum (black solid), the spectrum resulting from Newtonian To describe such a mixture, we extend the macroscopic free-streaming (orange dashed), and the spectrum obtained reformulation of KFT to include two different particle from linear Eulerian perturbation theory (EPT, blue dashed). species. The long term goal hereby is to use this approach For reference, the initial BBKS spectrum [10] is shown as well to study the non-linear evolution of structures which con- (black dotted). The tree-level RKFT result interpolates between tain both baryonic and dark matter. In this formalism, the linear growth on large scales and free-streaming growth on collisionless character of dark matter is kept while at the small scales where the particles’ momentum variance becomes same time the fluid properties of baryonic matter needs relevant. to be taken into account. We demonstrate the functionality of this formalism on (∆) structure growth in the linear regime. Our aim hereby is At small wavenumbers, Pδ follows the linear EPT spectrum, before it starts to fall below the linear predic- a proof of concept showing that KFT is able to correctly 1 describe the co-evolution of mixtures. The foundations tion at wavenumbers k & 20h Mpc− . When going to even higher wavenumbers it eventually approaches and follows presented here can be used in the future to investigate 4 1 nonlinear structure growth. the free-streaming spectrum for k & 10 h Mpc− . First of all, this shows that the partial resumma- Furthermore, we demonstrate that our formalism can tion of Newtonian gravitational interactions captured qualitatively reproduce some of the key effects baryons by the macroscopic propagator allows to precisely re- have on linear structure formation. Firstly, the suppres- cover the linear growth of structures on the largest scales sion of structures on small scales due to the repulsive even though the structure growth resulting from non- effects of baryonic pressure is demonstrated. Secondly, by interacting Newtonian dynamics is orders of magnitude also taking interactions between photons and baryonic

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matter into account, essential features of baryon-acoustic particles characterised by a position, a momentum and oscillations are reproduced. an enthalpy. However, to facilitate the incorporation of mesoscopic particles into the RKFT framework, we approximate the 10.1 KFT with two particle types gas as being isothermal. That is, we approximate the baryon temperature as being spatially homogeneous To describe the interactions of two different particle types throughout the Universe, following the evolution of the in the framework of KFT we expand the macroscopically mean gas temperature T (t). In this case, every mesoscopic reformulated generating functional (160) to a system of particle has the same enthalpy two particle species in a straightforward fashion by re- ¡ ¢ γ b 2¡ ¢ placing the macroscopic fields and as well as the H T (t) m cs T (t) , (174) φρ φβ = γ 1 collective dressed response field F by two-component − vectors containing the respective field contributions for where mb is the mass of a mesoscopic particle, γ is the baryons and dark matter, adiabatic index of the gas and cs(T ) is its sound velocity. This effectively reduces the dynamical degrees of freedom ¡ b d¢ ¡ b d¢ ¡ b d¢ φ φ ,φ , φ φ ,φ , F F ,F . (170) ρ → ρ ρ β → β β → to just a position and a momentum, allowing us to treat baryons like microscopic particles experiencing a gravita- Proceeding in a completely analogous fashion to Sect.9, tional potential we obtain a macroscopic propagator that takes exactly the same form as in the single-particle species case (166). The 3 a(t) 1 v˜g(k,t) (175) only difference being that all 2-point functions appearing = −2 ρ¯g(t) k2 in the propagator are now replaced by 2 2 matrices, × as well as an additional effective repulsive interaction à (0,bb) (0,bd)! à (0,bb) (0,bd)! potential modelling the effects of pressure (0) Gρρ Gρρ (0) GρF GρF Gρρ (0,db) (0,dd) , GρF (0,db) (0,dd) , 2¡ ¢ 2 → Gρρ Gρρ → G G cs T (t) a (t) ρF ρF v˜p(k,t) , (176) = b 2 g(t) (171) ρ¯ Hi

I (1,2) (2π)3δ (k k )δ (t t )I . (172) where ρ¯b is the mean number density of the mesoscopic → D 1 + 2 D 1 − 2 2 particles [65]. The different components of the potential (0) dd bd db The individual components of Gρρ describe the auto- and matrix (172) are then given by v v v vg and bb = = = cross-correlations between the density fields of baryonic v vg vp. To arrive at the expressions (175) and (176), (0) = + and dark matter, while the components of GρF describe we have set the masses of both dark matter and meso- the response of these density fields to the interactions scopic particles equal. We can do this without loss of gen- between particles of either the same or different species. erality since we are dealing with effective particles. We are These interactions are accordingly collected in a symmet- now able to obtain quantitative results for the evolution ric 2 2 potential matrix, of mixed dark and isothermal baryonic matter. × Ã ! vbb vbd v . (173) → vdb vdd 10.3 Results at late times

To be able to calculate the power spectrum for matter Their density fluctuation power spectra are evolved from mixtures, we still need to find the appropriate interaction the epoch when matter and radiation decoupled at a red- potential vbb, which is the only entry containing not only shift of z 1100 to a redshift of z 100. We choose a = = gravitational interactions. relatively high redshift for the final time since we cur- rently perform the calculations only in the linear order of the resummed theory. Furthermore, the beginning of the 10.2 Baryon interactions reionization era makes it difficult to rigorously track the mean temperature evolution of the gas. Unlike collisionless dark matter, baryonic matter is well- We use the same initial BBKS power spectra [10] for described by an ideal gas, and hence its dynamics are both dark and baryonic matter. However, in Fig. 12, we set governed by the ideal fluid dynamics discussed in Sect. the initial baryonic power spectrum to zero on wavenum- 1 8. Accordingly, we treat baryons in KFT as mesoscopic bers k 1h Mpc− as a crude approximation for the Silk ≥

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damping, which suppressed baryonic structures at small 1.2 scales during radiation decoupling [66]. We plot the results in Figs. 11 and 12, where we have 1.0 divided the obtained power spectra by the power spec- trum for pure dark matter to show the relative deviations 0.8 caused by the incorporation of baryonic matter. DMO

The results show that the repulsive pressure interac- P 0.6 / δ

tions suppress structure growth at small scales. The fact P that the dark-matter power spectrum is also suppressed 0.4 shows that RKFT is able to capture the interplay between the baryonic pressure interactions and the gravitational 0.2 DM + Baryons interactions between the two particle species. This also Baryons DM allows to describe the build-up of baryonic structures on 0.0 scales where the initial baryonic power spectrum was set 0.001 0.01 0.1 1 10 wave number k [h Mpc 1] to zero, as can be seen in Fig.12. Even the subsequent − suppression of the dark-matter spectrum caused by the pressure of these new baryonic structures is captured. The next steps towards gaining further insight into Figure 12 Relative deviation of the baryon power spectrum baryonic effects on structure formation after radiation (red), the dark matter power spectrum (black) and the total decoupling are to drop the isothermal approximation and power spectrum (blue) of a mixed system as compared to a to develop effective potentials capturing additional phys- system consisting of dark matter only, evolved from the time of ical properties of baryonic matter, e.g. radiative cooling radiation decoupling until z 100, using the same cosmologi- = 1 cal parameters as in Fig. 11. At wavenumbers k 1h Mpc− effects. ≥ we set the initial baryonic power spectrum to zero as a crude model for the effect of Silk damping. Due to the gravitational 1.2 interaction with the dark matter, baryonic structures are never- theless build up on these scales. 1.0

0.8 10.4 Mesoscopic particles with photons DMO

P 0.6 /

δ To investigate structure formation before radiation decou- P pling at z 1100, it is necessary to also take into account 0.4 ≈ that the pressure forces acting on the baryons are mainly due to photons. Therefore, we modify the set-up of the 0.2 DM + Baryons Baryons mesoscopic particle by constructing it in such a way that DM it contains both baryons and photons (cf. Fig.13). 0.0 0.001 0.01 0.1 1 10 When investigating structure growth between matter- 1 k [h Mpc− ] radiation equality and photon decoupling, we assume that the mass of the mesoscopic particle and hence its gravitational potential are dominated by its baryonic con- Figure 11 Relative deviation of the baryon power spectrum tent. Hence, the gravitational potential of the particle is (red), the dark matter power spectrum (black) and the total the same as in (175). However, the enthalpy of the particle power spectrum (blue) of a mixed system as compared to a is dominated by the photons since they are much more system consisting of dark matter only, evolved from the time abundant – the ratio in the number densities being given 10 of radiation decoupling until z 100, using slightly different by η 6 10− – as well as due to their ultrarelativistic = ≈ · cosmological parameters Ω 0.315, Ω 0.685, Ω nature. m = Λ = b = 0.049 and h 0.673. On small scales, the growth of baryonic The resulting pressure potential in a baryon-photon = as well as dark matter structures is suppressed by the baryon fluid is given by pressure. 4 π kBTCMB a(t) v˜ (k,t) (177) p,γ = b 2 180ζ(3) ρ¯ ηmPH0 Ωm,0 g(t)

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1 10−

2 10− ] 3

Mpc 3 3 10− − h [ δ P

4 10−

KFT Eisenstein-Hu (1998) 5 10−

1 0.0 −

0.2 BBKS − P / δ 0.4 P − Figure 13 Illustration of mesoscopic particles containing both 0.001 0.01 0.1 1 1 baryons and photons. The color of the baryons and photons k [h Mpc− ] indicates to which mesoscopic particle they belong.

Figure 14 (Upper panel) Comparison between the power spec- trum at recombination obtained in our model (red) and the power spectrum obtained using the Eisenstein-Hu transfer function (blue), using the same cosmological parameters as in Fig. 11. (Lower panel) The relative deviation of both spectra with kB being the Boltzmann constant, mP the proton from the linearly evolved initial BBKS spectrum [10] is shown. mass, TCMB the current temperature of the CMB, H0 the The KFT spectrum reproduces oscillatory features close to current value of the Hubble function and Ωm,0 the current the positions of the BAOs in the Eisenstein-Hu spectrum, but dimensionless matter density parameter. shows a stronger pressure-induced suppression of structure growth on intermediate and small scales. Furthermore, the Assuming an initial BBKS power spectrum, the result- effect of Silk damping is not included in our model. ing matter power spectrum can be seen in Fig. 14. It is compared to the power spectrum obtained using the Eisenstein-Hu transfer function [67]. We clearly see that our model qualitatively reproduces 11 Structure formation with modified features of BAOs close to their positions in the semi- theories of gravity analytic Eisenstein-Hu model. The main difference be- tween the two models is a stronger suppression of struc- 2 1 General relativity successfully describes the physics of tures above a wavenumber of k 10− h Mpc− . ≈ gravitational interactions on a vast range of scales under This suppression is a consequence of assuming a sim- the assumption that the total matter and energy budget of plified initial BBKS spectrum, which does not take the the Universe is to be augmented by the presence of three effect of photons on the transfer function into account. In unknown ingredients, namely dark matter, dark energy the future, we will correct for this to improve the quanti- in form of a cosmological constant, and the inflaton field. tative agreement between our results and Eisenstein-Hu. It is understood that the theory can only be used as an Beyond this, the mesoscopic particles need to be modified effective field theory valid up to the Planck scale, which is to also take into account the dissolution of structures at not renormalizable. Even though it is observationally in small scales due to Silk damping. This can be achieved by good agreement with cosmological data, general relativity treating the enthalpy as an additional dynamical property suffers from severe and persistent theoretical obstacles, of the particles. one of them being the cosmological constant itself. These

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have motivated the construction of alternative theories of 1.16 gravity, both in the large- and in the small-scale regimes. q = 2.00 1.14 q = 1.50 In the same spirit as general relativity, one can con- q = 1.00 1.12 q = 0.50 struct other effective field theories for the underlying grav- q = 0.10 itational interactions. Keeping the fundamental proper- q = 0.05 1.10 ties of general relativity intact (for instance unitarity, lo- GR P cality and invariance under Lorentz symmetry), any mod- / 1.08 ification of gravity will inevitably introduce new propa- Proca P 1.06 gating degrees of freedom. They usually come in form of additional scalar, vector, or tensor fields. Therefore, the 1.04 majority of alternative theories of gravity can be classified 1.02 by means of three important classes (see [68] for a recent review on alternative theories of gravity) 1.00 0.001 0.01 0.1 1 10 – scalar-tensor theories: quintessence, Brans-Dicke, k- 1 wave number k [h Mpc− ] essence, Galileon, Horndeski, beyond-Horndeski, and DHOST-type theories; – vector-tensor theories: vector-Galileon, generalized Figure 15 The non-linear power spectrum of generalized Proca theories, beyond-generalized Proca, or multi- Proca theories compared to the standard spectrum in gen- Proca theories; eral relativity as a function of the wave number k. The different – tensor-tensor theories: massive gravity, bigravity, or models within this class of gravity theories are represented by multi-gravity theories; q . If present, these additional degrees of freedom weaken v gravity on cosmological scales. They either form a con- densate whose energy density sources self-acceleration of the cosmic expansion, or they contribute as a conden- tary tool to probe the evolution of the inhomogeneous sate whose energy density compensates the cosmologi- perturbations. However, these probes are not sufficiently cal constant. The latter is used to solve the cosmological- powerful to disentangle certain degeneracies between constant problem, while the former is useful for applica- different alternative gravity theories. Hence, testing al- tions to dark energy and inflation. The presence of some ternative theories of gravity on cosmological scales will of these degrees of freedom, for instance cosmic vector require a thorough understanding of non-linear cosmic fields, could even give rise to a violation of the cosmologi- structure formation. Nevertheless, high-resolution simu- cal principle. Quite generically, these alternative theories lations of cosmic structures in the non-linear regimes for of gravity will modify the many existing alternative theories of gravity would not – the background evolution by altering the Hubble func- be affordable or almost impossible. Therefore, an essen- tion H; tial task for future inference from cosmological data will – perturbations by enforcing altered structure formation be to develop analytical approaches to non-linear cosmic due to: an altered time sequence of gravitational cluster- structure formation. ing; the evolution of peculiar velocities and the number KFT offers a promising framework for this purpose. density of collapsed objects; modifications in the gauge- Modifications of gravity theories do not alter the KFT for- invariant matter density contrast and its relation to the malism itself. Many alternative theories of gravity can thus gravitational potential; changes in the gravitational slip easily be incorporated into the fundamental KFT equa- parameter, the effective gravitational potential, and the tions, and e.g. non-linear power spectra obtained from growth rate; KFT can be directly compared to observations. The few These alternative theories come with free parameters or necessary adaptations concern even with free functions. Restricting their allowed param- – the background evolution, encoded in the Hubble func- eter space is an indispensable task in order to test them tion H; against general relativity. At the background level, geomet- – the Hamiltonian of the system, including additional rical probes measuring the angular-diameter distance as degrees of freedom; a function of redshift (CMB and BAO) and the distance- – a possible evolution of the effective gravitational cou- redshift relation of supernovae can be used to constrain pling constant Geff; the allowed field space. Concerning linear perturbations, – the linear growth factor D ; and + the CMB temperature anisotropies provide a supplemen- – changes to the gravitational potential.

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As a proof of concept, we show here results obtained within a specific dark-energy model taken from the class of generalized Proca theories [69–72]. Concrete self-accelerating solutions have successfully been con- structed [73–75] and compared with background probes [76]. The non-linear power spectrum obtained for this specific modification of gravity is shown in Fig.15. One immediate observation is the increase of power on scales 1 k 2h Mpc− , which would give rise to enhanced struc- ∼ ture formation on these scales. Direct comparisons of such results with N-body simulations would be highly interesting.

Figure 16 Illustration of the Rydberg blockade in a gas. The ground-state atoms (blue) are exited into Rydberg states (red). 12 Beyond cosmological applications However, due to the Rydberg blockade, there is a radius Rb around each Rydberg atom where no other ground state atom As stated at the beginning of this article, Kinetic Field The- can be excited. ory can be applied to any classical ensemble of particles in or out of equilibrium. To illustrate this, an interesting exercise is to consider an ensemble of Rydberg atoms out of equilibrium. In the past decade Rydberg atoms – atoms excited into a very high principal quantum number n – have enjoyed increasing popularity in theoretical as well as experimen- tal physics. Due to the large separation of the outer elec- tron and the nucleus, Rydberg atoms have a strong dipole moment. Thus, the interaction of atoms in Rydberg states with each other even when they are separated by a micro- scopic distance is still strong, whereas the van der Waals forces between two ground-state atoms separated by a macroscopic distance would be negligible. In addition, Ry- dberg atoms have a long lifetime of 100µs. This makes ∼ them excellent candidates for the study of interactions in many-body systems [77,78]. Figure 17 Illustration of the energy shift due to the Rydberg blockade. A laser tuned to resonance with the excitation of The two properties of Rydberg atoms that make them one atom is not resonant with the excitation of the second such interesting systems for KFT are their strong interac- atom inside the Rydberg blockade radius, given that the line tions and the Rydberg blockade [79,80] illustrated in Fig. width of the excitation is smaller than ∆E . When in the 16 and Fig. 17. The Rydberg blockade provides a natural vdW Rydberg-blockade regime, the system undergoes Rabi oscilla- (anti-)correlation function for these systems, as no two tions between the collective states g g...g and g g...r ...g Rydberg atoms can be closer than two times the Rydberg | 〉 | j 〉 at a frequency pNΩ with the single-atom Rabi frequency Ω blockade radius Rb initially. characterizing the coupling between the ground and Rydberg Strictly speaking, a system of Rydberg atoms has to be state of a single atom. treated in the scope of quantum mechanics. However, if treated as a classical system with given anti-correlation function due to the Rydberg blockade radius, they can be useful to distinguish between quantum and classical cor- 12.1 Rydberg systems and KFT relations in strongly-correlated many-body systems. Find- ing the barrier at which quantum effects can no longer be The initial correlation function is set up with the following ignored is a central task in the field of quantum mechan- picture in mind: N ground-state atoms with a high pack- ics and may help to understand the nature of correlations ing fraction are simultaneously excited into the Rydberg due to quantum-mechanical effects. state. Due to the Rydberg blockade, no two Rydberg atoms

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are closer than 2Rb. Thus, the Rydberg atoms can initially t = 0.0 tc well be approximated as hard spheres. The excitation is as- 2.0 t = 0.1 tc sumed to be a Gaussian random process, even though, in t = 0.2 tc t = 0.3 t reality, Gaussianity is broken precisely because of the Ry- c 1.5 t = 0.4 tc dberg blockade effect due to a strong interaction between t = 0.5 tc the atoms. It stands to reason that, although Gaussianity is broken on very small scales around each Rydberg atom, 1.0 it is again restored on scales outside the Rydberg blockade, RDF where interactions between ground-state atoms are neg- 0.5 ligible and atoms are excited at random. Since the initial conditions are formulated at the scales of the Rydberg- blockade radius, and the volume of the Rydberg blockade 0.0 is very small compared to the relevant scales, Gaussianity can be safely assumed. Thus, the initial distribution of Ry- 0 5 10 15 20 r [µm] dberg atoms will be described by a multivariate Gaussian with a correlation matrix determined by anti-correlations due to the Rydberg blockade. The initial correlations can thus be formulated just as in Sect. 3.3. Figure 18 Comparison between predictions from KFT (dashed) and MD simulations (solid) for the free evolution, The system of N Rydberg atoms is assumed to be ho- i.e. no interaction potential between particles, of a system of mogeneous and isotropic, with the particle positions ini- initially anti-correlated Rydberg atoms. The dashed lines are tially being correlated due to the Rydberg blockade, but barely visible due to the good agreement with the, slightly momentum correlations being absent except for a mo- noisy, solid lines from the simulation. Note that the radial dis- mentum dispersion due to a physical temperature that tribution function RDF ξ(r ) 1 is shown here instead of the will be set externally. In Fig.18 the initial correlation func- = + two-point correlation function ξ(r ). The results shown here tion which is directly sampled from a molecular-dynamics 25 are for n 32000 particles of mass m 1.44 10− kg in a (MD) simulation is shown. When treating Rydberg atoms = = · box of V 8 106 µm3 and a Rydberg radius of R 5µm. as classical particles, their trajectories will be subject to = · b = The results are given in terms of the collision time-scale Hamilton’s equations of motion. ¯ ¯ tc d/vth 64µs with mean particle distance d and ther- The result for the free evolution, i.e. no interaction po- = ≈ mal velocity vth. tential between particles, of the system is shown in Fig. 18. The predictions from KFT for the free evolution of par- ticles agree with those from the MD simulations on all complete generating functional can be defined such that scales. The initial structures due to the Rydberg blockade the statistical properties of the particle ensemble at any are gradually washed out by thermal motion. The inclu- later time can then be derived by applying suitable func- sion of an interaction potential between particles is now tional derivatives to this generating functional. Up to this straightforward in the scope of the KFT formalism. For point, the theory is applicable to wide classes of classical this system the resummation scheme introduced in Sect. particle ensembles, or, more generally, to ensembles of 9 is the most efficient approach to compute the density- classical degrees of freedom. fluctuation power spectrum with linear effects. With the tree-level results from Sect.9, it is already possible to cap- The focus of this review has been the application ture all linear effects in the initial power spectrum to infi- of KFT to cosmology. In contrast to other analytic ap- nite order in the interaction potential. proaches to cosmic structure formation, KFT has the de- cisive advantage of avoiding the notorious shell-crossing problem by construction. Compared to the structurally similar statistical quantum field theories, the Hamilto- 13 Summary and conclusions nian equations of motion for classical particles are deter- ministic and, by their symplectic nature, have unit func- Kinetic field theory applies the principles and concepts of tional determinant. The initial state for the particles in a statistical field theory to ensembles of classical particles cosmology is empirically well-defined as a statistically ho- in or out of equilibrium. The particle ensemble is charac- mogeneous and isotropic, Gaussian random field. Based terized by the probability distribution of its phase-space on these conditions, applying KFT to cosmology is quite coordinates and by the equations of motion. An exact and straightforward.

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Particle interactions have to be approximated, though. We have described three ways for doing so; first, by ex- panding the interaction operator into a Taylor series, lead- ing to the equivalent of Feynman rules; second, by ap- plying Born’s approximation relative to suitably chosen inertial particle trajectories; and third, by a resummation scheme that emerges from a reformulation of KFT purely in terms of macroscopic fields. We have shown how non- linear power spectra for the cosmic density and velocity fields can be derived from KFT and how KFT can be used to explain the internal structure of gravitationally-bound, dark-matter structures. We have further clarified the relation between KFT and the BBGKY hierarchy of more conventional kinetic theory. The application of KFT to fluids is possible by introducing mesoscopic pseudo-particles which, in ad- dition to their microscopic interaction potential, acquire a repulsive potential for mimicking pressure forces. This approach then allows to describe mixtures between dark matter and gas, and we have shown how effects like the baryonic acoustic oscillations can be described in this way. As a further extension of KFT within cosmology which is quite straightforwardly possible, we have given a first example for its application to generalizations of general relativity [68]. Finally, we have shown an application of KFT to a completely different physical system, composed of cold Rydberg atoms, to illustrate the flexibility of the KFT approach. KFT itself, but in particular its applications to cos- mology, are in an early stage of development, and much work remains to be done. The main purpose of this re- view is three-fold: (1) It should demonstrate how natural the foundations of KFT are and how straightforward its generating functional can be derived. (2) It should sketch formal extensions of the theory, in particular in view of different ways to include particle-particle interactions and resummation schemes. (3) It should describe possi- bilities for manifold applications of KFT to cosmological problems, including the joint evolution of gas and dark matter, and give an outlook on applications of KFT to systems entirely unrelated to cosmology. Fundamental clarifications of statistical properties of self-gravitating systems are also in reach and have been initiated in terms of fluctuation-dissipation relations [81]. Apart from this, calculating higher-order spectra of cosmological quanti- ties and quantifying the accuracy of KFT are among the next important steps to be undertaken.

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Appendix

Appendix A: Guide to calculating the power spectrum in the perturbative approach

This appendix is intended as a comprehensive guide to provide the reader with a concise picture of the steps needed to arrive at the power spectrum presented in Fig.3 of this review. 2 We begin with (40) and apply (1) a Taylor expansion of the exponential factor in terms of the interaction operator to first order; and (2) a Taylor expansion of the exponential factor in the initial conditions (64) in terms of the initial correlations up to second order. Step 1 results in the following expression for the power spectrum P (k) £1 iSˆ ¤ Z [L,0] . (178) δ ≈ + I 0 To prepare for step 2 we split the momentum correlations (65) into a damping term 2 σ1 ­ ® QD : Lp ,Lp (179) = 3 and a term contaning momentum correlations between any two distinct particles, h i Q : L> C E L . (180) = p p j pk ⊗ jk p The Taylor expansion up to second order of step 2 then results in the following expressions, Ã ! Z Q 1 (1) N QD /2 X ~ X i Lq ,q Z0 [L,0] : V − e− dq 1 i Cδj pk Lpk Cδj δk e 〈 〉 , = − 2 + j + 2 j k 6= V N Z (2) − QD /2 2 i Lq ,q Z [L,0] : e− dq Q e . (181) 0 = 8 〈 〉 (1) (2) Z0 [L,0] contains all terms up to first order in initial correlations and Z0 [L,0] only the second order terms. We can then write the free generating functional as Z [L,0] Z (1)[L,0] Z (2)[L,0] . (182) 0 ≈ 0 + 0

Note that we only consider higher-order momentum correlations here and ignore cross terms of the form QCδj δk ~ and QCδj pk Lpk since terms containing momentum correlations dominate at late times due to the time dependence of the momentum propagator g (τ): g (τ,0). The initial momentum correlations and, of course, the interaction qp = qp potential between different particles is responsible for the growth of structures while the damping term is responsible for dissolving those structures. If we take initial momentum correlations as well as interactions into account order by order, we must make sure that the damping term does not enter exponentially into the low-order contributions since this would lead to an overestimation of the damping effect. We shall therefore later approximate the damping term exp( Q /2) consistently at one order less than the term exp( Q/2). This implies that damping will only be included in − D − terms of at least second-order in the initial momentum correlations. To first and second order in the initial correlations, with the improved Zel’dovich propagator gqp (τ) defined in (61) and suitably approximated damping terms, we thus find the following expressions for the power spectrum in our free theory 2 P (1)(k,τ) P (i) (k)¡1 g (τ)¢ , δ = δ + qp g 4 (τ) Z Ã~ ~ !2 Ã~ ~ ~ !2 (2) qp (i) ¡ ¢ (i) ³ ´ k k0 k (k k0) P (k,τ) P k0 P ~k ~k0 · · − , (183) δ = ³ σ2 ´ δ δ − k 2 ~ ~ 2 2 1 1 g 2 (τ)k2 k0 0 (k k0) + 3 qp −

2 Note that all vectors in this appendix will be indicated by an arrow to help avoid possible confusion during implementation.

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(i) where we have specified for clarity that the power spectra on the right-hand sides are the power spectra Pδ character- ising the initial particle distribution. In our perturbative approach the interaction potential is given by

A(τ) µ 1 ν¯ ¶ v˜(k,τ) , (184) = − ρ¯ k2 + k where we have introduced an additional term as compared to (78) which corresponds to an adhesion term following the idea of the adhesion approximation [82–84]. This approximation was introduced in order to compensate the effect of free streaming in the Zel’dovich approximation once particle trajectories have crossed. The necessity for such a term in the perturbative scheme arises due to the fact, that to first order in the gravitational potential the overshooting of particle trajectories due to the improved Zel’dovich propagator is not adequately compensated, causing a loss of power on small scales, where structures are wiped out. This overshooting is compensated by going to second-order perturbation theory, but has not yet been implemented in our code. In our more advanced schemes to include particle interactions which we present in Sect. 4.2 and 4.3 the adhesion term becomes obsolete, since the choice for the free propagator describing inertial motion and the form of the force term are consistently linked.

In (184), ν¯ is an amplitude with the dimension of a length scale. As a suitable length scale, we choose the velocity dispersion σv , propagated to the time τ by the improved Zel’dovich propagator gqp (τ),

ν¯ g (τ)σ . (185) = qp v We approximate ν¯ by its late-time behaviour, setting g (τ)σ 2. qp v ≈ With the interaction potential (184), the first-order perturbation contributions to the non-linear power spectrum are Z τ (1) (1) ¡ ¢ (i) ¡ ¢ δ Pδ (k,τ) 2 1 gqp (τ) Pδ (k) dτ0 A(τ0)gqp (τ,τ0) 1 gqp (τ0) , (186) = + 0 + Z τ Z (1) (2) 2 δ P (k,τ) dτ0 A(τ0)gqp (τ,τ0) (ZA ZB ZC ZD), δ = ³ σ2 ¡ ¢´ + + + 1 1 T 2 T 2 T~ T~ 0 k0 + 3 + 0 − · 0 where we introduced the notations T : g (τ)~k and T 0 : g (τ0)~k0 for brevity. The terms Z , Z , Z , and Z have = qp = qp A B C D been derived in detail in [42] so that we merely state the results here,

~ ~ 2~ ~ ~ ~ ~ ~ ~ ³ ´ ³ ´ 2 (k k0) k (k k0)(T T 0) (k k0) ~ ~ ~ ZA gqp (τ)gqp (τ0) · · − − · − Pδ k k0 Pδ k0 , (187) = k 2 (~k ~k )4 − 0 − 0 ~ ~ 2~ ~ ~ ~ ~ ~ ~ ³ ´ ³ ´ 2 (k k0) k0 (k k0)(T T 0) (k k0) ~ ~ ~ ZB gqp (τ)gqp (τ0) · · − − · − Pδ k Pδ k k0 , (188) = − k2 (~k ~k )4 − − 0 (~k ~k0)~k (T~ T~0)~k0 (T~ T~0) ³ ´ ³ ´ · · − · − ~ ~ ZC gqp (τ)gqp (τ0) 2 2 Pδ k Pδ k0 , (189) = − k k0 ¡ ¢2 (~k ~k )2 ~k (~k ~k ) ³ ´ ³ ´ 1 2 2 2 0 0 ~ ~ ~ ZD gqp (τ)gqp (τ0)k · · − Pδ k k0 Pδ k0 . (190) = 2 k 4 (~k ~k )4 − 0 − 0 (1) (1) (1) The terms Pδ and δ Pδ that are proportional to the initial power spectrum must reproduce the linear growth of 2 (i) the initial power spectrum that can be given as D Pδ (k). We therefore replace the terms linear in the initial power 2 (i) + spectrum in (183) and (186) by D Pδ (k). The final result for the non-linear density-fluctuation power spectrum to first order in the interaction potential+ and to second order in the initial correlations is then given by

2 (i) (2) (1) (2) Pδ(k,τ) D (τ)Pδ (k) Pδ (k,τ) δ Pδ (k,τ) . (191) = + + +

We still need to evaluate the integrals over τ0 and k0 that are left in (191) which is done numerically.

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The final ingredients still required are the cosmological model of our Universe and an initial density-fluctuation power spectrum. The cosmological parameters are needed for the computation of the expansion function of the Universe which, in turn, is needed for the growth factor D (a) that serves as a time coordinate for us. We have, of + course, now the complete freedom to take the cosmological model of our choosing; we can choose any initial power spectrum that we like, start our calculations at the appropriate scale factor a at which our initial power spectrum is defined and can then compute the non-linear power spectrum (191) at a desired final scale factor. To arrive at Fig.3 we made the following choices: – ΛCDM Universe with Ω 0.3, Ω 0.7, Ω 0.04, h 0.7, σ 0.8. m = Λ = b = = 8 = – The initial power spectrum was generated with CAMB [45] at redshift z 1100. = – Our final time is chosen to be today (corresponding to z 0). = There are no other parameters in KFT.

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