Newtonian and Relativistic Cosmologies
Cosmological -body simulations are now being performed using Newtonian gravity on scales larger than the Hubble radius. It is well known that a uniformly expanding, homogeneous ball of dust in Newtonian gravity satisfies the same equations as arise in relativistic FLRW cosmology, and it also is known that a correspondence between Newtonian and relativistic dust cosmologies continues to hold in linearized perturbation theory in the marginally bound/spatially flat case. Nevertheless, it is far from obvious that Newtonian gravity can provide a good global description of an inhomogeneous cosmology when there is significant nonlinear dynamical behavior at small scales. We investigate this issue in the light of a perturbative framework that we have recently developed Green and Wald (2011), which allows for such nonlinearity at small scales. We propose a relatively straightforward “dictionary”—which is exact at the linearized level—that maps Newtonian dust cosmologies into general relativistic dust cosmologies, and we use our “ordering scheme” to determine the degree to which the resulting metric and matter distribution solve Einstein’s equation. We find that, within our ordering scheme, Einstein’s equation fails to hold at “order 1” at small scales and at “order ” at large scales. We then find the additional corrections to the metric and matter distribution needed to satisfy Einstein’s equation to these orders. While these corrections are of some interest in their own right, our main purpose in calculating them is that their smallness should provide a criterion for the validity of the original “dictionary” (as well as simplified versions of this dictionary). We expect that, in realistic Newtonian cosmologies, these additional corrections will be very small; if so, this should provide strong justification for the use of Newtonian simulations to describe relativistic cosmologies, even on scales larger than the Hubble radius.
With the improvements in computational abilities that have taken place in recent years, it is now feasible to do numerical simulations of structure formation in cosmology on scales comparable to—or even larger than—the Hubble radius. Such simulations are being carried out by a number of groups Evrard et al. (2002); Warren et al. (2006); Teyssier et al. (2009); Crocce et al. (2010). However, these simulations are being carried out using Newtonian gravity. Although it would appear reasonable to expect Newtonian gravity to yield a good description of phenomena on scales much smaller than the Hubble radius—except, of course, in the immediate vicinity of strong field objects—at first thought, it might seem absurd that it could be expected to yield a reliable description of phenomena on scales comparable to, or larger than, the Hubble radius. After all, Newtonian gravity posits forces that act instantaneously over arbitrarily large distances, whereas the dynamical evolution laws of general relativity assert that all influences propagate causally and that the distribution of matter outside of one’s past light cone is irrelevant. Similarly, the Newtonian gravity description of the Hubble expansion involves relative motion of bodies, whereas the general relativistic description involves the expansion of space. Why should Newtonian gravity give an accurate description of behavior on scales comparable to—or greater than—the Hubble radius, when the relative velocity of bodies is comparable to—or greater than—the speed of light?
Nevertheless, as we shall review in the next section, it is well known (see, e.g., Peebles (1980)) that under the assumptions of spatial homogeneity and isotropy, the equations for a uniformly expanding pressureless fluid (“dust”) in Newtonian gravity are identical to the dynamical equations for a dust filled Friedmann-Lemaître-Robinson-Walker (FLRW) universe in general relativity—even in the case of nonvanishing spatial curvature. An explanation for this remarkable correspondence can be found from the fact that in both Newtonian gravity and general relativity, in the presence of spherical symmetry, the behavior of a co-moving ball of dust does not depend upon the distribution of matter outside of the ball111This fact is closely related to the fact that there is no gravitational field/curvature inside a spherical shell of matter in Newtonian gravity (by Newton’s theorem) and general relativity (by Birkhoff’s theorem).. Thus, in both Newtonian gravity and general relativity, the dynamical behavior of a co-moving ball of dust in a homogeneous, isotropic universe is the same as it would be if that ball were placed in an empty, asymptotically flat spacetime. However, for a sufficiently small ball of dust in an otherwise empty spacetime, Newtonian gravity should be an excellent approximation to general relativity. Thus, for a sufficiently small ball, the density and co-moving radius of the ball must satisfy the same dynamical equations in Newtonian gravity and general relativity. This implies that the equations for the density and comoving radius in a homogeneous, isotropic Newtonian dust cosmology must coincide with the equations for the density and scale factor in a FLRW dust cosmology. This correspondence continues to hold in the presence of a cosmological constant term in the Newtonian and general relativistic equations.
The above argument relies crucially on exact spherical symmetry. Thus, one might expect that no such correspondence between Newtonian and relativistic cosmologies would hold if one perturbs the homogeneous, isotropic solutions away from spherical symmetry. Remarkably, however, the correspondence between Newtonian and relativistic cosmologies extends into the regime of linearized perturbation theory in the case of perturbations off of a spatially flat FLRW dust cosmology. More precisely, as pointed out by Bardeen Bardeen (1980) and will be reviewed in the next section, the scalar gauge-invariant variables of linearized relativistic perturbation theory obey exactly the same equations as the variables describing linearized irrotational dust perturbations of the corresponding Newtonian cosmology. Furthermore, it is not difficult to see that this correspondence extends to the vector case as well, i.e., vector gauge-invariant variables of linearized relativistic perturbation theory obey exactly the same equations as the corresponding Newtonian variables describing vorticity perturbations. Thus, the scalar and vector sectors222The tensor modes correspond to additional degrees of freedom present only in general relativity, and they have no Newtonian correspondence. of linearized relativistic perturbation theory off of a spatially flat FLRW dust model are in exact correspondence with arbitrary Newtonian perturbations off of the corresponding Newtonian dust cosmology.
Further justification for the validity of the Newtonian approximation in cosmology is provided by the work of Oliynyk Oliynyk (2010a, b) (see also Futamase Futamase (1996)). Oliynyk rigorously proved that for a given -torus Newtonian cosmology, there exists a one-parameter family of general relativistic solutions that limits to this Newtonian cosmology, thus showing that there are general relativistic solutions that are arbitrarily close to the Newtonian solution. However, for Oliynyk’s one-parameter families, the ratio of the size of the universe to the Hubble radius goes to zero in the limit333Oliynyk formulated his limit as one in which the -torus remains of fixed size, but the speed of light—and the Hubble radius—goes to infinity. By re-scaling the spatial coordinates, his limit can be reformulated as one in which the speed of light remains constant and the Hubble radius goes to a well defined limit, but the size of the -torus then approaches zero in the limit.. Thus, the general relativistic solutions proven by Oliynyk to be very close to a Newtonian solution have size small compared with the Hubble radius, and thus have no “long wavelength part”. Thus, Oliynyk’s results do not directly address the issue of whether Newtonian simulations on scales comparable to the Hubble radius correspond closely to a general relativistic solution, but it can be viewed as providing additional justification for the validity of Newtonian gravity on scales small compared with the Hubble radius.
Taken together, the above considerations strongly suggest that for a universe that is sufficiently close to a spatially flat FLRW dust model, Newtonian gravity should provide a good description of structure formation on all scales. However, the situation is far from straightforward for the following reasons: (i) Although, as described above, there is a correspondence at linearized order between Newtonian theory and general relativity, the “dictionary” needed to translate a linearized Newtonian solution into metric and matter perturbations in any particular gauge is nontrivial, and it is not obvious how this dictionary compares with standard dictionaries used for the Newtonian and post-Newtonian approximations to general relativity on small scales. Thus, it is not obvious how to produce a “global dictionary” that works on all scales. (ii) If one has a candidate global dictionary, it is not obvious how to formulate criteria to determine whether the resulting general relativistic spacetime is “sufficiently close” to a solution to Einstein’s equation to trust its predictions. The main complication here is that the failure to take post-Newtonian corrections into account on small scales will cause the general relativistic spacetime to fail to satisfy Einstein’s equation by a larger amount than the failure to properly account in any way for the long wavelength perturbations. For most applications in cosmology, the tiny post-Newtonian corrections to the metric and matter motion on small scales are of no interest, but the leading order deviation of the metric and matter density from a FLRW model on large scales is of great interest. Thus, the proper criteria for being “sufficiently close” to a solution to Einstein’s equation must take into account the distinction between small scales and large scales. (iii) One would like to know explicitly what the dominant corrections to Newtonian cosmology are, both to be able to quantitatively judge its reliability and to be able to make its predictions more accurate.
The difficulties in addressing the above issues stem from the fact that the approximations of Newtonian gravity (which, a priori, is expected to be good on small scales) and linearized perturbation theory (which, a priori, is expected to be good on large scales) are incompatible. Specifically, in the Newtonian gravity approximation certain nonlinear terms in the equations are kept (as they must be at small scales), but it is essential that time derivatives of quantities be small compared with space derivatives Oliynyk (2010b, a); Futamase and Schutz (1983); Futamase and Itoh (2007). By contrast, linearized perturbation theory allows time derivatives of quantities to be comparable to their space derivatives (as they must be at large scales), but it is essential that all nonlinear terms be negligible. In order to properly treat phenomena on all scales, one needs an approximation scheme that can accommodate nonlinear phenomena on small scales but treats time derivatives on the same footing as space derivatives on large scales. We recently proposed an approach that accomplishes this Green and Wald (2011), and we will apply this approach here444Our approach is closely related to Baumann et al. (2010); see also Carbone and Matarrese (2005)..
The main questions we wish to address in this paper can now be stated concretely as follows. Suppose that a Newtonian cosmological simulation has been performed on a -torus (i.e., periodic boundary conditions), where the size of the -torus may be larger than the Hubble radius. For convenience, we assume that the Newtonian solution has been presented as a continuum solution—i.e., that suitable smoothing has been done if the solution was produced from an -body simulation. We would like to know the following: (1) What general relativistic spacetime and dust matter distribution should we associate to this Newtonian cosmology, i.e., what “global dictionary” should we use? (2) To what extent is this spacetime a solution to Einstein’s equation, i.e., what are the leading order terms in Einstein’s equation that fail to be satisfied? (3) What are the leading order corrections to the metric and dust distribution that improve the accuracy of this solution, and how large are these corrections?
Our approach will be to use the framework of Green and Wald (2011) to provide a “counting scheme” for the sizes of terms in Einstein’s equation. We will start with a candidate “global dictionary,” which is suggested by the known correspondences between Newtonian gravity and general relativity in the exactly homogeneous and isotropic case and at the linearized level. We will then see that in our counting scheme, the resulting general relativistic spacetime fails to satisfy Einstein’s equation to at small scales and to at large scales. The main effort in our paper will then be to find the corrections to the metric and dust distribution that, within our counting scheme, improve the accuracy of the solution to at small scales and to at large scales. It should be emphasized that we shall not prove existence of a one-parameter family of solutions to Einstein’s equation with the properties we desire—a far more difficult task than solving for leading order corrections. Nevertheless, if the leading order corrections we obtain are small compared with terms appearing in the original global dictionary, we believe that this provides a strong indication that there is a general relativistic solution that corresponds closely to the Newtonian cosmology. Conversely, if these corrections are not negligibly small, then either the Newtonian cosmology is not providing a sufficiently accurate representation of the general relativistic spacetime or the dictionary being used will have to be significantly modified.
Our analysis also addresses concerns that have been expressed with regard to the use of a “Newtonianly perturbed FLRW metric,” which corresponds to the using the “abridged dictionary” given by (46)–(48) below. Ishibashi and Wald Ishibashi and Wald (2006) have argued that this metric should provide an excellent description of our universe. However, several authors van Elst and Ellis (1998); Buchert et al. (2009) have objected to the use of this metric on the grounds that, if taken literally, and dust peculiar velocities are ignored, then strong constraints relating to exact solutions of Einstein’s equation apply, and the metric is only able to describe a spatially homogeneous continuum. Other concerns have been raised by Rasanen Rasanen (2010). The spacetime metric and dust matter distribution that we produce in this paper—as summarized in section IV—solves Einstein’s equation to a much higher degree of accuracy than the Newtonianly perturbed FLRW metric does, and, in particular, fully takes into account peculiar velocities and leading nonlinear terms in the Einstein equation. No inconsistencies of any kind are encountered in obtaining this much more accurate solution. Thus, the approximate solution considered in Ishibashi and Wald (2006) should be fully justified provided only that the corrections to (46)–(48) given in section IV are negligibly small, as we argue is the case.
We remark that if one has an equation , one must draw a clear distinction between having a quantity that approximately solves this equation (i.e., ) as compared with having a quantity that is an approximate solution (i.e., for some exact solution ). If the equation is suitably well posed, if , and if is the exact solution with the same initial data as , then and will remain close to each other for sufficiently early times. However, may fail to remain close to at late times because of the build-up of secular effects. For example, the Newtonian solution for the motion of Mercury solves the general relativistic equations of motion to an excellent approximation at all times, but provides a very poor approximation to the general relativistic solution for the position of Mercury after years. In this paper, we are concerned with the issue of obtaining general relativistic spacetimes that solve Einstein’s equation to an excellent approximation at all times, but we will not be concerned with the issue of whether these spacetimes provide good global-in-time approximations to exact solutions of Einstein’s equation.
In the next section, we shall review the correspondence between homogeneous, isotropic Newtonian cosmology with dust matter and FLRW models in general relativity, as well as the correspondence at the linearized level between these models in the marginally bound/spatially flat case. On the basis of this correspondence, we will propose a dictionary (40)–(44) to translate Newtonian cosmologies into general relativistic spacetimes. In section III, we will apply our counting scheme Green and Wald (2011) to analyze how well Einstein’s equation is being satisfied, and we will obtain corrections needed to satisfy Einstein’s equation to . We will then obtain the further modifications to the metric and dust distribution needed to obtain a solution to Einstein’s equation to at large scales. These corrections are of some interest in their own right. For example, as we shall see in Appendix B, there are small modifications of some global properties of the cosmology, such as a slight modification of the expansion rate and the introduction an (even smaller) anisotropic expansion. However, our main purpose in determining these corrections is to provide a criterion for the validity of the Newtonian cosmology as translated into a general relativistic spacetime via the dictionary (40)–(44) and/or its abridgment (46)–(48) or its simplification (49)–(51): The full set of metric and matter corrections to our original dictionary are given by eqs. (97)–(109) and these corrections can be computed straightforwardly for any given Newtonian cosmology. The smallness of these corrections should provide a reliable criterion for judging the validity of using a Newtonian simulation with the dictionary (40)–(44) (or its abridgment or simplification) to describe a relativistic cosmology.
Ii Background and Linearized Correspondence
In this section we shall review the correspondence between homogeneous, isotropic Newtonian dust cosmology and FLRW models, as well as the correspondence between linearized perturbations of these models. We shall then propose a dictionary—valid to linearized order—the accuracy of which will be evaluated and improved upon in the following section.
In Newtonian gravity, the gravitational field is described by a Newtonian potential , and the dust matter is described by a mass density and a velocity field . The Newtonian field potential is related to the mass density by the Poisson equation
which we have generalized to allow for the presence of a cosmological constant . In addition, the matter variables must satisfy mass conservation and Euler equations, which, for dust matter, take the form
In these equations, the flat Euclidean metric of space is used to contract indices.
ii.1 Background Correspondence
As a cosmological ansatz, we seek a solution to the above equations of Newtonian gravity in which the density is spatially uniform, , and the velocity field is uniformly expanding, . Note that is related to the radius, , of any comoving ball by
Since , (2) implies that
from which it follows that
Using mass conservation, we can eliminate from the Euler equation,
The Poisson equation, (1), has the non-singular solution
Substituting for and , we obtain
which is one of the Friedmann equations. To obtain the other Friedmann equation, we rewrite this equation as
Integrating once, we obtain
where is a constant of integration. By choosing the size of the comoving ball appropriately, we may choose to take the values . When , the value of determines whether the universe is unbound and expands forever (), is marginally bound and expands forever but with expansion velocity approaching zero (), or is bound and will recollapse within finite time (). Of course, in Newtonian gravity, does not have any interpretation in terms of spatial curvature; space is always Euclidean.
ii.2 Linearized Correspondence
We first re-write the exact Newtonian equations relative to some (arbitrarily chosen) background solution of the previous section. We introduce comoving coordinates
We then define the velocity variable by
so measures the velocity relative to the Hubble flow of the background solution. We also define density and potential deviations from the background, and , by
In terms of these quantities, the Newtonian equations are
From now on, since we will always work in comoving coordinates, we shall drop the primes.
Next, we re-write these equations using “conformal time” defined by
We also denote derivatives with respect to conformal time with overdots. In terms of the conformal time variable, the Newtonian background equations are
and the Newtonian equations for the quantities describing the deviations from the background are
where we have now added a subscript so that these Newtonian quantities can be easily distinguished from the corresponding general relativistic quantities that we will introduce later. We emphasize that (23)–(25) are exact. We shall assume below that these equations are solved on a -torus, i.e., a “box” at fixed comoving coordinates (of the background solution) with periodic boundary conditions.
We now compare the linearized Newtonian equations with the linearized general relativistic equations about a dust FLRW model. In Bardeen (1980), Bardeen decomposed linearized metric and stress-energy perturbations into their scalar, vector, and tensor parts, which evolve independently. He then introduced gauge invariant quantities to describe these perturbations. In the case of a perfect fluid, the two scalar gauge invariant variables describing metric perturbations are related—in his notation, —so the scalar perturbations are fully described by (or ), the scalar part of the velocity perturbation, , and a density perturbation variable (defined by eqs. (3.9)–(3.11) and (3.13) of Bardeen (1980)). We can similarly decompose a linearized Newtonian perturbation: and are scalar quantities, and the velocity perturbation can be decomposed as
where can be written as a gradient and . Newtonian perturbations have no tensor part. It is then straightforward to see that, as pointed out by Bardeen Bardeen (1980), under the correspondence
the linearized Newtonian equations become identical to the equations describing scalar perturbations of a spatially flat dust cosmology as given by eqs. (4.3), (4.5) and (4.8) of Bardeen (1980). Note that this correspondence holds only for perturbations of spatially flat models, i.e., there are additional terms in the linearized Einstein equation when the background solution has nonvanishing spatial curvature.
It is not difficult to see that the exact correspondence between linearized Newtonian gravity and general relativistic perturbations of spatially flat models extends to vector perturbations as well with
where the gauge invariant quantity is defined by Bardeen’s eq. (3.23). Specifically, the vector part of the linearized Euler equation (28) for is identical to Bardeen’s eq. (4.13) for . There are no additional Newtonian equations for vector perturbations. However, there is an additional general relativistic equation [Bardeen’s eq. (4.12)],
which is a Poisson equation for a quantity not present in Newtonian theory. On a torus, this equation has a solution if and only if there is no spatially homogeneous part of , i.e.,
This equation must hold for all for all times , but it is easily checked that this equation is preserved under time evolution, so it suffices to impose it at any one time. Thus, (35) is a constraint that must be imposed upon a linearized Newtonian solution in order that it correspond to a linearized solution of Einstein’s equation under the correspondence (33)
In summary, provided only that the constraint (35) is satisfied, there is an exact correspondence, given by (30)–(34), between the complete linearized Newtonian equations for dust matter off of a homogeneous and isotropic background and the scalar and vector parts of the linearized Einstein equation off of a spatially flat dust FLRW background. As previously noted, there are no counterparts to tensor perturbations in Newtonian gravity, i.e., general relativity has these additional degrees of freedom not present in Newtonian gravity.
ii.3 A Proposed Dictionary
Based upon the results of the previous subsections, we now shall propose a dictionary that translates a solution of the exact Newtonian equations (23)–(25) into a general relativistic spacetime metric and dust matter stress-energy tensor . In the next section, we shall investigate the extent to which satisfies Einstein’s equation as well as what further corrections need to be made to to make it solve Einstein’s equation to higher accuracy.
First, the Newtonian equations (23)–(25) were written relative to a “background solution” of (9) and (11) with “scale factor” and mass density . Since these Newtonian background equations are identical to the equations for a dust filled FLRW general relativistic spacetime, we define our dictionary so that it associates the corresponding FLRW spacetime to this background solution. Thus, we have defined our dictionary for the case . In the following, we shall restrict consideration to the case where for the background solution, since this is the only case where we expect a good dictionary to exist when deviations from homogeneity and isotropy occur. We shall assume that periodic boundary conditions have been imposed on the Newtonian background solution, so that the corresponding FLRW background solution has -torus spatial slices. For convenience, we assume that the co-moving spatial coordinates of the Newtonian and FLRW background solutions range between and .
In order for our dictionary to produce a definite general relativistic spacetime, we must make a choice of gauge for the metric. In the context of linearized perturbation theory, a natural and very useful gauge choice is the longitudinal gauge, in which the metric takes the form
where , and , and spatial indices are raised and lowered with the background flat Euclidean metric . In the context of linearized perturbation theory, the quantities , , , and represent the metric perturbation, and it can be shown that an arbitrary metric perturbation can be put in the form (36) by an infinitesimal gauge transformation. It also can be shown that this gauge is essentially unique, i.e., there is essentially no additional gauge freedom that maintains the form (36). However, linearized perturbation theory is not adequate for our purposes, since our dictionary is required to map Newtonian solutions that differ by a finite amount from the Newtonian background solution into metrics that differ by a finite amount from an FLRW model. Nevertheless, it should be possible to show via the implicit function theorem that for metrics that differ from an FLRW model by a sufficiently small but finite amount, the metric form (36)—with , and —always can be imposed by a (nonlinear) gauge transformation. We shall not attempt to prove such a result here, and will merely take (36) as an ansatz for the metric in constructing our dictionary. However, we believe that imposition of the metric form (36) does not involve any loss of generality if the metric is sufficiently close to an FLRW model.
The stress-energy tensor of dust in the general relativistic spacetime takes the form
We define the three-velocity, , of the dust to be such that the components, , of the four-velocity in our gauge are proportional to . Normalizing using the metric form (36), we obtain
Thus, this equation gives the formula for the -velocity appearing in (37) in terms of the -velocity that will be specified by our dictionary below. We define the fractional density perturbation in the general relativistic model via
As already stated above, the Newtonian solution is specified by . With the above gauge choice, the general relativistic spacetime and matter distribution is specified by . Our proposed dictionary will therefore be defined by providing formulas for , , , , , and in terms of the Newtonian variables. To obtain this dictionary, we start by taking the formulas that hold at linearized order under the correspondence of the previous section, which we obtain by expressing the Bardeen variables appearing in (30)–(33) in terms of . Then we improve our definition for (and correspondingly for ) by requiring consistency with the nonlinear momentum constraint at small scales, leading to the replacement of by . We thereby propose the following dictionary:
and is the solution to the equation
with555The condition can be imposed by using the coordinate freedom . , where the overbar denotes spatial average, i.e.,
At large scales, one would expect the vector part of to be negligible because in linear perturbation theory, vector modes are known to decay Bardeen (1980). In addition, comparing the Poisson equation for with the Poisson equation for , one would expect to be smaller than by order at small scales, and thus should be negligible compared with and at all scales. Thus, (41) should yield a negligibly small correction to the equation . Thus, under normal circumstances, it should be acceptable to replace our proposed global dictionary with the following abridged version of the dictionary:
together with . This abridged dictionary corresponds to a continuum version of the dictionary666Their definition of differs from ours by a term involving the perturbed volume element. given by Chisari and Zaldarriaga Chisari and Zaldarriaga (2011).
Finally, on small scales should dominate the other terms appearing on the right side of (42). Thus, on scales much smaller than the Hubble radius, it should be possible to use the following simplified version of the dictionary:
together with . This simplified dictionary is very commonly assumed. However, on scales comparable to the Hubble radius, all terms in (42) should be of comparable size, so if one is interested in investigating behavior on large scales, the full dictionary or abridged dictionary should be used.
As explained above, our dictionary (40)–(44) has been constructed so as to produce a solution of Einstein’s equation to linearized order in . A Newtonian cosmology that corresponds to our universe should have and , but will normally have on small scales. Therefore, it is not obvious, a priori how accurate our dictionary is in producing a solution to Einstein’s equation. In fact, it is clear that there may be difficulties in this regard because the dictionary should produce a spacetime that nearly satisfies the linearized Einstein equation on small scales, but the linearized Einstein equation is incompatible (via the linearized Bianchi identity) with the nonlinear dynamical behavior of matter777In many references (see, e.g., Bertschinger (1993); Chisari and Zaldarriaga (2011)), the linearized Einstein equation is written down together with the nonlinear dynamical equations for matter. This combined system of equations is mathematically inconsistent. that occurs on small scales. We now investigate how close (40)–(44) comes to producing a solution to Einstein’s equation.
Iii Counting scheme and improved dictionary
As stated at the end of the previous section, we wish to determine how close our dictionary comes to producing a solution of Einstein’s equation when the Newtonian cosmology has and , but may have at small scales. To analyze this issue, we need a consistent approximation scheme that can take advantage of the fact that the deviation of the metric from an FLRW model is small on all scales, but permits very large deviations of the stress-energy tensor from an FLRW model to occur on small scales. Such an approximation scheme was recently developed by us in Green and Wald (2011) and used to analyze the backreaction effects of small scale inhomogeneities on large scale dynamics. We refer the reader to that reference for the precise mathematical formulation of the approximation. For our purposes here, it suffices to observe that, as in ordinary perturbation theory, in our approximation scheme there is a “small parameter” (denoted in Green and Wald (2011)) that measures the deviation, , of the metric from a background metric , so . However, unlike ordinary perturbation theory, first spacetime derivatives of are allowed to be , and second spacetime derivatives of —and, hence, the deviations of the stress-energy tensor from the background stress-energy—are allowed to be . In particular, the quadratic products and that appear in Einstein’s equation are , so our approximation scheme allows small scale inhomogeneities to affect the dynamics of the background metric. One of the main results of Green and Wald (2011) is that, in fact, the only possible effect that these nonlinear terms can have on the dynamics of the background metric is to contribute an effective stress-energy that is traceless and has positive energy, corresponding to the presence of gravitational radiation. For the present work, we assume that the universe contains a negligible amount of gravitational radiation, so that this effective stress-energy tensor can be set to zero, and the background metric (which has FLRW symmetry) therefore obeys the ordinary Einstein equation with dust stress energy tensor.
In addition to analyzing the effects of small scale inhomogeneities on the dynamics of the background metric, in Green and Wald (2011) perturbation theory was generalized to allow for significant nonlinearity at small scales, while at the same time maintaining a linearized description at large scales (see also Baumann et al. (2010)). In order to ascribe different behavior to perturbations at different scales these notions must of course be defined. In Green and Wald (2011), the notion of the “long wavelength part” of quantities was defined in a mathematically precise manner by considering the weak limit of these quantities as . As explained in Green and Wald (2011), at sufficiently small but finite , this should correspond closely to taking an average over a spatial scale888For the present universe, should meet these criteria. that is small compared with the background curvature (i.e., the Hubble radius) but sufficiently large that at this scale and beyond we have . For the present work, we shall identify the long wavelength part, , of a tensor field, , with the spatial average999In Green and Wald (2011) averages over (short) time intervals were also performed. The purely spatial averaging we perform here with a suitable “window function” corresponds precisely to the averaging done in Baumann et al. (2010). of its components
using a suitable “window function,” , of size , i.e., a smooth function which is equal to 1 for , and which smoothly falls to 0 outside of this region101010Equivalently, one could work in Fourier transform space and multiply the Fourier transform, , by , where interpolates between (for ) and for .. The requirement that be much smaller than the background curvature scale ensures that this averaging process is well-defined, whereas the requirement that be sufficiently large that should ensure that the long wavelength parts of perturbations behave linearly.
We define the “short wavelength part” of by
thereby providing a decomposition of any quantity into its long and short wavelength parts. The framework of Green and Wald (2011) allows one to make different assumptions in a mathematically consistent manner about the long and short wavelength parts of the various quantities. In particular, derivatives of short wavelength parts can pick up the factors of described above, but derivatives do not increase the size of long wavelength parts.
Our framework can be straightforwardly applied to cosmological Newtonian gravity. It is natural in this case also to impose the additional conditions that velocities are suitably “small” and time derivatives of quantities are correspondingly small compared with space derivatives at small scales. Specifically, the sizes we assign111111The orders we assign to the quantities in Table 1 correspond to the post-Newtonian orderings of Futamase and Schutz Futamase and Schutz (1983) up to a rescaling of the spatial coordinates, but they differ from the post-Newtonian orderings of Oliynyk Oliynyk (2010a, b). to the short wavelength part of the quantities of the previous section are given in Table 1.
On the other hand, the long wavelength part of all of these quantities and their space and time derivatives are assumed to be . It should be noted that certain products of short wavelength quantities can have large scale average. In particular, long wavelength averages of nonlinear quantities corresponding Newtonian potential energy121212A priori, , but due to the fact that is bounded below by , in fact ; see the lemma of section II of Green and Wald (2011)., kinetic energy, and linear momentum enter the perturbation equation for .
Our aim here is simply to use the framework of Green and Wald (2011) as a “counting scheme” in powers of to see how close our dictionary comes to producing a solution to Einstein’s equation. Specifically, we assume that we have been provided with a Newtonian cosmological solution where the “sizes” of quantities correspond to Table 1. To complete our “counting scheme,” we must also assign an -order to . Since is obtained by solving the Poisson equation (44) and, according to Table 1, the source term is of order , we assign the order .
Having assigned -orders to all quantities, we may ask the following question: If we substitute the Newtonian solution into our dictionary (40)–(44) to produce a spacetime metric and dust stress-energy tensor , how close does come to satisfying Einstein’s equation?
If we are to have confidence that the dictionary is producing a good approximation to a solution to Einstein’s equation, we would want Einstein’s equation to be solved to at least in . This is a nontrivial requirement, since there are individual terms, such as , in Einstein’s equation that are in our counting scheme. As we shall see below, the dictionary (40)–(44) solves Einstein’s equation to but fails to yield a solution to Einstein’s equation at in . Nevertheless, we will then show that we can make further small corrections to the metric so that Einstein’s equation does hold to . As we shall see, these metric corrections should be at small scales and therefore should be negligible. If so, our original dictionary (40)–(44) should be producing an accurate relativistic cosmology in terms of its description of the metric and matter distribution on small scales.
In addition, if the dictionary is to be trusted for its description of large scale structure—including on scales comparable to (or larger than) the Hubble radius—we would want Einstein’s equation to hold to at least at large scales. As we shall see below, even after we have made the necessary corrections to the metric so that Einstein’s equation is satisfied to at small scales, Einstein’s equation will fail to hold to at large scales in our counting scheme. We will therefore make further large scale corrections to the dictionary so that Einstein’s equation holds to at large scales. As we shall see, although these corrections are formally of order , they would be expected to make negligible corrections to ordinary linearized perturbation theory at long wavelengths. If so, our original dictionary (40)–(44) should be producing an accurate relativistic cosmology in terms of its description of the metric and matter distribution on large scales.
The above corrections provide us with an improved dictionary that incorporates the dominant general relativistic corrections to (40)–(44). Although the improved dictionary is undoubtedly far more precise than would be needed for most applications, it is important as a matter of principle to know that corrections can be made so that Einstein’s equation holds to at all scales and at on large scales. Furthermore, for any given Newtonian cosmology, the correction terms appearing in the improved dictionary can be calculated straightforwardly, and their size should give a reliable indication of the accuracy of the original dictionary (40)–(44). If, as indicated above, these correction terms are negligibly small, then the Newtonian cosmology should provide—via the original dictionary (40)–(44) and/or its abridgment (46)–(48) or simplification (49)–(51)—an excellent description of what is predicted by general relativity.
iii.1 Solving Einstein’s equation to
iii.1.1 How well are Einstein’s equation solved by the original dictionary?
Appendix A presents the calculation of Einstein’s equation for the metric (36) and stress-energy tensor (37)–(39), keeping all terms that could potentially contribute to as well as all terms that could potentially contribute to at large scales. Given a Newtonian cosmological solution , we substitute it into the dictionary (40)–(44), and substitute the result into Einstein’s equation, freely using the Newtonian equations to simplify the resulting expressions. Equation (111) yields
where we used the Poisson equation (23) for in the second equality. Since the quantities , and are each in our counting scheme (and these terms do not cancel), we see that this component of Einstein’s equation is not satisfied to .
Equation (112) yields
On the torus this may be integrated, giving
Thus, we obtain
i.e., these components of Einstein’s equation are satisfied131313In fact, the precise forms of (41) and (44) were chosen so that no further corrections to and would be needed to satisfy Einstein’s equation to . to .
Finally, from eq. (113) we obtain the space-space components,
Thus, these components of Einstein’s equation are not satisfied to .
iii.1.2 Corrections to the dictionary needed to solve Einstein’s equation to
We will now show that all components of Einstein’s equation can be satisfied to by making the additional corrections , , and to the spacetime metric as follows:
with , , and all . However, we do not make any modifications to the original dictionary expressions for , , and , i.e., we continue to use
In particular, it should be emphasized that no additional corrections are made to the matter distribution variables and .
We have already seen that the original dictionary solved the time-space components of Einstein’s equation to and it is not difficult to see that these equations continue to hold with the above revisions. Thus, to solve Einstein’s equation to , we need only consider the space-space components (113) and the time-time component (111). To solve (113), we note that we can uniquely decompose any symmetric tensor field on a -torus with flat metric and flat derivative operator as
with , , and . This defines the decomposition of into its scalar (), vector (), and tensor () parts. Thus, we can solve an equation of the form by separately solving its scalar, vector, and tensor parts. To begin, we take the double divergence of the traceless part of (113). We obtain141414In substituting for we have neglected some terms proportional to , which can in fact be quite large in low density regions. When one makes a uniform momentum correction , this corresponds, in a low density region, to a very large velocity correction which is unphysical. If such a situation were to occur, then a fix would be to transfer some of this momentum to a higher density region.
Here we have dropped terms which are , since we have taken two spatial derivatives of an equation that we wish to satisfy to . We can solve (67) to the desired order by defining to be the solution to the following double Poisson equation:
A solution for exists on a torus because the source term is a divergence and therefore has no spatially constant piece. This solution is unique up to a spatially constant function of time, which we fix by requiring that its spatial average, , vanishes. Since the double divergence of the traceless part of (113) has now been solved to , the scalar part of the traceless part of (113) should now be solved to , as desired. Note that since the four spatial derivatives applied to yields a quantity that is , the short wavelength part, , of should be , so our assumption that is is self-consistent.
To see if this equation holds, we take its Laplacian. The double Laplacian of will then appear, and we can substitute for this quantity using (68). Since we want to solve (69) to , and each spatial derivative increases the small scale order by a factor of , we wish to solve the Laplacian of (69) to , so the equation we wish to solve is
so (70) is indeed solved to the desired order. We have thus fully solved the scalar parts of the space-space part of Einstein’s equation to .
Next, we consider the vector part of (113) by taking its divergence, using the fact that the scalar parts have already been solved. We obtain
Using the Newtonian Euler and mass conservation equations, along with the definition of , one can show that