A Some Useful Quantities

The missing link: a nonlinear post-Friedmann framework for small and large scales


We present a nonlinear post-Friedmann framework for structure formation, generalizing to cosmology the weak-field (post-Minkowskian) approximation, unifying the treatment of small and large scales. We consider a universe filled with a pressureless fluid and a cosmological constant , the theory of gravity is Einstein’s general relativity and the background is the standard flat CDM cosmological model. We expand the metric and the energy-momentum tensor in powers of , keeping the matter density and peculiar velocity as exact fundamental variables. We assume the Poisson gauge, including scalar and tensor modes up to order and vector modes up to terms. Through a redefinition of the scalar potentials as a resummation of the metric contributions at different orders, we obtain a complete set of nonlinear equations, providing a unified framework to study structure formation from small to superhorizon scales, from the nonlinear Newtonian to the linear relativistic regime. We explicitly show the validity of our scheme in the two limits: at leading order we recover the fully nonlinear equations of Newtonian cosmology; when linearized, our equations become those for scalar and vector modes of first-order relativistic perturbation theory in the Poisson gauge. Tensor modes are nondynamical at the order we consider (gravitational waves only appear at higher order): they are purely nonlinear and describe a distortion of the spatial slices determined at this order by a constraint, quadratic in the scalar and vector variables.

The main results of our analysis are as follows: (a) at leading order a purely Newtonian nonlinear energy current sources a frame-dragging gravitomagnetic vector potential, and (b) in the leading-order Newtonian regime and in the linear relativistic regime the two scalar metric potentials are the same, while the nonlinearity of general relativity makes them different.

Possible applications of our formalism include the calculations of the vector potential Bruni et al. (2014a); Thomas et al. (2015) and the difference between the two scalar potentials from Newtonian N-body simulations, and the extension of Newtonian approximations used in structure formation studies, to include relativistic effects.

98.80.-k; 98.80.Es; 95.35.+d; 95.36.+x

I Introduction

The CDM model Peebles (1984); Efstathiou et al. (1990) has emerged in the last few decades as the standard “concordance” model of cosmology Tegmark et al. (2004). Beyond photons and baryons, the main components of CDM are cold dark matter (CDM), able to cluster and form structures, and the cosmological constant , responsible for the observed acceleration of the Universe’s expansion.

CDM is based on Einstein general relativity (GR), and on the cosmological principle, i.e. a cosmological version of the Copernican principle Ellis et al. (2012); Maartens (2011). This request for the Universe to be, on average, homogeneous and isotropic translates, in the language of spacetime, into assuming a Robertson-Walker metric. With this, Einstein equations reduce to the Friedmann equations; the solutions of these equations are the Friedmann-Lemaitre-Robertson-Walker (FLRW) models. Cosmic microwave background (CMB) anisotropy measurements de Bernardis et al. (2000) have established that the Universe is, to a great degree, spatially flat, as confirmed by recent Planck Ade et al. (2014) and baryon acoustic oscillations (BAO) data Aubourg et al. (2014). In this standard scenario, small primordial inflationary perturbations on top of the FLRW background grow and produce the CMB fluctuations and the large-scale structure that we observe at low redshift.

The theoretical tools that we use to study the growth of the large-scale structure are basically two: i) relativistic perturbation theory Bertschinger (1996); Malik and Wands (2009) is used to describe fluctuations in the early Universe, in the CMB, and in the matter density field on very large scales; ii) Newtonian methods, notably N-body simulations Bertschinger (1998), are used to study the growth of structures in the nonlinear regime, at late times and small scales. Lagrangian perturbation theory (LPT) Ehlers and Buchert (1997); Bernardeau et al. (2002) (i.e. the Zel’dovich approximation or its second-order extension, 2LPT) is typically used to set up initial conditions for N-body simulations; LPT and other Newtonian approximations are also used to model nonlinear scales, e.g. BAO and CDM halos Crocce and Scoccimarro (2008); Matsubara (2008); Taruya et al. (2010); Wang and Szalay (2014); McCullagh and Szalay (2015); Seljak and Vlah (2015).

Observational cosmology has now reached an unprecedented precision, allowing stringent tests on models of the Universe. The tightest constraints come from the CMB Ade et al. (2015). A number of probes exist at low redshift, such as supernovae Betoule et al. (2014), the local measurements of the expansion rate Riess et al. (2011) and the three-dimensional mapping of the large-scale structure Eisenstein et al. (2011); Parkinson et al. (2012). In particular, rich clusters, lensing and redshift space distortion allow the measurement of the growth of clustering Song and Percival (2009); Samushia et al. (2012). The standard CDM model is very well supported by all these observations. However, a tension emerges in the framework of the base CDM when parameter values measured from low-redshift probes are compared with the values obtained from the CMB, as recently pointed out (Macaulay et al., 2013; Verde et al., 2013; Battye et al., 2014; Ruiz and Huterer, 2014). In particular, this tension shows up in a different growth of clustering in LSS at different epochs Ade et al. (2014), as confirmed by the Planck release: “as in the 2013 analysis, the amplitude of the fluctuation spectrum is found to be higher than inferred from some analyses of rich cluster counts and weak gravitational lensing” (see Ade et al. (2015) and Refs. therein). Measurements of redshift space distortion also suggest less clustering than the CMB Macaulay et al. (2013); Battye et al. (2014). Notably, adding a variable total neutrinos mass as extra parameter to the base CDM model relieves the tension Battye et al. (2014); Verde et al. (2013).

Given that the presence of a CDM component is widely accepted, while a cosmological constant has its own problems Weinberg (1989), much work is gone into exploring alternatives to . A number of possibilities have been proposed which, in essence, can be divided in three groups. One is to keep GR and modify the energy-momentum content, the dark sector in particular, replacing with some form of dark energy Amendola and Tsujikawa (2010). A second alternative that has been widely explored in the last decade is to replace GR with a modified theory of gravity Nojiri and Odintsov (2006); Clifton et al. (2012). A third, more radical, option is that of abandoning the cosmological principle Clarkson and Maartens (2010), considering inhomogeneous models and the possibility that the observed acceleration of the Universe is the result of backreaction Buchert and Räsänen (2012); Clarkson et al. (2011), either dynamical or optical Kolb et al. (2010).

Galaxy surveys are now aiming at the same 1% precision as CMB measurements: for instance, the detection of BAO by the Sloan Digital Sky Survey has recently allowed the first 1% level cosmological constraint by a galaxy survey Anderson et al. (2014). In addition, future surveys such as Euclid Laureijs et al. (2011); Amendola et al. (2013); Scaramella et al. (2015) and The Square Kilometer Array Jarvis et al. (2015); Schwarz et al. (2015); Kitching et al. (2015) will reach scales of the order of the Hubble horizon. It is, therefore, crucial that the theory used to make predictions and to interpret observations is developed with a matching accuracy. In particular, while exploring alternatives to the standard CDM scenario is an interesting challenge, it seems timely to refine the theoretical modelling of CDM, bridging the above mentioned gap between the Newtonian treatment of nonlinear small scales and the relativistic description of large scales. Ultimately, going beyond the Newtonian approximation in simulations of large-scale structure should be important in order to take into account causal, retardation and other GR effects that may be non-negligible for simulations that aim at accuracy, in view of future surveys such as Euclid Laureijs et al. (2011), on scales of the order of the Hubble horizon.

A first step in this direction would be to include GR corrections in the initial conditions for simulations, using a dictionary based on first-order perturbation theory Chisari and Zaldarriaga (2011), cf. Green and Wald (2012); also, first-order GR effects on horizon scales must be taken into account in interpreting bias and non-Gaussianity Bruni et al. (2012). However, while the Poisson equation in Newtonian gravity establishes a linear relation between the gravitational potential and the matter density field, the intrinsic nonlinearity of GR unavoidably generates new effects, even when perturbations are small. For instance, an initially Gaussian curvature inflationary perturbation translates into an effective non-Gaussianity of the density field in the matter era Bartolo et al. (2005, 2010); Bruni et al. (2014b, c), a nonlinear effect that should be taken into account in initial conditions for simulations. But the Poisson equation embodies action-at-the-distance; i.e., it is the mathematical representation of the acausal nature of Newtonian gravity. Thus the ultimate step forward would be that of investigating the effects of relativistic nonlinearity in cosmological structure formation, including GR corrections in the evolution of the matter density field in N-body simulations, as well as in approximate treatments of the nonlinear regime Chisari and Zaldarriaga (2011); Green and Wald (2012); Bartolo et al. (2005, 2010); Bruni et al. (2012, 2014a); Kopp et al. (2014); Bruni et al. (2014b, c); Adamek et al. (2014); Green and Wald (2014); Villa et al. (2014); Rampf et al. (2014); Thomas et al. (2015); Yoo (2014).

Aim of this paper is to present a new nonlinear relativistic post-Friedmann (PF) formalism which, in essence, is a generalization to cosmology of the post-Minkowski (weak-field) approximation, married with the fundamental assumption of the post-Newtonian (PN) approximation that velocities are small. Our goal is a relativistic framework valid on all scales, and including the full nonlinearity of Newtonian gravity at small scales. In this framework - assuming a flat CDM background and a fluid description of matter - the exact equations of Newtonian cosmology Peebles (1980); Peacock (1999) appear as a consistent approximation of the full set of Einstein equations, determining leading-order terms (which we call 0PF) in the metric. GR corrections (which we call 1PF) appear next, and are of two types: terms quadratic in the Newtonian variables (we may call these proper PN terms), and proper linear GR terms. Once linearised, the nonlinear approximate equations we obtain for a set of appropriately resummed variables are those for the scalar and vector sectors of first-order relativistic perturbation theory Ma and Bertschinger (1994). Tensor modes are purely nonlinear and nondynamical; i.e., they do not describe propagating gravitational waves, but rather a distortion of the spatial slices.

Let us clarify how our PF formalism differs from the traditional PN approach Weinberg (1972) and its various applications in cosmology. In a contemporary perspective Poisson and Will (2014), the correct derivation of the post-Newtonian approximation on the flat background spacetime follows consistently from the post-Minkowski approximation once the assumption is made, which also implies to neglect time derivatives with respect to space derivatives1. However, in cosmology the background is a FLRW solution, in our case the flat CDM model, and what we can assume to be small are peculiar velocities, not the change with time of the physical distance between two arbitrary observers. To illustrate the point, adopting a Newtonian perspective, in the absolute space of Newtonian cosmology one uses the background comoving coordinates as the Eulerian grid, and the physical position of a fluid element is . Then , where is the physical (or proper) peculiar velocity Bertschinger (1996), representing the deviation from the Hubble flow; is the peculiar velocity with respect to the comoving grid Peacock (1999). Then, should we assume that , we would end up with an approximation only valid at small scales, well below the Hubble horizon, . In addition, traditionally the PN formalism has been developed to study GR corrections to the orbits of isolated objects. Thus, the focus is on the equations of motion, rather than on a consistent approximation of the full set of Einstein equations. In developing his PN treatment of relativistic hydrodynamics, having in mind applications to relativistic stars, Chandrasekhar Chandresekhar (1965) also focused on the equations of motion for the fluid, as derived from the conservation equations. In both cases, the metric and other variables are expanded2 in powers of , then the expansion is applied to the equations of motion. As it is well known, a correction to the time-time component of the Minkowski metric is all is needed to obtain the Newtonian equations of motion. All other corrections are then considered post-Newtonian. We will discuss this point in detail in Sec. VI.2, where, following Misner et al. (1973), we call this approach focused on the equations of motion ”passive”. In cosmology, however, we find desirable to consider an ”active” approach, where the space-space and time-time components of the metric are equally weighted, leading directly to a consistent treatment, order by order in the expansion parameter , of the full set of Einstein equations3. In view of applications, a useful byproduct of this active approach is that the equal weighing of the different metric components is what is needed to obtain the correct photon trajectories, i.e. the null geodesics that are at the base of the causal structure of the spacetime.

Various early and more recent works have applied the traditional PN expansion in cosmology: in Futamase (1988); Tomita (1988, 1991); Futamase (1996) the PN equation of motion for particles in the expanding Universe are derived; in Kofman and Pogosian (1995) the authors clarify the role of the electric and magnetic part of the Weyl tensor in the Newtonian approximation; in Takada and Futamase (1999) and Matarrese and Terranova (1996) the PN analysis is given in Lagrangian coordinates using, respectively, the 3+1 and 1+3 framework; in Hwang et al. (2008) the authors derive a complete set of field and hydrodynamic equations in Newtonian-like forms. Other remarkable works following a PN method are those of Szekeres Szekeres and Rainsford (2000); Szekeres (2000). Finally, Carbone and Matarrese (2005) follows a ”hybrid approximation scheme”, somehow closer in spirit to our PF approach, a mix between standard cosmological perturbation theory and PN approximation, also obtaining equations for the generation of gravitational waves.

All these works apply the standard iterative approach of the PN expansion. Here instead we focus on defining a set of resummed 1PF variables, and on deriving the set of nonlinear evolution and constraint equations they satisfy. Specifically, assuming the Poisson (or conformal-Newtonian Ma and Bertschinger (1994); Bertschinger (1996); Matarrese et al. (1998); Malik and Wands (2009)) gauge, we expand the metric in powers of and in the equations we retain all scalar terms up to order (and all vector terms up to ), instead of peeling-off the different orders, cf. Szekeres (2000); Szekeres and Rainsford (2000). This leads to a final set of equations that differs from those of the works mentioned above. Partly following the PN tradition, the 0PF and 1PF orders, respectively refer to terms and , relevant for scalar potentials. The 0PF terms are Newtonian, the 1PF terms contain the GR corrections. Finally, we stress that, in order to obtain a system of equations with a well-posed Cauchy problem, we should consider terms of the next 2PF order, i.e.  Szekeres (2000); Szekeres and Rainsford (2000).

In summary, the main goal of this paper is to present a set of nonlinear resummed equations up to 1PF order which retain in full the nonlinearity of Newtonian theory on small scales and all linear relativistic perturbation theory on large scales. The 1PF scheme is, therefore, capable of describing, in a unified framework and at the relevant leading orders, the evolution of large-scale structure on all scales of cosmological interest. Our two main results simply follow from the analysis of our nonlinear equations: at leading order a purely Newtonian nonlinear energy current sources a frame-dragging gravitomagnetic vector potential; in the leading-order Newtonian regime and in the linear relativistic regime the two scalar metric potentials are the same, while the nonlinear 1PF equations for the resummed scalar potentials imply that the nonlinearity of GR makes them different.

The paper is organized as follows. After defining the various metric terms in Sec. II, we obtain the stress energy tensor in Sec. III, the field equations in Sec. IV and mass and momentum conservation equations in Section V. We then consider our equations in two opposite limits: the Newtonian regime on small scales, neglecting terms (see Sec. VI) and the linear regime on large scales through the linearization of the equations (see Sec. VII). In Sec. VIII, we first define suitable resummed variables, then we obtain a consistent set of nonlinear equations describing their evolution. In Sec. IX, we draw our main conclusions. Finally, in Appendix A, we apply the PF expansion to the Riemann and Ricci tensors.

Ii Newtonian and post-Friedmann metric variables

We consider a homogeneous and isotropic FLRW flat background where two kinds of perturbation terms are added to the metric, representing two different levels of accuracy. The first will give the Newtonian regime, i.e. an approximate solution of Einstein equations such that the dynamics is described by the exact nonlinear equations of Newtonian cosmology for pressureless matter. The second will give the relativistic corrections, adding terms to the Newtonian equations. In our post-Friedmann framework the goal is to calculate these relativistic terms; in this spirit, we shall refer to the Newtonian approximation as the 0PF (or leading) order, and to the first relativistic corrections as 1PF order.

The expanding parameter for the inhomogeneous perturbations is formally given by , so that the components of the metric tensor in the line element


can be written as


where is proper time and is the scale factor of the FLRW background. Note that we assume that both the scale factor and the metric are dimensionless, while the coordinates have dimension of a length. Greek indices take the values and refer to spacetime coordinates, Latin indices refer to the spatial coordinates. In particular in our post-Friedmann scheme, for a proper “powers of ” order counting, it is important to note that the time coordinate is . Here the Kronecker represents the metric on the flat spatial slices of the background; the spatial Cartesian coordinates are understood as an Eulerian system of reference, cf. Rampf (2014); Rampf and Wiegand (2014).

Our metric is a generalization to cosmology of Chandrasekhar’s metric for post-Newtonian hydrodynamics Chandresekhar (1965) (cf. Hwang et al. (2008) for a cosmological application). However, it is important to remark the difference of our post-Friedmann approach from the standard post-Newtonian one Blanchet (2006); Will (1987); Weinberg (1972); Poisson and Will (2014). In the latter, the focus is on the equation of motion for matter, hence only the leading order metric perturbation is Newtonian, while the is post-Newtonian. In our approach the focus will instead be on the complete set of Einstein equations. We aim at a self-consistent approximate set of equations at each order, similarly to the post-Minkowski (weak field) approximation Bertschinger (2000); Poisson and Will (2014). Then, consistency of the Einstein equations dictate that the and metric perturbations must be of the same order (see Sec. VI for the Newtonian approximation). Anticipating these results, the indices and label Newtonian and post-Friedmann quantities. The quantities are the only relevant one at the 0PF order of approximation of Einstein equations, i.e. in the Newtonian regime. The quantities appear at the 1PF order.

We assume the Poisson (or conformal Newtonian) gauge Ma and Bertschinger (1994); Bertschinger (1996); Matarrese et al. (1998); Malik and Wands (2009); Uggla and Wainwright (2013), so that the three-vectors and are divergenceless, and , and is transverse and tracefree (TT); i.e., it represents pure tensor modes . Commas in front of indices have the standard meaning of partial derivatives.

Having completely fixed the gauge leaves us with six degrees of freedom at each order: the two scalars and at leading order, and and at 1PF order; the two independent components of ( at 1PF); the two independent components of . However, the latter only appears in the equations at 1PF order. At leading order, Einstein equations impose , i.e. a single scalar gravitational potential in the Newtonian regime, as expected. At leading order, is determined by the (vector part of the) Newtonian energy current, and cannot be set to zero. It is not dynamical; i.e, it doesn’t contribute to matter motion. However it appears in the metric and does affect null geodesics and observables. It can, therefore, be extracted from Newtonian N-body simulations Bruni et al. (2014a); Thomas et al. (2015) and it contributes to lensing Thomas et al. (2014).

Note that the 1PF corrections consist of quadratic combinations of Newtonian quantities and intrinsic 1PF variables. We have also included tensorial TT modes but, at 1PF order, they cannot be interpreted as gravitational waves; they satisfy a constraint equation rather than an evolution equation (see Sec. IV.3.3, c.f. Rampf and Wiegand (2014)).

Iii Matter variables

Having defined the metric variables and their weight with respect to the expansion parameter , we now look at the matter quantities. The dimensionless 4-velocity is defined as


and it satisfies the usual relation


i.e. is a unitary timelike vector field. In Newtonian cosmology one uses the background comoving coordinates as the Eulerian grid, the physical position of a fluid element is , and and respectively are is the physical (or proper) peculiar velocity Bertschinger (1996) (the deviation from the Hubble flow) and the peculiar velocity with respect to the comoving grid Peacock (1999). With this in mind, it is then natural to define the physical peculiar velocity as . Therefore, we also define .



and, using (4) and keeping terms up to order , the 4-velocity components are


We consider a Universe filled by cold dark matter (CDM), described by a single pressureless (dust) component with energy-momentum tensor


where is the mass density.

In the Poisson gauge the components and trace of then are


All these quantities are written in order to explicitly show the different contributions in powers of , where in each expression the first term beyond the background term (if present) represents the leading Newtonian order, the second the first 1PF correction, etc. Note that there is no approximation in the trace: indeed, the mass density plays the role of a fundamental exact quantity that is not expanded into contributions at different orders. In the following it will be useful to use the density contrast , defined as usual: , where denotes the background matter density.

Iv Einstein equations

iv.1 Expansion of Einstein equations at 1PF order

We now consider Einstein field equations for the metric (2), including the cosmological constant :


Expanding in powers of , in all equations we retain the first two terms of the expansion. We then obtain the following equations, where the dot denotes partial differentiation with respect to coordinate time .

Time-time component

Spatial component


Time-space component


Finally, neglecting in Eqs. (IV.1) and (11) all terms representing inhomogeneities, we obtain the background equations:


These are recast into the standard Friedmann and Raychaudhuri equations for the flat FLRW background


after substituting the Hubble expansion scalar .

iv.2 1PF equations for the inhomogeneities

We now subtract the background parts (13) from Eqs. (IV.1) and (11), in order to obtain equations for the inhomogeneous quantities. The time-time component of the field equations then gives a generalized Poisson equation:


Note that the cosmological constant disappears from the equations above; it only directly contributes to the background dynamics, i.e. Eqs. (14). Thus perturbations are only affected by through their coupling with the Hubble (background) expansion.

The trace of the space-space component (11) gives


while the trace-free part is


iv.3 Scalar, vector and tensors parts

Scalar equations

It is useful to recast the previous equations in order to isolate, in the linear part of the equations, the scalar, vector and tensor contributions. For the scalar sector let us apply the divergence operator on Eq. (12):


Multiplying Eq. (IV.3.1) by , and applying on Eq. (15), we obtain the following constraint equation:


Now, we can obtain a second scalar constraint by applying the operator on both sides of Eq. (IV.2). This then gives:

These equations will be useful in Sec. VIII.

Vector equations

The vectorial part of equation (12) can be found using the curl operator:


Alternatively, we can also obtain a constraint equation for the vector part by applying the operators to Eq. (12) and to Eq. (IV.3.1), finding


These vectorial equations also depend on scalar quantities, because of nonlinearity; however, the divergence of these equations would give zero, as it should.

An evolution equation for can be obtained by applying the operator on Eq. (IV.2) and subtracting the equation obtained applying on Eq. (IV.3.1). This procedure leads to


This shows that an evolution term for only appears at order .

Tensor equations

In order to isolate the TT part of the metric we define the following nonlinear quantities:


With these definitions, Eq.(IV.3.2) becomes


Finally, using Eqs. (IV.2), (IV.3.1) and (26), we obtain the following constraint equation for :