Non-equilibrium statistical field theory for classical particles: Linear and mildly non-linear evolution of cosmological density power spectra
We use the non-equlibrium statistical field theory for classical particles, recently developed by Mazenko and Das and Mazenko, together with the free generating functional we have previously derived for point sets initially correlated in phase space, to calculate the time evolution of power spectra in the free theory, i.e. neglecting particle interactions. We provide expressions taking linear and quadratic momentum correlations into account. Up to this point, the expressions are general with respect to the free propagator of the microscopic degrees of freedom.
We then specialise the propagator to that expected for particles in cosmology treated within the Zel’dovich approximation and show that, to linear order in the momentum correlations, the linear growth of the cosmological power spectrum is reproduced. Quadratic momentum correlations return a first contribution to the non-linear evolution of the power spectrum, for which we derive a simple closed expression valid for arbitrary wave numbers. This expression is a convolution of the initial density power spectrum with itself, multiplied by a mode-coupling kernel. We also derive the bispectrum expected in this theory within these approximations and show that its connected part reproduces almost, but not quite, the bispectrum expected in Eulerian perturbation theory of the density contrast.
Within the framework of the non-equilibrium field theory for classical particles, developed by Mazenko and Das and Mazenko Mazenko (2010); Das and Mazenko (2012), we have recently derived a free generating functional for canonical, microscopic ensembles of classical particles whose positions and momenta are initially correlated in phase space Bartelmann et al. (2014). While the motivation for such initial ensembles was already derived from cosmological structure-formation theory, this generating functional may be of substantially wider use.
The conventional approaches to the theory of cosmic structure formation perturb the cosmic density field and study the evolution of the perturbations in the Eulerian or Lagrangian pictures (see Peebles (1980); Bernardeau and Valageas (2008) for reviews). Proceeding into the regime of non-linear density perturbation is notoriously difficult in this approach, and numerous ways have been studied to cope with and to overcome these difficulties, a substantial fraction of which is caused by the dissipation-free nature of the dark-matter particles: Once particle trajectories cross, velocity fields become non-unique, and the density becomes locally arbitrarily large (see Buchert (1992); Buchert and Ehlers (1993); Buchert (1994); Bouchet et al. (1995); Ma and Bertschinger (1995); Ehlers and Buchert (1997); Seljak (2000); Valageas (2001); Cooray and Sheth (2002); Ma and Bertschinger (2004); Valageas (2004); Valageas and Munshi (2004); Bernardeau and Valageas (2008); Pietroni (2008); Anselmi et al. (2011); Anselmi and Pietroni (2012); Carrasco et al. (2012); Pietroni et al. (2012); Valageas et al. (2013); Carrasco et al. (2014); Porto et al. (2014) for a highly incomplete and admittedly subjective selection of studies).
The non-equilibrium statistical field theory of classical particles avoids these difficulties altogether. It begins with an ensemble of classical particles, whose initial phase-space distribution needs to be suitably correlated in view of cosmological applications. The statistical evolution of this ensemble is fully described by a generating functional. Very much like in statistical quantum field theory, this generating functional consists of operators describing particle interactions, applied to a free generating functional. In its application to classical particle ensembles, operators are switched in between which enable the extraction of collective fields from the free generating functional (see Mazenko (2010, 2011); Das and Mazenko (2013, 2012) for the pioneering studies).
The free generating functional is fully determined once the initial particle distribution in phase space is specified, and once the free particle propagators are chosen. For linear equations of motion, such as the Hamiltonian equations, the free propagators are easily given.
This approach to cosmic structure formation has two main advantages compared to the conventional approach. First, the density (or any other collective field) is read off the microscopic degrees of freedom in the generating functional when needed, but no dynamical equation for the density needs to be solved. Second, crossing particle trajectories do not affect the theory. Third, even mild perturbations of particle trajectories can lead to substantial perturbations of the density, allowing to intrude deeply into the non-linear regime of density fluctuations, and fourth, the theory naturally contains a damping term which prevents the notorious runaway evolution of non-linear power spectra at large wave numbers.
Here, we use this theory combined with the initially correlated, free generating functional from Bartelmann et al. (2014) to derive two- and three-point cumulants of the cosmic particle density. We begin with a brief summary of the theory in Sect. 2. In Sect. 3, we use the formalism to derive closed expressions for the density power spectra, taking momentum correlations into account in linear and quadratic order. As an interlude, we calculate cross-spectra between the density and response fields in Sect. 4.
Cosmological results are derived in Sect. 5, where we first show that the well-known linear evolution of the cosmological power spectrum is recovered in this theory if linear momentum correlations are taken into account, and if the free propagator of the microscopic degrees of freedom is taken from the Zel’dovich approximation Zel’dovich (1970). For momentum correlations of quadratic order, we obtain a simple and closed expression for a first contribution to the non-linear evolution of the power spectrum, valid for arbitrary wave numbers. Finally, we derive the bispectrum expected in this theory and show that its connected part almost, but not quite reproduces the result from Eulerian perturbation theory of the density field. In Sect. 6, we briefly summarise our results.
ii.1 Brief summary of the theory
We begin by summarising briefly the non-equilibrium field theory for classical particles developed by Mazenko and Das and Mazenko Mazenko (2010, 2011); Das and Mazenko (2012, 2013) together with the generating functional derived in Bartelmann et al. (2014) for a canonical ensemble of classical particles whose initial phase-space coordinates are correlated.
A canonical ensemble of classical point particles can statistically be described by the partition function
where is the free generating functional
Here, is an operator representing any interaction terms in the particle action. The phase-space trajectory of the free point particle with index , , is given by
with the matrix-valued retarded Green’s function of the equations of motion for the free particles. The initial phase-space position of the particle is . The phase-space coordinates of all particles are collected in the tensor-valued structure
with being an -dimensional vector which is unity in its -th component and zero elsewhere. Likewise, we introduce
to collect the Green’s functions acting on all particles in one object. Finally, the source fields
similarly collect the source-field components for the positions and momenta of all particles. The angular brackets denote the scalar product
where summation over repeated indices is implied.
into (2), we can factorise the free generating functional into a factor depending on the initial phase-space positions and a factor depending on the sources and only,
Since the initial phase-space positions can be pulled out of the time integral in (10), we introduce the time-integrated source vectors and by
defined by means of the projectors
Here, is the unit matrix in three dimensions, and is a matrix whose elements are all zero. In terms of ,
The integral measure in phase space,
weighs the individual phase-space points with an initial probability distribution . Below, we shall specify to be the phase-space probability distribution
derived in Bartelmann et al. (2014) for a point set whose initial correlations are expressed by the correlation operator containing correlations between spatial positions as well as between spatial positions and momenta, and by the momentum correlation matrix .
ii.2 Density cumulants
Based on the free generating functional for the microscopic degrees of freedom, all required collective fields are expressed by the operator tuple . At least, has two components, a density operator and an operator for the response field required for describing the interaction between particles. In Fourier space, the operator for the contribution of particle to the density, acting at the arbitrary position , is
where is the Fourier wave vector conjugate to the position . The operator for the complete density is the sum over the one-particle density operators,
When applied to the free generating functional , the one-particle density operator simply translates the source ,
by the amount
The application of further density operators leads to further translations. The result of applying any sequence of an arbitrary number of one-particle density operators will thus lead to
Once all necessary density operators have been applied, the source can be switched off. Writing in the representation (9), we can then write
In other words, the application of an arbitrary sequence of density operators results in replacing by in the phase-space integrated generating functional given in (14). The time-integrated source then needs to be replaced by the time-integrated shift tensor with the components
Inserting the shift (22) into these equations, we find
According to (21), an -point density cumulant is found by summing over all particle indices,
The further procedure now consists in specifying the shift tensor and its components, and to insert the results in place of the source components and into the expressions derived in Bartelmann et al. (2014) for the contributions to the free generating functional. Specifically, we have
Thus, the time-integrated shift vectors have non-vanishing components only where the component index agrees with the particle index set by the one-particle density operator applied to the free generating functional. For any shift tensor specified by a complete set of particle indices , we write
ii.3 Contributions from linear and quadratic momentum correlations
We now proceed to work out the terms for and . For doing so, we need the result
from Bartelmann et al. (2014) for the terms contributed by linear momentum correlations to the free generating functional. Here, the abbreviations
were used. The prime in (31) indicates that and are to be excluded from the integration over all spatial particle positions .
Of the terms contributed by quadratic momentum correlations, we only need the expressions
With those expressions, the generating functional is
where the damping term
Iii Two-point density cumulants
iii.1 Linear momentum correlations
Since the particles cannot be distinguished, it suffices to select any set of out of the particles to evaluate the remaining sum in (29). These particles can be labelled with indices from to without loss of generality. The generating functional then needs to be calculated for this specific selection of particles, and the resulting terms multiplied by the number of possibilities for the particular subset of particles selected from the canonical ensemble of particles.
If we wish to calculate density cumulants taking momentum correlations into account to first or second order, we need to evaluate the expressions from Eq. (128) and given in Eqs. (132), (133) and (134) of Bartelmann et al. (2014).
For a two-point cumulant, , we can choose . Since the particles have to be different for the correlation terms in and not to vanish, we set . Then,
and no other components of appear. Then, the remaining integrals over all positions except and simply give
If we set , i.e. if the cumulant is taken synchronously, we further have
taking into account that the remaining delta distribution ensures . Therefore,
This expression is symmetric under the permutation . Since the index pair can be selected in ways from the particles, we immediately find
with the damping term
where was used once more.
iii.2 Quadratic momentum correlations
Proceeding to the contribution of quadratic momentum correlations to the two-point cumulant, we see immediately that only terms of the form
derived in Bartelmann et al. (2014) can contribute because terms with three or four different particle indices must vanish for a two-point cumulant. Setting again , and evaluating the factors with the appropriate momentum shift vectors (40), we immediately arrive at
As derived in Bartelmann et al. (2014), the terms are symmetric in the first and second index pair and under exchanges of the two index pairs, and the indices in the first and second index pairs must be different. Under these requirements, the term (47) appears times in the sum over particle indices: terms with the index combinations , , and are all equivalent, and others do not appear. Furthermore, we have to multiply with the number of ways for selecting a pair from the particles. Thus, we arrive at the contribution
for the contribution of quadratic momentum correlations to the two-point density cumulant. The damping term equals that given in (45).
Iv Response-field cumulants
iv.1 General expressions and mean response field
It was shown in Bartelmann et al. (2014) that the response field can be obtained in Fourier space by applying an operator to the free generating functional whose one-particle representation can be expressed by a one-particle density operator,
multiplied by the factor
This expression, valid for a single response-field operator applied to the free generating functional, is easily generalised. Suppose we apply operators in total, of which are density and are response-field operators. Since each response-field operator contains a density operator to be executed first, we will have to apply density operators in total. The result will have to be multiplied by response-field factors. Thus, we have
Decomposing the density operators and the response-field into their single-particle contributions, we obtain the shift tensor from (22). After inserting this into from (11), any single-particle response-field operator returns the factor
Applying a single response-field operator to the generating functional, we obtain the mean response field. In the general approach sketched above, we set . Then, from (52), we have
if the propagator vanishes for , as it will usually do. Then, the mean response field vanishes identically.
iv.2 Response-field power spectra
For , we have the density-response cumulant
in this case and
taking into account, we find
after summing over all . Changing the order of and in (54) only changes the ordering of the times and , thus leading to
The cross-spectra and will obviously vanish if . Applying both and will return a product of propagators with time orderings and , which must vanish for causality, hence
V Cosmological power spectra
v.1 Hamiltonian vs. Zel’dovich propagators
It was shown in Bartelmann (2014) that a free classical point particle in an expanding space-time has the propagator
where the growth factor is used as the time coordinate ,
set to zero at some early time. The scale factor of the expanding space-time is set to unity at the initial epoch, , and the growth factor is normalised such as to equal unity at . The function in (60) is
with the Hubble function , its value at the initial time, and
In an Einstein-de Sitter universe, , thus , and . Therefore, , hence and
Even though the finite bound on was derived for an Einstein-de Sitter model here, it remains true in general cosmological models at least in the matter-dominated phase that the propagator remains finite even for and .
Clearly, therefore, inserting the free Hamiltonian propagator (60) into the free linear power-spectrum term (60) cannot reproduce the result well-known from ordinary cosmological perturbation theory that the matter power spectrum evolves linearly as .
We can, however, easily achieve this behaviour if we replace the Hamiltonian propagator (60) by the Zel’dovich propagator
Then, the time-evolution factor in (60) turns into
and the free linear power-spectrum contribution (60) scales as as it should.
This replacement of the free Hamiltonian by the Zel’dovich propagator is justified and advantageous because the Zel’dovich approximation already contains part of the gravitational interaction between the particles. With the free Hamiltonian propagator, the power spectra shown here would develop without any gravitational interaction taken into account. Instead, by the gravitational interaction it already includes, the Zel’dovich propagator reproduces the linear growth of the density power spectrum at the lowest order, i.e. with the free generating functional and the momentum correlations taken into account at linear order only.
With the Zel’dovich propagator, the free contribution (47) to the power spectrum with quadratic momentum correlations is proportional to the fourth power of , i.e. both convolved power spectra evolve linearly.
The damping term
present in both contributions (44) and (48) requires a separate consideration. It arises from the streaming of the particles with the initial root mean-square velocity quantified by . In cold dark matter, free streaming is suppressed by the gravitational interaction between the particles. Since gravitational interaction is not included yet beyond the part contained in the Zel’dovich approximation at the level of the theory presented here, the damping is both inevitable and unrealistic. We take account of this fact by ignoring the damping term completely at the lowest order of the theory, i.e. when considering the free theory with the Zel’dovich propagator and the linear momentum correlations.
At the next higher order of the free theory, i.e. when the momentum correlations are taken into account quadratically, the damping term needs to be switched on in an appropriate way. If we could include the complete momentum-correlation hierarchy, the complete exponential damping term from (67) would also have be included. However, since the quadratic momentum correlation goes only one step beyond the lowest order, it is appropriate to approximate the factor (67) also at lowest non-trivial order. We write
and multiply the power-spectrum contribution (47) by this approximate damping factor instead of the complete exponential.
v.3 Power-spectra contributions from the free generating functional
We thus obtain the power-spectrum contributions in the free theory with the Zel’dovich propagator and the suitably approximated damping terms
where we have specified for clarity that the power spectra on the right-hand sides are the power spectra characterising the initial particle distribution. The two contributions (69) are shown in Fig. 1 for an initial CDM power spectrum evolved in an Einstein-de Sitter universe such that its present normalisation reaches . The fluctuations visible there in the term are numerical noise from a Monte-Carlo integration of the convolution remaining in (69). With the Monte-Carlo integrator contained in the Gnu Science Library, evaluating the integrand in (69) times, the curves in Fig. 1 require seconds on a single core of a slightly outdated desktop PC.
The contribution to the bispectrum due to linear momentum correlations can be easily read off (29). Ignoring the damping term and focussing on the correlated contribution to the free generating functional in (29), we first notice that again neither nor must vanish since the result would otherwise be zero. Therefore, at least one each of the particle indices must be set to and , while the third particle index available for the bispectrum remains free.
If we set this third index to or as well, the multiplicity of the resulting term is , which is lower by a factor of than the multiplicity required for the bispectrum. This term is thus negligible. Only terms with the third index set to will remain. Adoping implies
Moreover, for a synchronous bispectrum, . Using the Zel’dovich propagator,
where the latter step follows because one of the remaining delta distributions ensures . Combining results, we find
The index combination adds the same expression. Taking the remaining cyclic index permutations into account leads to the bispectrum contribution
for the bispectrum contribution from linear momentum correlations.