Why there is no Newtonian backreaction

TL;DR

The paper argues that Newtonian backreaction does not alter the cosmological scale factor, reaffirming the need for dark energy and clarifying misconceptions about backreaction effects in Newtonian gravity.

Contribution

It clarifies that the conventional equations are exact in Newtonian gravity and refutes claims that structure formation can eliminate the need for dark energy.

Findings

Newtonian backreaction is negligible for large volumes.

The generalized Friedmann equation differs from Buchert's approach.

Backreaction does not eliminate the need for dark energy.

Abstract

In the conventional framework for cosmological dynamics the scale factor $a(t)$ is assumed to obey the `background' Friedmann equation for a perfectly homogeneous universe while particles move according to equations of motions driven by the gravity of the density fluctuations. It has recently been suggested that the emergence of structure modifies the evolution of $a(t)$ via Newtonian (or `kinematic') backreaction and that this may avoid the need for dark energy. Here we point out that the conventional system of equations is exact in Newtonian gravity and there is no approximation in the use of the homogeneous universe equation for $a(t)$. The recently proposed modification of Racz et al.\ (2017) does not reduce to Newtonian gravity in the limit of low velocities. We discuss the relation of this to the `generalised Friedmann equation' of Buchert and Ehlers. These are quite different things; their formula describes individual regions and is obtained under the restrictive assumption that the matter behaves like a pressure-free fluid whereas our result is exact for collisionless dynamics and is an auxiliary relation appearing in the structure equations. We use the symmetry of the general velocity autocorrelation function to show how Buchert's $\cal Q$ tends very rapidly to zero for large volume and that this does not simply arise `by construction' through the adoption of periodic boundary conditions as has been claimed. We conclude that, to the extent that Newtonian gravity accurately describes the low-$z$ universe, there is no backreaction of structure on $a(t)$ and that the need for dark energy cannot be avoided in this way.

Equations

$$\ddot{a}+\frac{4\pi }{3}G\overline{\rho }a=0$$

$$\dot{\mathbf{v}}+H\mathbf{v}=-\nabla \varphi /a$$

$${\nabla }^2\varphi =4\pi G(\rho -\overline{\rho })a^2.$$

$$3\ddot{a}/a+4\pi GM/a^3=\mathcal{Q}.$$

$${\ddot{\mathbf{r}}}_i=Gm\sum _{j\ne i}\frac{{\mathbf{r}}_j-{\mathbf{r}}_i}{|{\mathbf{r}}_j-{\mathbf{r}}_i{|}^3}.$$

$${\ddot{\mathbf{x}}}_i+2\frac{\dot{a}}{a}{\dot{\mathbf{x}}}_i=\frac{Gm}{a^3}\sum _{j\ne i}\frac{{\mathbf{x}}_j-{\mathbf{x}}_i}{|{\mathbf{x}}_j-{\mathbf{x}}_i{|}^3}-\frac{\ddot{a}}{a}{\mathbf{x}}_i.$$

$$\mathbf{x}=\mathbf{r}/a\mathrm{and}\dot{\mathbf{x}}=((H-\dot{a}/a)\mathbf{r}+\delta \dot{\mathbf{r}})/a.$$

$$\begin{array}{rl}{\ddot{\mathbf{x}}}_i+2\frac{\dot{a}}{a}{\dot{\mathbf{x}}}_i-& \frac{Gm}{a^3}\int d^{3}x\delta n(\mathbf{x})\frac{\mathbf{x}-{\mathbf{x}}_i}{|\mathbf{x}-{\mathbf{x}}_i{|}^3}\\ & =-\left(\frac{\ddot{a}}{a}+\frac{4\pi Gm\overline{n}}{3a^3}\right){\mathbf{x}}_i.\end{array}$$

$$\mathcal{Q}=\frac{1}{V}\int \mathbf{dA}\cdot (\mathbf{u}(\nabla \cdot \mathbf{u})-(\mathbf{u}\cdot \nabla )\mathbf{u})-\frac{3}{2V^2}{\left[\int \mathbf{dA}\cdot \mathbf{u}\right]}^2$$

$$\begin{array}{rl}{\mathbf{dA}}^{\prime }\cdot {\mathbf{v}}^{\prime }& =dA(1+(v_{xx}+v_{yy})\delta t)\\ & \times (\mathbf{n}-(\hat{\mathbf{x}}{v}_{zx}+\hat{\mathbf{y}}{v}_{zy})\delta t)\cdot (\mathbf{v}+\dot{\mathbf{v}}\delta t)\end{array}$$

$$\begin{array}{rl}d(\mathbf{dA}\cdot \mathbf{v})& /dt=({\mathbf{dA}}^{\prime }\cdot {\mathbf{v}}^{\prime }-\mathbf{dA}\cdot \mathbf{v})/\delta t\\ & =[{\dot{v}}_z+v_z{v}_{xx}+v_z{v}_{yy}-v_x{v}_{zx}-v_y{v}_{zy}]dA.\end{array}$$

$$\ddot{V}=\int \mathbf{dA}\cdot [\dot{\mathbf{v}}+\mathbf{v}(\nabla \cdot \mathbf{v})-(\mathbf{v}\cdot \nabla )\mathbf{v}].$$

$$\mathcal{Q}=\frac{1}{V}\int \mathbf{dA}\cdot (\mathbf{v}(\nabla \cdot \mathbf{v})-(\mathbf{v}\cdot \nabla )\mathbf{v})-\frac{2}{3V^2}{\left[\int \mathbf{dA}\cdot \mathbf{v}\right]}^2$$

$$\begin{array}{rl}\mathcal{Q}=& \frac{H}{V}\int \mathbf{dA}\cdot (\mathbf{r}(\nabla \cdot \mathbf{u})-(\mathbf{r}\cdot \nabla )\mathbf{u}-2\mathbf{u})\\ & +\frac{1}{V}\int \mathbf{dA}\cdot (\mathbf{u}(\nabla \cdot \mathbf{u})-(\mathbf{u}\cdot \nabla )\mathbf{u})\\ & -\frac{2}{3V^2}{\left[\int \mathbf{dA}\cdot \mathbf{u}\right]}^2\end{array}$$

Full text

1 Introduction

It is usually assumed that the spacetime in an inhomogeneous cosmology may be described by a metric which is that of a homogeneous FRW model $ds^{2}=-dt^{2}+a(t)^{2}dx^{2}$, where $x$ is conformal position, with additional very small ‘weak field’ metric perturbations. This does not require that the density perturbations be small; only that the velocities associated with structures are small compared to $c$. The scale factor for the background is assumed to obey the Friedmann equation

[TABLE]

for a homogeneous background universe with density ${\overline{\rho}}\propto a^{-3}$, and where dot denotes time derivative. The density here may be augmented by additional terms $\rho+3P/c^{2}$ for homogeneous dark energy or radiation backgrounds satisfying the appropriate continuity equations. The peculiar (i.e. non-Hubble) motions of non-relativistic particles such as dark matter or galaxies obey the ‘structure evolution’ equations

[TABLE]

where ${\bf v}\equiv a\dot{\bf x}$, $H=\dot{a}/a$ and the spatial derivative is with respect to comoving coordinates ${\bf x}$, and where $\phi$ is a solution of Poisson’s equation sourced by the density perturbation, i.e.

[TABLE]

This system of equations, which may also be obtained in Newtonian cosmology (Peebles 1980), may be used to find the evolution of linearised perturbations and solved in N-body codes to obtain non-linear structure.

Some, however, going back at least to Ellis (1984), have questioned the validity of this as (1) is derived assuming that the Universe is homogeneous, which is obviously not the case. To address this, Buchert and Ehlers (1997), modelling the matter as a Newtonian pressureless fluid (only a crude approximation once multi-streaming occurs, but valid in the linear and quasi-linear regime), have found that the size $a\equiv V^{1/3}$ of a region of volume $V$ containing mass $M$ obeys

[TABLE]

Here ${\cal Q}=2\langle(\theta-\langle\theta\rangle)^{2}\rangle/3+2\langle\omega^{2}-\sigma^{2}\rangle$ where $\theta$ is the volume expansion rate $\sigma^{2}$ and $\omega^{2}$ are the shear and rotations squared, and $\langle\ldots\rangle$ denotes an average over the volume. As with (1) this may be augmented by including a cosmological constant.

This is highly suggestive. The equation of motion (4) for the linear size $a$ is strikingly similar to (1) but has an extra term containing quantities that are second order in the velocity shear $d{\bf v}/d{\bf r}$. The quantities being averaged in ${\cal Q}$ are, like the individual terms in (1), generally of order the inverse squared dynamical time $G\rho$, so one might naively think this would be a strong effect. However, as Buchert and Ehlers point out, for large volumes the actual effect is less than this. As regards the implications for cosmology, they say that ‘the average motion may be approximately given by a Friedmann model on a scale which is larger than the largest existing inhomogeneities.’, but they also argue that ‘the “conspiracy assumption” that ${\cal Q}=0$ […] must be considered a strong restriction on generality’.

Equation (4) is the basis of Newtonian (or ‘kinematic’) backreaction’; the idea that there is a modification of the expansion rate caused by the emergence of structure. It has been studied by Buchert, Kerscher & Sicka 2000, who make some interesting claims; explored in N-body simulations by Kazimierczak 2016; and has been widely discussed in reviews of backreaction (e.g. Buchert & Räsänen, 2012).

In a similar vein, Racz et al. (2017) have proposed that the successes of the $\Lambda$CDM concordance cosmology can be obtained without the need for dark energy. They say ‘Cosmological N-body simulations integrate Newtonian dynamics with a changing GR metric that is calculated from averaged quantities.’ but that ‘There is a choice in how the averaging is done.’ They propose to maintain equations (2) and (3) but obtain $a(t)$ by averaging the local expansion rate $\dot{a}/a$ computed from the local density under some simplifying assumptions and then using this to update $a(t)$ at each time-step. Performing N-body calculations using this algorithm and with matter only they find $a(t)$ very similar to the solution of the Friedmann equation in $\Lambda$CDM. They argue that the successes of the concordance cosmology can thereby be retained without the need for dark energy through this ‘strong backreaction’ effect.

But is it really legitimate to assume that backreaction from structure causes $a(t)$ to deviate at all from the solution of (1)? We can address this in the context of Newtonian gravity. This is relevant because Newtonian gravity should provide a very accurate description of the local universe since all velocities – Hubble and peculiar – are small. And it is in the relatively local universe that the current expansion rate – a problem for matter dominated cosmology in the conventional framework – is measured. As we shall discuss in more detail below, at $z<0.1$ for example, where $H_{0}$ is reliably measured, the background can be treated as Newtonian to a precision of order $z^{2}\simeq 0.01$ and corrections to lowest order weak field gravity perturbations are suppressed by at least a factor $v_{\rm pec}/c$, the ratio of peculiar velocities to the speed of light. Also, the absolute value of the curvature radius, which is arguably a non-Newtonian construct and which may be identified with $a$, is not relevant here. All that counts is the expansion rate $\dot{a}/a$ and how $a(t)$ changes with time. In a homogeneous model these are determined locally. The question of how inhomogeneity affects the expansion might seem to be more complex, but it would seem bizarre indeed if the expansion rate of the local universe were affected by the emergence of structure in the distant universe. So if backreaction is at all important it should be revealed in a Newtonian analysis.

We will now show that, despite the apparently questionable assumption of homogeneity in (1), the system of equations (1-3) is actually precisely equivalent to the full Newtonian equations of motion.

2 Newtonian Cosmology in Scaled Coordinates

For $N$ particles of mass $m$ interacting under their mutual gravitational attraction there are $3N$ second order differential equations

[TABLE]

These may be solved numerically provided initial positions ${\bf r}_{i}$ and velocities $\dot{\bf r}_{i}$ for the particles.

Writing this in terms of arbitrarily re-scaled coordinates ${\bf r}=a(t){\bf x}$, so $\dot{\bf r}=\dot{a}{\bf x}+a\dot{\bf x}$ and $\ddot{\bf r}=\ddot{a}{\bf x}+2\dot{a}\dot{\bf x}+a\ddot{\bf x}$, (5) becomes

[TABLE]

What we are interested in is the motion of particles with initial conditions that are close to being in uniform Hubble expansion with some initial expansion rate $H$ (very close if we start at early times). So we might lay down particles on a regular grid in ${\bf r}$-space within some very large spherical boundary centred on the origin and give the particles small displacements $\delta{\bf r}$ and velocities $\dot{\bf r}=H{\bf r}+\delta\dot{\bf r}$ with ‘peculiar’ velocities $\delta\dot{\bf r}$ chosen to excite the growing mode. The corresponding initial conditions in terms of ${\bf x}$-coordinates are

[TABLE]

The sum in (6) will have two components: A ‘zeroth order’ acceleration that, in the limit that the grid spacing becomes very small, is the same as the gravitational acceleration of a uniform density sphere, which grows linearly with ${\bf x}_{i}$, plus a perturbation determined by the displacements from the grid (we may think of the source of the gravity being that of the unperturbed grid of particles plus that of a set of dipole sources). If we define the number density of particles in ${\bf x}$-space $n({\bf x})\equiv\sum_{i}\delta({\bf x}-{\bf x}_{i})$ and $\delta n\equiv n-{\overline{n}}$ with ${\overline{n}}$ the inverse of the grid cell volume in ${\bf x}$-space, equations (6) become

[TABLE]

It is interesting to compare this with the conventional equations. Those are $3N+1$ equations (3 per particle plus the Friedmann equation for $a$) whereas here we have only $3N$ equations, just as in (5).

But since $a(t)$ is arbitrary we may assert that $a(t)$ is such that the RHS of (8) vanishes – i.e. that $a(t)$ is a solution of (1) – in which case the vanishing of the LHS is equivalent to the conventional structure equations (2) and (3).

Moreover, if we set the initial conditions for (1) to be $\dot{a}/a=H$ then we see from the second of (7) that $\dot{\bf x}=\delta\dot{\bf r}/a$; the initial velocity in ${\bf x}$-space is a pure perturbation with no Hubble-flow component.

Alternatively, if one does not require (1) one obtains modified ‘structure’ equations with a large-scale radial acceleration that would drive a Hubble-like flow to compensate. The results for all physical quantities such as positions, velocities, density etc. however are all invariant with respect to the choice of $a(t)$.

We thus obtain the original conventional system of equations, in which there is no feedback (or ‘backreaction’) from the structure equations on the expansion. But this is no longer open to the challenge that (1) is only an approximation. Equation (8) is precisely equivalent to (5); we are simply using the freedom in choice of $a(t)$ to impose (1) as an identity. We emphasise that the resulting system of equations – the basis of ‘Newtonian cosmology’ – is not novel. It was first obtained by Dmitriev & Zeldovich (1964) and is what is integrated in essentially all modern N-body simulations. Equivalent equations of motion were also obtained by Peebles (1989) in the context of reconstruction of local group orbits from the action principle. The difference here is mainly one of perspective. We have shown that, in principle, the scale factor is arbitrary and need not obey (1) but, in that case, one must then also modify the ‘structure’ equations accordingly. We note that Newtonian cosmology with point mass particles was also considered by Ellis & Gibbons (2015) who considered a model in which there a population of ‘background’ particles with no peculiar motions and ‘galaxies’ that respond to their own peculiar gravity and the mean-field gravity of the background particles.

As discussed by Dmitriev and Zel’dovich (1964), Newtonian cosmology is obtained by considering perturbations to a large uniform density expanding sphere. The radius $R$ of this sphere may be taken to infinity within Newtonian physics as all the physically observable quantities are regular in that limit. In that sense the results are insensitive to the ‘boundary conditions at infinity’. It is however required that one consider a sphere as any other geometry would not expand isotropically and homogeneously. Within the infinite sphere the structure equations (8) may be used to describe structure that is periodic within some finite size box of side $L$, in which case the peculiar potential, velocity and displacements may be expanded as Fourier series as is commonly done.

3 Discussion

We have tried to clarify the meaning of the conventional equations of Newtonian cosmology. We have expressed the usual Newtonian equations (5) in terms of re-scaled (or what cosmologists call ‘comoving’) coordinates ${\bf x}$ to obtain (8). But in these equations the scale factor $a(t)$ is completely arbitrary and has no physical impact so there is no dynamical equation that $a(t)$ must obey. This reflects the fact that the universe we live in can, if one so wishes, be considered to be a perturbation of some hypothetical ‘background’ cosmology, but there is freedom in choosing the background. Exploiting this freedom, the Friedmann equation (1) may be asserted as an identity, and with the initial conditions set to $\dot{a}/a=H_{0}$ we have shown that we then obtain the conventional equations of cosmological dynamics. In these equations (1) should not be considered a dynamic equation so much as an auxiliary relation that determines the form of the equations of motion of the particles.

Newtonian dynamics does not strictly require that the scale factor obey the conventional Friedmann equation. But if $a(t)$ is chosen not to obey the Friedmann equation this results in an additional long-range radial force in the equations of motion in ${\bf x}$-coordinates; the RHS of (8). This is required in order that physical quantities like the expansion rate be independent of the choice of scale factor. Similarly, if the initial $\dot{a}/a$ is not taken to agree with the initial physical expansion rate this implies initial conditions where there will be a net expansion or contraction in comoving coordinates. So if (1) is violated, or the initial conditions are not set appropriately, the solutions of the ‘structure’ equations no longer just describe the emerging structure; they also include part or all of the ‘background’ evolution.

The fully non-linear dynamics of the local universe are exactly described using the standard equations in the Newtonian limit. In these the evolution of the scale factor is decoupled from the evolution of structure, and is fixed by the initial density and expansion rate and, of course, the presence of dark energy. There is no Newtonian backreaction on $a(t)$ from structure.

Specifically, one cannot, as Racz et al. have proposed, keep (2) and (3) but modify (1). These equations are seen from (8) to be intimately linked together. To modify (1) alone results in a theory that does not reduce to Newtonian gravity in the limit of small velocities as does Einstein’s gravity.

To remind ourselves why this is important, this means that a matter only universe, with baryon and dark matter densities (in relation to radiation density) set at values that are acceptable for big-bang nucleosynthesis and CMB acoustic peaks, cannot be successfully matched to observations. As is well known, if the density parameter is taken to be unity this will result in an unacceptably small final expansion rate and if a low $\Omega$ is chosen this would result in global hyperbolic spatial curvature that would mess up the angular scale of the CMB ripples.

How does this relate to the ‘generalised Friedmann equation’ (4) of Buchert & Ehlers (1997)? It is important to realise that their formula has a very different meaning to the Friedmann equation that appears with the structure equation in the conventional framework. Their $a$ is the cube root of a particular volume $V$ and their equation describes the relationship between $\ddot{a}/a$ and the density within that volume. It is not at all surprising that the $\ddot{a}/a$ for some particular volume would differ at some level from $-4\pi GM/3a^{3}$ if there is inhomogeneity. The acceleration is some combination of the background plus fluctuation and the mass density is similarly the background density plus the density fluctuation. But these two fluctuations need not be the same. Indeed, it is perhaps surprising that the deviations would appear only at second order in the fluctuations and not be already present in linear theory. But one would hardly call this ‘backreaction’ of structure on the global expansion rate; it is simply inhomogeneity affecting the local expansion rate and local density but in slightly different ways. The key question is really whether there is a systematic difference. If the combination of quantities being volume averaged in ${\cal Q}$ has a non-zero expectation value then this would imply deviations from Friedmann behaviour even in the limit that $V\rightarrow\infty$ and one would have to reject (1) in favour of (4).

But this is not the case. A strong indication of this, as shown by Buchert & Ehlers, is that ${\cal Q}$ can also be expressed as a surface integral. They obtained this by decomposing the total velocity into a Hubble flow plus perturbation ${\bf v}=H{\bf r}+{\bf u}$ with the expansion rate being that of the region in question. More relevant is to consider the peculiar velocity with respect to the global expansion rate. As shown in the appendix, this gives

[TABLE]

where the first term is equation 14 of Buchert & Ehlers and the second term appears in their appendix B.

An obvious, but largely unanswered, question is: How does $\cal Q$ in (9) depend on $V$? And how large is it typically? An under-appreciated feature of (9) is that, as discussed in the appendix, the expectation value of the integrand of the first term vanishes by symmetry. Consequently, the average of this term, taken over an ensemble of volumes of any size, also vanishes. The typical value of the fluctuation in this contribution to $\cal Q$ for a volume of size $r$ is $|{\cal Q}|\sim v^{2}/r^{2}$, independent of the ‘coherence length’ $\lambda$ of the peculiar velocity field. This tends to zero as $r\rightarrow\infty$, and should be considered to be a ‘cosmic variance’ fluctuation. The second term has a non-zero expectation value, but this is of order $\langle{\cal Q}\rangle\sim v^{2}\lambda^{2}/r^{4}$ and falls to zero even faster.

Thus the quantitative answer to the question that Ellis posed and Buchert & Ehlers addressed is that, averaged over large volumes, the scale factor does obey the Friedmann equation and there is no backreaction on $a(t)$ from the emergence of structure, consistent with what we have found above.

t is reasonable to ask how, if at all, the conclusions here differ from the current position of experts in the backreaction community. In the first paragraph of Buchert & Räsänen (2012) they say that ‘In standard linear theory, the effect vanishes on average by construction. In Newtonian gravity, this turns out to be true also in the non-perturbative regime.’ This is not in conflict with what we have found here. However, a key phrase here is ‘by construction’. Expanding on this they say ‘When we impose periodic boundary conditions in Euclidean space, the backreaction variable $\cal Q$ is strictly zero on the periodicity scale (a three-torus has no boundary)’. Similarly, Buchert, Kerscher & Sicka (2000) say ‘Note that both the numerical and analytic approaches enforce a globally vanishing backreaction by imposing periodic boundary conditions’. This connection between vanishing of $\cal Q$ and periodic BCs is repeated, and later, in their discussion of N-body simulations one reads that ‘Most cosmological N body simulations solve [….] with periodic boundary conditions. Hence, the boundary of $C$ is empty [….], and from Eq. (10) we directly obtain ${\cal Q}_{C}=0$.’ and, following this, ‘It will be a challenge to incorporate backreaction effects in N body simulations.’

We think it might be possible for a reader of these papers to come away with the impression that the N-body simulations and analytic calculations are missing some extra non-negligible backreaction physics ‘by construction’ through the special choice of periodic boundary conditions. This might be further reinforced by Buchert & Ehlers statement that for $\cal Q$ to be zero would require a ‘conspiracy’.

What we have shown here that is $\cal Q$ tends to zero very rapidly in the limit of large volumes regardless of whether the structure is assumed to be periodic. This is based solely on the symmetry properties of statistically homogeneous and isotropic velocity fields. Another minor novelty of our approach is to show how the surface integral form for $\cal Q$ may be obtained directly rather than through the intermediary step of Raychaudhuri’s equation.

The analysis leading to our (8) represents a significant advance over the approach followed in e.g. Buchert & Ehlers and later backreaction studies where it is assumed that the matter can be modelled as a pressure-free fluid. Uncondensed baryonic gas may, if the cooling time is sufficiently short, approximate such a fluid. Collisionless dark matter at very early times before non-linear structures form may behave a lot like such a fluid. But in the non-linear regime that is relevant here this assumption is, at the very least, highly questionable. Once multi-streaming occurs, collisionless dark matter and galaxies develop pressure. The same is true for the bulk of the baryonic gas which cannot cool efficiently. It is only in this way that realistic equilibrated (i.e. ‘virialised’) or quasi-equilibrated structures can form. The only equilibrium state for a pressure-free gas is, in contrast, a dense rotationally supported disk.

The analysis here has been entirely Newtonian. It is certainly true that there must be genuine relativistic effects that will modify the expansion rate. One such effect is that of intergalactic pressure. It is known that most galaxies harbour black holes and it is thought that these merge in the process of the merging of their hosts. The rapidly time varying gravitational potential will inevitably result in expulsion of a small amount of stars and dark matter at high velocities. This results in non-zero kinematic pressure in intergalactic space which, owing to the expansion, will do $PdV$ work. According to special relativity, $\delta E=\delta mc^{2}$, so this loss of energy results in a decrease in mass and therefore a modification to the continuity equation; i.e. there will be a non-zero, and positive, pressure $P$ in $\dot{\rho}=-3H(\rho+P/c^{2})$. This pressure will also appear in the Friedmann $\ddot{a}$ equation and will result in a modification to the expansion rate. Simple estimates, however, suggest that this is negligible for all practical purposes.

One might naively question whether the pressure inside bound stellar systems or in stars themselves might need to be included in some average sense in the Friedmann equations. That is not the case, as was shown by Einstein & Straus (1945). The title of their paper was ‘The Influence of the Expansion of Space on the Gravitation Fields Surrounding the Individual Stars’ and they concluded that there is none. The fact that distant matter is expanding away from stars does not affect them; their gravitational mass – the parameter defining the Schwartzschild geometry that surrounds them – is fixed. Consequently the gravitational mass density of a population of stars, black holes or other compact objects must dilute as $1/a^{3}$ so the pressure $P$ in the continuity equation (and consequently also in the acceleration equation) must vanish.

It has been proposed (e.g. Buchert & Räsänen 2012) that there may be strong GR backreaction on the expansion. We would argue that something quite radical is required for this to be the case. Imagine that we live in an ‘island universe’ much like ours, but extending only to say about $z=0.1$, thus including the region where the expansion rate $H_{0}$ is reliably established to be approximately 70 km/s/Mpc. For a homogeneous sphere, the errors incurred in the Newtonian approximation – the difference between the proper and gravitational mass for instance – is on the order of $v^{2}/c^{2}$ or about one percent.

Adding structure within the island excites the usual scalar metric perturbation with only one spatial degree of freedom as this is driven by the density. Beyond this lowest order weak field are ‘gravitomagnetic’ effects, driven by the matter 3-current associated with structure, but the metric perturbations arising from motions are suppressed relative to the Newtonian term by a factor $v_{\rm pec}/c$. That such post-Newtonian effects are small is supported by direct weak-field calculations of Adamek et al (2013). Beyond the 4 degrees of freedom associated with the matter 4-current, all that is left are the two metric degrees of freedom of gravitational waves, but these do not affect the expansion rate as they are traceless. The errors involved in modelling the expansion of such an island universe with Newtonian physics should therefore be very small.

What then is the effect of adding the external universe? If this is spatially homogeneous and isotropic then, as Einstein & Straus showed, there is no effect. The challenge for backreaction proponents is to explain how the emergence of structure at great distances can affect the local dynamics and make any appreciable changes to the local expansion rate (and thus e.g. reconcile the large observed $H_{0}$ with that expected in a flat universe without dark energy). There are local tidal influences from distant structures, but these are small and, like gravitational waves, do not affect the expansion. The problem with believing that this occurs in GR is that a cornerstone of the theory is that spacetime is locally flat. This means that in the local universe it is the local matter that controls the dynamics through the 1st law of thermodynamics (energy conservation), expressed in the Friedmann continuity equation, and the conservation of momentum expressed in the Friedmann acceleration equation.

Finally, and returning to Newtonian dynamics, we mention another probably small but not obviously vanishing cause of backreaction; that of ‘tidal torques’. It is well known that, in conventional models for galaxy formation, galaxies acquire their angular momentum through non-linear effects as they depart from the linear regime but before they decouple. This can be thought of as a kind of ‘mode-mode’ coupling between the galaxy scale fluctuations and a larger-scale motion; the global expansion. It does not seem entirely obvious that this has vanishing effect on the expansion of the universe. But our main result here shows that, to the extent that the structure is a statistically homogeneous and isotropic random process, there can be no such effect.

4 Acknowledgements

I am grateful to Kevin Croker, John Learned and Istvan Szapudi for stimulating discussions on this topic. It is also a pleasure to acknowledge useful feedback from Pierre Fleury, Thomas Buchert, George Ellis, Syksy Räsänen, an anonymous referee and, in particular, Anthony Challinor.

Appendix A Acceleration in terms of surface averages

Raychaudhuri’s equation leads to the Freidmann-like (4) containing the additional term ${\cal Q}$ that is a volume average of quadratic scalars constructed from the velocity shear tensor. Decomposing the velocity into a Hubble-flow plus perturbation, Buchert & Ehlers obtained a surface integral expression for ${\cal Q}$. Here we show how this may be obtained directly.

The rate of change of the volume at some time $t$ is $\dot{V}=\int{\bf dA}\cdot{\bf v}$ where ${\bf v}$ is the velocity and ${\bf dA}={\bf n}dA$ is an outward directed surface area element. At some slightly later time $t^{\prime}=t+\delta t$ the rate of change of the volume will be $\dot{V}^{\prime}=\int{\bf dA}^{\prime}\cdot{\bf v}^{\prime}=\int dA^{\prime}{\bf n}^{\prime}\cdot{\bf v}^{\prime}$.

The ratio $dA^{\prime}/dA$ is the determinant of the 2D matrix describing the mapping from positions in the initial surface to the final surface. Since ${\bf r}^{\prime}={\bf r}+{\bf v}\delta t$ this is easily found to be $dA^{\prime}/dA=1+(v_{xx}+v_{yy})\delta t$ to first order in $\delta t$ where we have erected coordinates so the $z$-axis is parallel to ${\bf n}$ and where $v_{xx}\equiv\partial v_{x}/\partial x$ etc. Similarly, the unit normal changes if $v_{z}$ varies across the area element: ${\bf n}\rightarrow{\bf n}^{\prime}={\bf n}-(\hat{\bf x}v_{zx}+\hat{\bf y}v_{zy})\delta t$. Thus

[TABLE]

and therefore

[TABLE]

The coordinate frame independent expression of this is easily found by noting that, in this frame, this is the same as ${\bf dA}\cdot(\dot{\bf v}+{\bf v}({\boldsymbol{\nabla}}\cdot{\bf v})-({\bf v}\cdot{\boldsymbol{\nabla}}){\bf v})$.

The second time derivative of the volume $\ddot{V}=(\dot{V}^{\prime}-\dot{V})/\delta t$ is therefore

[TABLE]

Gauss’s law tells us the first term is $\int{\bf dA}\cdot\dot{\bf v}=-4\pi GM$.

To convert this into an expression involving $a=V^{1/3}$ we use $3\ddot{a}/a=\ddot{V}/V-(2/3)(\dot{V}/V)^{2}$ to obtain (4) – i.e. equation 9 of Buchert & Ehlers – but now with

[TABLE]

Writing the velocity as ${\bf v}=H{\bf r}+{\bf u}$ we find

[TABLE]

The integrand in the first line is ${\boldsymbol{\nabla}}\times({\bf r}\times{\bf u})$, so its integral vanishes. The second line is identical to equation 14 of Buchert & Ehlers. That was obtained assuming that $3H$ is the volume average of the volume expansion rate within the particular volume considered and, as they discuss, the last term here, which appears in their appendix B, enters if $H$ is taken to be the global expansion rate.

An under-appreciated feature of the surface integral term is that the expectation value of the integrand vanishes by symmetry. This is because it involves products of the velocity and its spatial derivative like $\langle v_{x}v_{zx}\rangle$. This is the derivative with respect to lag ${\bf r}^{\prime}$, at ${\bf r}^{\prime}=0$, of the correlation function $\langle v_{x}({\bf r})v_{z}({\bf r}+{\bf r}^{\prime})\rangle$. But for a velocity field that is statistically homogeneous and isotropic – no further assumptions are required – these functions are even functions of ${\bf r}^{\prime}$ so $\langle v_{x}v_{zx}\rangle$ vanishes (Monin & Iaglom, 1975; see also Gorski 1988).

For an individual volume the contribution to ${\cal Q}$ will not vanish. If the velocity field has coherence length $\lambda$ the integrand is on the order of $u^{2}/\lambda$. The mean square is $\langle{\cal Q}^{2}\rangle\sim N(\Delta Au^{2}/\lambda)^{2}/V^{2}$ where $\Delta A\sim\lambda^{2}$ and $N=A/\Delta A$. It follows that the typical contribution is $|{\cal Q}|\sim u^{2}/r^{2}$, independent of $\lambda$. This becomes very small for large volumes.

The last line in (14), also of 2nd order in ${\bf u}$, differs from the second in that it has a non-vanishing (negative) expectation value. But it is on the order of ${\cal Q}\sim u^{2}\lambda^{2}/r^{4}$ and so is even smaller than the second term for large $V$. We believe this is what Kazimierczak (2016) has measured in N-body simulations.