# The Price of Abandoning Dark Matter Is Nonlocality

Cedric Deffayet, R.P Woodard

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Cedric Deffayet, R.P Woodard. The Price of Abandoning Dark Matter Is Nonlocality. Journal of Cosmology and Astroparticle Physics, 2024, 2024 (05), pp.042. 10.1088/1475-7516/2024/05/042 hal-04483176

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# The price of abandoning dark matter is nonlocality

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# The price of abandoning dark matter is nonlocality

C. Deffayet and R.P. Woodard

<sup>a</sup>Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS Sorbonne Université, Université Paris Cité, F-75005 Paris, France

<sup>b</sup>Department of Physics, University of Florida, Gainesville, FL 32611, U.S.A.

E-mail: cedric.deffayet@phys.ens.fr, woodard@phys.ufl.edu

**ABSTRACT:** We consider the classic question posed by Pardo and Spergel about the price of abandoning dark matter in the context of an invariant, metric-based theory of gravity. Our answer is that the price is nonlocality. This has been known for some time in the context of the quasi-static regime. We show that it also applies for cosmology and we exhibit a model which reproduces standard CDM successes such as perturbations in the cosmic microwave background, baryon acoustic oscillations and structure formation.

**KEYWORDS:** dark matter theory, modified gravity, cosmological perturbation theory

ARXIV ePRINT: 2402.11716

<figure><img src="image_2.png" /></figure>
# Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>2</td><td>Reconstructing the effective action</td><td>2</td></tr><tr><td>2.1</td><td>Linearized perturbations</td><td>2</td></tr><tr><td>2.2</td><td>Eliminating CDM</td><td>3</td></tr><tr><td>2.3</td><td>Reconstructing the effective action</td><td>3</td></tr><tr><td>3</td><td>Invariant formulation</td><td>4</td></tr><tr><td>3.1</td><td>The ADM metric</td><td>4</td></tr><tr><td>3.2</td><td>Representing the full CDM stress tensor</td><td>5</td></tr><tr><td>3.3</td><td>Nonlocal effective action</td><td>6</td></tr><tr><td>4</td><td>Conclusions</td><td>6</td></tr></table>

## 1 Introduction

Despite increasingly aggressive efforts at direct detection [1–7], the evidence for dark matter is still restricted to its impact on gravity. It is therefore worthwhile considering whether or not the same gravitational phenomena might be instead signal some modification of gravity. Milgrom’s MOdified Newtonian Dynamics (MOND) [8–10] does an excellent job of explaining, and even predicting observed features of galactic structure which are usually attributed to dark matter [11–23]. MOND has mixed success explaining galactic clusters [24–26], but its real challenge is to reproduce the successes of dark matter in cosmology, including the anisotropies in the cosmic ray microwave background (CMB), baryon acoustic oscillations (BAO) and linearized structure formation.

If dark matter is abandoned in favor of modified gravity, it must be that MOND represents the static limit of some larger, relativistic theory. So it is perhaps not surprising that extrapolations of MOND should encounter problems as one approaches the time-dependent regime of cosmology. And it is significant that relativistic extensions of MOND such as Bekenstein’s TeVeS [27] do a much better job of reproducing cold dark matter (CDM) cosmology [28–31]. In particular, the AeST model of Skordis and Zlosnik [32] is internally consistent and agrees with those phenomenological checks which have so far been made [33–36].

The only metric-based modification of gravity which is both stable and generally coordinate invariant consists of changing the Einstein-Hilbert Lagrangian from $R$ to some nonlinear function $f(R)$ [37]. This induces a new scalar degree of freedom which could be regarded as a sort of dark matter [38]. Within the context of locality, the other options for modifying gravity involve either abandoning the metric as the sole gravitational field variable — such as TeVeS [27] and AeST [32] — or else abandoning full general coordinate invariance [39, 40]. We will here explore the option of abandoning locality.

Our result can be regarded as an answer to the question posed in the title of the recent paper by Pardo and Spergel, “What is the price of abandoning dark matter?” [41]. We
demonstrate that the price is nonlocality, provided one maintains both general coordinate invariance and the metric's status as the sole gravitational field. This has already been shown for gravitationally bound systems [42, 43], but cosmological extensions of that model involve fitting a free function just to recover the correct expansion history [44], and in any case fail to provide the extra gravitational force need for structure formation without dark matter [45]. In this new effort we exploit the equations which describe cold dark matter (CDM) cosmology in the regime of linearized perturbations, in particular, the fact that there is no strong evidence for self-interactions [46–48]. This means that the CDM stress tensor must be separately conserved, which permits us to express it in terms of linearized scalar gravitational perturbations. We accomplish this in section 2, and also infer a gravitational effective action whose variation gives the CDM stress tensor. In section 3 we devise a generally coordinate invariant expression for this effective action. This approach is guaranteed to reproduce the cosmology of CDM. In our Conclusion (section 4) we discuss the possibility that this action arises from secular interactions between inflationary gravitons which became nonperturbatively strong [49]. We also discuss the prospects for a single effective action which describes both cosmology and gravitationally bound systems.

## 2 Reconstructing the effective action

The purpose of this section is to derive a purely gravitational effective action whose variation is guaranteed to reproduce CDM phenomenology in the linearized regime. We begin by giving the metric perturbations and the two variables which characterize the CDM stress tensor. The fact that the CDM stress tensor is separately conserved permits us to express the density contrast and momentum divergence in terms of the metric. The section closes by constructing a gravitational effective action whose variation gives the CDM stress tensor.

### 2.1 Linearized perturbations

It is simple to reproduce the CDM contribution to the expansion history because this is just a single degree of freedom. The real challenge is getting the infinite number of degrees of freedom characterized by linearized perturbations. Although there are four scalar perturbations about the Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, gauge transformations can be used to set any two to zero. We choose the two nonzero perturbations as,

$$ds^2 = -[1 + 2\Psi(t, \vec{x})] dt^2 + 2\partial_i B(t, \vec{x}) dt dx^i + a^2(t) d\vec{x} \cdot d\vec{x}. \quad (2.1)$$

The first order connections are,

$$\Gamma^0_{00} = \dot{\Psi}, \qquad \Gamma^0_{0i} = \partial_i [\Psi + HB], \qquad \Gamma^0_{ij} = Ha^2\delta_{ij} [1 - 2\Psi] - \partial_i \partial_j B, \quad (2.2)$$

$$\Gamma^i_{00} = \frac{\partial_i}{a^2} [\Psi + \dot{B}], \qquad \Gamma^i_{j0} = H \delta_{ij}, \qquad \Gamma^i_{jk} = -H \partial_i B \delta_{jk}. \quad (2.3)$$

The CDM stress tensor $\Delta T_{\mu\nu}$ is characterized (at linearized order) by its background energy density $\bar{\rho}(t)$, the density contrast $\delta(t, \vec{x}) \equiv \delta\rho(t, \vec{x}) / \bar{\rho}(t)$ and the momentum divergence $\theta(t, \vec{x})$,

$$\Delta T_{00} = \bar{\rho}[1 + \delta + 2\Psi + \dots], \quad \Delta T_{0i} = \bar{\rho}\partial_i[0 - \theta + \dots], \quad \Delta T_{ij} = \bar{\rho}[0 + 0 + \dots]. \quad (2.4)$$
## 2.2 Eliminating CDM

We now exploit the fact that the CDM stress tensor is separately conserved. Relations (2.2)–(2.3) can be used to derive two equations,

$$0 = D^{\mu} \Delta T_{\mu 0} = -(\partial_t + 3H)\bar{\rho} + \bar{\rho} \left\{ -\dot{\delta} + \frac{\nabla^2}{a^2} [B - \theta] \right\} + \dots, \quad (2.5)$$

$$0 = D^{\mu} \Delta T_{\mu i} = 0 + \bar{\rho} \partial_i \{ \dot{\theta} + \Psi \} + \dots \quad (2.6)$$

The 0th order part of equation (2.5) implies that the background CDM energy density has the equation of state of pressureless matter,

$$\bar{\rho}(t) = \frac{\rho_0}{a^3(t)}, \quad (2.7)$$

where $\rho_0$ is a constant. Equation (2.6) allows us to express the momentum divergence as the time integral of the Newtonian potential,

$$\theta(t, \vec{x}) = -\frac{1}{\partial_t} \Psi(t, \vec{x}) = -\int_0^t dt' \Psi(t', \vec{x}). \quad (2.8)$$

Substituting (2.8) in equation (2.5) gives the density contrast,

$$\delta = \frac{1}{\partial_t} \left[ \frac{\nabla^2}{a^2} \left( B + \frac{1}{\partial_t} \Psi \right) \right]. \quad (2.9)$$

## 2.3 Reconstructing the effective action

We seek an effective action $\Delta S$ whose variation gives the CDM stress tensor (2.4), considered as a functional of the metric. The general relation between an induced stress tensor and variations is,

$$\Delta T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta \Delta S}{\delta g^{\mu\nu}} \iff \frac{\delta \Delta S}{\delta g^{\mu\nu}} = -\frac{1}{2} \sqrt{-g} \Delta T_{\mu\nu}. \quad (2.10)$$

Because our metric (2.1) depends only on the two scalar perturbations $\Psi(t, \vec{x})$ and $B(t, \vec{x})$ we must specialize relation (2.10) to variations with respect to these two fields,

$$\frac{\delta \Delta S}{\delta \Psi} = -2 \frac{\partial g^{\mu\nu}}{\partial g_{00}} \times \frac{\partial \Delta S}{\partial g^{\mu\nu}} = -\sqrt{-g} g^{0\mu} g^{0\nu} \Delta T_{\mu\nu}, \quad (2.11)$$

$$\frac{\delta \Delta S}{\delta B} = -2 \partial_i \left\{ \frac{\partial g^{\mu\nu}}{\partial g_{0i}} \times \frac{\partial \Delta S}{\partial g^{\mu\nu}} \right\} = -\partial_i \left\{ \sqrt{-g} g^{0\mu} g^{i\nu} \Delta T_{\mu\nu} \right\}. \quad (2.12)$$

Substituting (2.1) and (2.4), and retaining only terms up to first order gives,

$$\frac{\delta \Delta S}{\delta \Psi} = -a^3 \bar{\rho} \{1 - \Psi + \delta + \dots\}, \quad \frac{\delta \Delta S}{\delta B} = -a^3 \bar{\rho} \frac{\nabla^2}{a^2} \{-B + \theta + \dots\}. \quad (2.13)$$

There are two ways to reconstruct an effective Lagrangian $\Delta \mathcal{L}$ whose variations reproduce (2.13). The first is to consider $\delta(t, \vec{x})$ and $\theta(t, \vec{x})$ as auxiliary fields whose equations of motion enforce relations (2.8)–(2.9). In that case the result is,

$$\Delta \mathcal{L}_1 = -\rho_0 \left\{ 1 + \Psi - \frac{1}{2} \Psi^2 - \left[ B - \theta \right] \frac{\nabla^2}{2a^2} \left[ B - \theta \right] + \delta(\dot{\theta} + \Psi) + \dots \right\}. \quad (2.14)$$
Note that we have used $\bar{\rho}(t) \times a^3(t) = \rho_0$, and that the variation of (2.14) with respect to $\delta$ enforces (2.8) while the variation with respect to $\theta$ gives (2.9).

The second way to realize (2.13) is through a nonlocal Lagrangian which depends only on $\Psi$ and $B$,

$$ \Delta \mathcal{L}_2 = -\rho_0 \left\{ 1 + \Psi - \frac{1}{2} \Psi^2 - \left[ B + \frac{1}{\partial_t} \Psi \right] \frac{\nabla^2}{2a^2} \left[ B + \frac{1}{\partial_t} \Psi \right] + \dots \right\}. \quad (2.15) $$

In this case one must interpret the variation of $\frac{1}{\partial_t} \Psi$ in the sense of nonlocal cosmology [50]. That means replacing the “advanced” integral, which is what really results from the variation, with the “retarded” integral which obeys the same differential equation,

$$ \int_t^\infty dt' F(t') \rightarrow - \int_0^t dt' F(t'). \quad (2.16) $$

The actual genesis of a nonlocal model would be through the Schwinger-Keldysh formalism [51–56], with (2.16) rigorously justified through the interference between contributions from forward and backwards propagating effective fields [57–59].

## 3 Invariant formulation

The purpose of this section is to construct an invariant effective action whose variation produces the full CDM stress tensor. We begin with a convenient representation for the general metric. Then it is shown how the full CDM stress tensor can be expressed in terms of two scalar fields which obey simple, first order differential equations. The section closes by giving an invariant Lagrangian whose variation reproduces the full CDM stress tensor.

### 3.1 The ADM metric

Because of the crucial role played by time evolution, it is convenient to carry out this analysis using the representation for the full metric introduced by Arnowitt, Deser and Misner [60],

$$ ds^2 = -N^2 dt^2 + \gamma_{ij} (dx^i - N^i dt) (dx^j - N^j dt). \quad (3.1) $$

The various components are known as the lapse $N(t, \vec{x})$, the shift $N^i(t, \vec{x})$ and the 3-metric $\gamma_{ij}(t, \vec{x})$. In these variables the measure and the inverse metric are,

$$ \sqrt{-g} = N \sqrt{\gamma}, \qquad g^{\mu\nu} = \begin{pmatrix} -\frac{1}{N^2} & -\frac{N^n}{N^2} \\ -\frac{N^m}{N^2} & \gamma^{mn} - \frac{N^m N^n}{N^2} \end{pmatrix}. \quad (3.2) $$

For the scalar perturbation geometry (2.1) we have,

$$ N^2 = 1 + 2\Psi + \frac{\partial_i B}{a} \frac{\partial_i B}{a}, \qquad N^i = -\frac{\partial_i B}{a^2}, \qquad \gamma_{ij} = a^2 \delta_{ij}. \quad (3.3) $$
## 3.2 Representing the full CDM stress tensor

Because cold dark matter has zero pressure and is collisionless, its stress tensor can be written in terms of an energy density $\rho(t, \vec{x})$ and a timelike 4-velocity field $u_\mu(t, \vec{x})$

$$ T_{\mu\nu} = \rho u_{\mu} u_{\nu}, \qquad g^{\mu\nu} u_{\mu} u_{\nu} = -1. \qquad (3.4) $$

Comparison with (2.4) and (2.8) implies an intriguing expansion for the 4-velocity,

$$ u_0 = -1 - \Psi + \dots = -\partial_t \left[ t + \frac{1}{\partial_t} \Psi + \dots \right], \qquad (3.5) $$

$$ u_i = 0 + \partial_i \theta + \dots = -\partial_i \left[ t + \frac{1}{\partial_t} \Psi + \dots \right]. \qquad (3.6) $$

This suggests that $u_\nu$ can be written as minus the gradient of a scalar whose expansion resides within the square brackets.

We can understand the expansions (3.5)–(3.6) using the fact that the CDM stress tensor is separately conserved,

$$ D_{\mu} T^{\mu}_{\nu} = D_{\mu} (u^{\mu} \rho) u_{\nu} + \rho u^{\mu} D_{\mu} u_{\nu} = 0. \qquad (3.7) $$

Conservation implies two distinct equations,

$$ \partial_{\mu} \left( \sqrt{-g} g^{\mu\nu} u_{\nu} \rho \right) = 0, \qquad (3.8) $$

$$ g^{\mu\rho} u_{\mu} D_{\rho} u_{\nu} = 0. \qquad (3.9) $$

Relation (3.9) can indeed be solved as $u_\nu = -\partial_\nu \phi$,<fnref n="1" /> where the scalar field $\phi(t, \vec{x})$ obeys the nonlinear, first order differential equation,

$$ g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi = -1. \qquad (3.10) $$

The correctness of this solution can be seen by substituting $u_\nu = -\partial_\nu \phi$ in relation (3.9) and then using (3.10),

$$ g^{\mu\rho} u_{\mu} D_{\rho} u_{\nu} = g^{\mu\rho} \partial_{\mu} \phi D_{\rho} \partial_{\nu} \phi = g^{\mu\rho} \partial_{\mu} \phi D_{\nu} \partial_{\rho} \phi = \frac{1}{2} D_{\nu} \left[ g^{\mu\rho} \partial_{\mu} \phi \partial_{\rho} \phi \right] = 0. \qquad (3.11) $$

Expressions (3.10) and (3.8) give well-posed evolution equations for $\phi(t, \vec{x})$ and $\rho(t, \vec{x})$, respectively. To see this for $\phi(t, \vec{x})$, substitute (3.1) into (3.10) and then solve for $\dot{\phi}$ with an irrelevant sign choice,

$$ \dot{\phi} = N \sqrt{1 + \gamma^{ij} \partial_i \phi \partial_j \phi} - N^i \partial_i \phi. \qquad (3.12) $$

Equation (3.12) obviously has a unique solution which will agree with our weak field expansions (3.5)–(3.6) if we choose the initial condition $\phi(0, \vec{x}) = 0$. Substituting (3.1) into (3.8) and then eliminating $\dot{\phi}$ using (3.12) gives a similarly well-posed evolution equation for $\rho(t, \vec{x})$,

$$ \partial_t \left[ \rho \sqrt{\gamma} \sqrt{1 + \gamma^{jk} \partial_j \phi \partial_k \phi} \right] = \partial_i \left[ N \rho \sqrt{\gamma} \gamma^{ij} \partial_j \phi - N^i \rho \sqrt{\gamma} \sqrt{1 + \gamma^{jk} \partial_j \phi \partial_k \phi} \right]. \qquad (3.13) $$

<footnote n="1">¹A normalized, timelike 4-velocity has three degrees of freedom, so our representation in terms of a single scalar represents the minimal solution.</footnote>
This also has a unique solution which will agree with (2.9) if we begin with,

$$ \rho(0, \vec{x}) = \frac{\rho_0}{\sqrt{\gamma(0, \vec{x})}}. \qquad (3.14) $$

Note finally that the method of characteristics gives a solution for $\rho(t, \vec{x})$ in terms of $\phi(t, \vec{x})$ and $g_{\mu\nu}(t, \vec{x})$.

## 3.3 Nonlocal effective action

We can reproduce the full CDM stress tensor (3.4), for a general metric, using a local effective Lagrangian with auxiliary fields $\rho(t, \vec{x})$ and $\phi(t, \vec{x})$,

$$ \mathcal{L}_1 = -\frac{1}{2} [\partial_{\mu}\phi\partial_{\nu}\phi g^{\mu\nu} + 1] \rho\sqrt{-g}. \qquad (3.15) $$

The variation with respect to $\phi$ gives the $\rho$ equation (3.8), while the variation with respect to $\rho$ implies the $\phi$ equation (3.10),

$$ \frac{\delta S_1}{\delta \phi} = \partial_{\mu} [\rho \sqrt{-g} g^{\mu\nu} \partial_{\nu} \phi] = 0, \qquad (3.16) $$

$$ \frac{\delta S_1}{\delta \rho} = -\frac{1}{2} [\partial_{\mu}\phi\partial_{\nu}\phi g^{\mu\nu} + 1] \sqrt{-g} = 0. \qquad (3.17) $$

The variation with respect to the metric reproduces the full CDM stress tensor (3.4) when equation (3.17) is used and we recall that the 4-velocity is $u_{\mu} = -\partial_{\mu}\phi$,

$$ T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S_1}{\delta g^{\mu\nu}} = \rho \partial_{\mu}\phi \partial_{\nu}\phi = \rho u_{\mu} u_{\nu}. \qquad (3.18) $$

Lagrangian (3.15), localized in terms of auxiliary scalars $\phi$ and $\rho$, is analogous to the weak field result (2.14) which was localized in terms of auxiliary scalars $\theta$ and $\delta$. We stress that the fields $\phi(t, \vec{x})$ and $\rho(t, \vec{x})$ in the CDM stress tensor (3.18) are not independent dynamical variables but rather nonlocal functionals of the metric which solve the well-posed evolution equations (3.12) and (3.13), with initial value data $\phi(0, \vec{x}) = 0$ and (3.14), respectively. It would be possible to derive a purely gravitational effective action, analogous to the weak field expression (2.15), as a surface term. However, it is really equation (3.18) which defines the model.

## 4 Conclusions

We have derived a nonlocal, metric-based effective action which exactly reproduces cold dark matter. For linearized scalar perturbations in cosmology (2.1) our result can be expressed either as a localized form (2.14), using auxiliary scalars $\theta$ and $\delta$, or else as a nonlocal, purely gravitational form (2.15). Either way, the effective CDM stress tensor takes the form (2.4), with (2.8)–(2.9). For a general metric (3.1) the localized form is (3.15), with auxiliary scalars $\phi$ and $\rho$. The nonlocal, purely gravitational form analogous to (2.15) would be a surface term. Either way, the effective CDM stress tensor takes the form (3.18), with the auxiliary scalars defined as nonlocal functionals for the metric which obey the well-posed evolution equations (3.12) and (3.13), starting from initial values,

$$ \phi(0, \vec{x}) = 0, \qquad \rho(0, \vec{x}) = \frac{\rho_0}{\sqrt{\gamma(0, \vec{x})}}. \qquad (4.1) $$
We stress that $\phi(t, \vec{x})$ and $\rho(t, \vec{x})$ are precisely defined, nonlocal functionals of the metric, without the independent initial value data which local fields would possess.

We stress that there can be no doubt about this model reproducing all the successes of CDM in cosmology. Those successes include the anisotropies of the cosmic microwave background, baryon acoustic oscillations, and linearized structure formation. All of this must come out right because this model was constructed by using the separate conservation of the CDM stress tensor to express it as a nonlocal functional of the metric. The only way this model can be falsified is by showing that CDM interacts with fields other than gravity, the evidence for which is weak [46–48]. This model should put to rest the frequent claims that no modified gravity theory can supplant dark matter. It also demonstrates that the answer to the question of Pardo and Spergel [41] about the price of abandoning dark matter: the price is nonlocality.

A key principle in constructing modified gravity models is that gravitational radiation in vacuum should move at the speed of light so as not to conflict with the simultaneous detection of optical and gravitational radiation from the binary neutron star merger GW170817 approximately 40 Mpc distant [61]. This is usually accomplished by making each term in the effective action involve at least two factors of the Ricci tensor. We did not need to do that here because our model exactly reproduces the result of the CDM stress tensor. So any change our model predicts in the kinematics of gravitational radiation is limited to one already predicted by cold dark matter.

One thing our model does not do is to agree with the nonlocal extension of MOND [42, 43] which was constructed to reproduce the baryonic Tully-Fisher relation [62], with sufficient weak lensing, in gravitationally bound structure. One can see this by simply counting derivatives and weak fields in the CDM stress tensor. At least in cosmological perturbation theory, our model’s change in the gravitational field equations is linear in the weak fields $\Psi(t, \vec{x})$ and $B(t, \vec{x})$, and every inverse differential operator is compensated by a positive power. In contrast, the model which works inside gravitationally bound structures is quadratic in the weak fields and involves three derivatives [42, 43]. It would be desirable to have a single formalism which connects both regimes.

It would also be desirable to derive these nonlocal models from fundamental theory. We believe they might arise from resumming the secular logarithms that are induced by loops of inflationary gravitons and which must eventually become nonperturbatively large during a prolonged period of inflation [49]. Such a resummation has been accomplished for the similar logarithms produced by nonlinear sigma models on de Sitter background [63]. The curvature-induced effective potentials which enable the resummation have been generalized from de Sitter to an arbitrary cosmological background which experiences primordial inflation [64]. And when the generalized effective potentials are used to evolve to late times, certain results do retain a strong memory of the inflationary epoch [65]. It remains to generalize the resummation technique from nonlinear sigma models to gravity.

Finally, one should keep in mind the possibility that MOND might be realized as a modification of inertia rather than as a modification of gravity [66]. In this regard it is worth noting that inflationary gravitons not only induce secular changes in the force of gravity [67], they also modify the kinematics (and hence the inertia) of massless fermions [68] and photons [69]. It would be interesting to explore changes to the inertia, and particularly the response to gravity, of massive particles [70].
Acknowledgments

We are grateful to C. Skordis for patiently explaining his model and the associated constraints
of CDM in cosmology. We are also grateful for discussions and correspondence with M.
Milgrom and S. Sarkar. This work was partially supported by NSF grant PHY-2207514 and
by the Institute for Fundamental Theory at the University of Florida.

Note added. As this paper was going to press we learned of “Mimetic Matter” models [71, 72] which have the same field equations (3.8) and (3.10) as the invariant model of section 3. The chief difference is that our auxiliary scalars ρ (t, ⃗x) and φ (t, ⃗x) are functionals of the metric, with fixed initial value data (4.1), whereas the mimetic fields have independent (and sometimes ghost) initial value data.

References

[1] PANDAX-II collaboration, *Dark matter results from 54-ton-day exposure of PandaX-II experiment*, Phys. Rev. Lett. **119** (2017) 181302 [arXiv:1708.06917] [INSPIRE].

[2] XENON collaboration, *Dark matter search results from a one ton-year exposure of XENON1T*, Phys. Rev. Lett. **121** (2018) 111302 [arXiv:1805.12562] [INSPIRE].

[3] PANDAX-4T collaboration, *Dark matter search results from the PandaX-4T commissioning run*, Phys. Rev. Lett. **127** (2021) 261802 [arXiv:2107.13438] [INSPIRE].

[4] ADMX collaboration, *Search for invisible axion dark matter in the 3.3-4.2 μeV mass range*, Phys. Rev. Lett. **127** (2021) 261803 [arXiv:2110.06096] [INSPIRE].

[5] LZ collaboration, *First dark matter search results from the LUX-ZEPLIN (LZ) experiment*, Phys. Rev. Lett. **131** (2023) 041002 [arXiv:2207.03764] [INSPIRE].

[6] ATLAS and CMS collaborations, *Dark matter searches at CMS and ATLAS*, in the proceedings of the 56th *Rencontres de Moriond on electroweak interactions and unified theories*, (2023) [arXiv:2301.10141] [INSPIRE].

[7] XENON collaboration, *First dark matter search with nuclear recoils from the XENONnT experiment*, Phys. Rev. Lett. **131** (2023) 041003 [arXiv:2303.14729] [INSPIRE].

[8] M. Milgrom, *A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis*, Astrophys. J. **270** (1983) 365 [INSPIRE].

[9] M. Milgrom, *A modification of the Newtonian dynamics: implications for galaxies*, Astrophys. J. **270** (1983) 371 [INSPIRE].

[10] M. Milgrom, *The MOND paradigm of modified dynamics*, *Scholarpedia* **9** (2014) 31410.

[11] M. Milgrom, *A modification of the Newtonian dynamics: implications for galaxy systems*, Astrophys. J. **270** (1983) 384 [INSPIRE].

[12] M. Milgrom, *On stability of galactic disks in the modified dynamics and the distribution of their mean surface brightness*, Astrophys. J. **338** (1989) 121 [INSPIRE].

[13] R. Brada and M. Milgrom, *Stability of disk galaxies in the modified dynamics*, Astrophys. J. **519** (1999) 590 [astro-ph/9811013] [INSPIRE].

[14] R. Brada and M. Milgrom, *The modified dynamics (MOND) predicts an absolute maximum to the acceleration produced by dark halos*, Astrophys. J. Lett. **512** (1999) L17 [astro-ph/9812117] [INSPIRE].
[15] M. Milgrom, *The central surface density of ‘dark halos’ predicted by MOND*, Mon. Not. Roy. Astron. Soc. **398** (2009) 1023 [arXiv:0909.5184] [INSPIRE].

[16] S. McGaugh and M. Milgrom, *Andromeda dwarfs in light of MOND*. II. Testing prior predictions, Astrophys. J. **775** (2013) 139 [arXiv:1308.5894] [INSPIRE].

[17] M. Milgrom, *MOND laws of galactic dynamics*, Mon. Not. Roy. Astron. Soc. **437** (2014) 2531 [arXiv:1212.2568] [INSPIRE].

[18] F. Lelli, S.S. McGaugh, J.M. Schombert and M.S. Pawlowski, *The relation between stellar and dynamical surface densities in the central regions of disk galaxies*, Astrophys. J. Lett. **827** (2016) L19 [arXiv:1607.02145] [INSPIRE].

[19] M. Milgrom, *Universal modified Newtonian dynamics relation between the baryonic and “dynamical” central surface densities of disc galaxies*, Phys. Rev. Lett. **117** (2016) 141101 [arXiv:1607.05103] [INSPIRE].

[20] S. McGaugh, F. Lelli and J. Schombert, *Radial acceleration relation in rotationally supported galaxies*, Phys. Rev. Lett. **117** (2016) 201101 [arXiv:1609.05917] [INSPIRE].

[21] F. Lelli, S.S. McGaugh, J.M. Schombert and M.S. Pawlowski, *One law to rule them all: the radial acceleration relation of galaxies*, Astrophys. J. **836** (2017) 152 [arXiv:1610.08981] [INSPIRE].

[22] S. McGaugh, *Predictions and outcomes for the dynamics of rotating galaxies*, Galaxies **8** (2020) 35 [arXiv:2004.14402] [INSPIRE].

[23] I. Banik and H. Zhao, *From galactic bars to the Hubble tension: weighing up the astrophysical evidence for Milgromian gravity*, Symmetry **14** (2022) 1331 [arXiv:2110.06936] [INSPIRE].

[24] A. Aguirre, J. Schaye and E. Quataert, *Problems for MOND in clusters and the Ly-α forest*, Astrophys. J. **561** (2001) 550 [astro-ph/0105184] [INSPIRE].

[25] M. Milgrom, *MOND in galaxy groups: a superior sample*, Phys. Rev. D **99** (2019) 044041 [arXiv:1811.12233] [INSPIRE].

[26] D. Clowe et al., *A direct empirical proof of the existence of dark matter*, Astrophys. J. Lett. **648** (2006) L109 [astro-ph/0608407] [INSPIRE].

[27] J.D. Bekenstein, *Relativistic gravitation theory for the MOND paradigm*, Phys. Rev. D **70** (2004) 083509 [Erratum ibid. **71** (2005) 069901] [astro-ph/0403694] [INSPIRE].

[28] C. Skordis, D.F. Mota, P.G. Ferreira and C. Boehm, *Large scale structure in Bekenstein’s theory of relativistic modified Newtonian dynamics*, Phys. Rev. Lett. **96** (2006) 011301 [astro-ph/0505519] [INSPIRE].

[29] G.W. Angus, B. Famaey and H.S. Zhao, *Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry*, Mon. Not. Roy. Astron. Soc. **371** (2006) 138 [astro-ph/0606216] [INSPIRE].

[30] S. Dodelson and M. Liguori, *Can cosmic structure form without dark matter?*, Phys. Rev. Lett. **97** (2006) 231301 [astro-ph/0608602] [INSPIRE].

[31] T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, *Modified gravity and cosmology*, Phys. Rept. **513** (2012) 1 [arXiv:1106.2476] [INSPIRE].

[32] C. Skordis and T. Zlosnik, *New relativistic theory for modified Newtonian dynamics*, Phys. Rev. Lett. **127** (2021) 161302 [arXiv:2007.00082] [INSPIRE].

[33] C. Skordis and T. Zlosnik, *Aether scalar tensor theory: linear stability on Minkowski space*, Phys. Rev. D **106** (2022) 104041 [arXiv:2109.13287] [INSPIRE].
[34] P. Verwayen, C. Skordis and C. Bœhm, *Aether Scalar Tensor (AeST) theory: quasistatic spherical solutions and their phenomenology*, arXiv:2304.05134 [INSPIRE].

[35] M. Bataki, C. Skordis and T. Zlosnik, *Aether scalar tensor theory: Hamiltonian formalism*, arXiv:2307.15126 [INSPIRE].

[36] A. Durakovic and C. Skordis, *Towards galaxy cluster models in Aether-Scalar-Tensor theory: isothermal spheres and curiosities*, arXiv:2312.00889 [INSPIRE].

[37] R.P. Woodard, *Avoiding dark energy with $1/r$ modifications of gravity*, Lect. Notes Phys. **720** (2007) 403 [astro-ph/0601672] [INSPIRE].

[38] J.A.R. Cembranos, *Dark matter from $R^2$-gravity*, Phys. Rev. Lett. **102** (2009) 141301 [arXiv:0809.1653] [INSPIRE].

[39] P. Horava, *Quantum gravity at a Lifshitz point*, Phys. Rev. D **79** (2009) 084008 [arXiv:0901.3775] [INSPIRE].

[40] A.O. Barvinsky et al., *Renormalization of Hořava gravity*, Phys. Rev. D **93** (2016) 064022 [arXiv:1512.02250] [INSPIRE].

[41] K. Pardo and D.N. Spergel, *What is the price of abandoning dark matter? Cosmological constraints on alternative gravity theories*, Phys. Rev. Lett. **125** (2020) 211101 [arXiv:2007.00555] [INSPIRE].

[42] C. Deffayet, G. Esposito-Farese and R.P. Woodard, *Nonlocal metric formulations of MOND with sufficient lensing*, Phys. Rev. D **84** (2011) 124054 [arXiv:1106.4984] [INSPIRE].

[43] C. Deffayet, G. Esposito-Farese and R.P. Woodard, *Field equations and cosmology for a class of nonlocal metric models of MOND*, Phys. Rev. D **90** (2014) 064038 [Addendum ibid. **90** (2014) 089901] [arXiv:1405.0393] [INSPIRE].

[44] M. Kim et al., *Determining cosmology for a nonlocal realization of MOND*, Phys. Rev. D **94** (2016) 104009 [arXiv:1608.07858] [INSPIRE].

[45] L. Tan and R.P. Woodard, *Structure formation in nonlocal MOND*, JCAP **05** (2018) 037 [arXiv:1804.01669] [INSPIRE].

[46] S. Tulin and H.-B. Yu, *Dark matter self-interactions and small scale structure*, Phys. Rept. **730** (2018) 1 [arXiv:1705.02358] [INSPIRE].

[47] S. Sarkar, *Is dark matter self-interacting?*, Nature Astron. **2** (2018) 856 [INSPIRE].

[48] S. Adhikari et al., *Astrophysical tests of dark matter self-interactions*, arXiv:2207.10638 [INSPIRE].

[49] R.P. Woodard, *The case for nonlocal modifications of gravity*, Universe **4** (2018) 88 [arXiv:1807.01791] [INSPIRE].

[50] S. Deser and R.P. Woodard, *Nonlocal cosmology*, Phys. Rev. Lett. **99** (2007) 111301 [arXiv:0706.2151] [INSPIRE].

[51] J.S. Schwinger, *Brownian motion of a quantum oscillator*, J. Math. Phys. **2** (1961) 407 [INSPIRE].

[52] K.T. Mahanthappa, *Multiple production of photons in quantum electrodynamics*, Phys. Rev. **126** (1962) 329 [INSPIRE].

[53] P.M. Bakshi and K.T. Mahanthappa, *Expectation value formalism in quantum field theory. 1*, J. Math. Phys. **4** (1963) 1 [INSPIRE].

[54] P.M. Bakshi and K.T. Mahanthappa, *Expectation value formalism in quantum field theory. 2*, J. Math. Phys. **4** (1963) 12 [INSPIRE].
[55] L.V. Keldysh, *Diagram technique for nonequilibrium processes*, Zh. Eksp. Teor. Fiz. **47** (1964) 1515 [INSPIRE].

[56] K.-C. Chou, Z.-B. Su, B.-L. Hao and L. Yu, *Equilibrium and nonequilibrium formalisms made unified*, Phys. Rept. **118** (1985) 1 [INSPIRE].

[57] R.D. Jordan, *Effective field equations for expectation values*, Phys. Rev. D **33** (1986) 444 [INSPIRE].

[58] E. Calzetta and B.L. Hu, *Closed time path functional formalism in curved space-time: application to cosmological back reaction problems*, Phys. Rev. D **35** (1987) 495 [INSPIRE].

[59] L.H. Ford and R.P. Woodard, *Stress tensor correlators in the Schwinger-Keldysh formalism*, Class. Quant. Grav. **22** (2005) 1637 [gr-qc/0411003] [INSPIRE].

[60] R.L. Arnowitt, S. Deser and C.W. Misner, *Dynamical structure and definition of energy in general relativity*, Phys. Rev. **116** (1959) 1322 [INSPIRE].

[61] S. Boran, S. Desai, E.O. Kahya and R.P. Woodard, *GW170817 falsifies dark matter emulators*, Phys. Rev. D **97** (2018) 041501 [arXiv:1710.06168] [INSPIRE].

[62] S.S. McGaugh, J.M. Schombert, G.D. Bothun and W.J.G. de Blok, *The baryonic Tully-Fisher relation*, Astrophys. J. Lett. **533** (2000) L99 [astro-ph/0003001] [INSPIRE].

[63] S.P. Miao, N.C. Tsamis and R.P. Woodard, *Summing inflationary logarithms in nonlinear sigma models*, JHEP **03** (2022) 069 [arXiv:2110.08715] [INSPIRE].

[64] E. Kasdagli, M. Ulloa and R.P. Woodard, *Coincident massless, minimally coupled scalar correlators on general cosmological backgrounds*, Phys. Rev. D **107** (2023) 105023 [arXiv:2302.04808] [INSPIRE].

[65] R.P. Woodard and B. Yesilyurt, *Remembrance of things past*, JHEP **08** (2023) 124 [arXiv:2305.17641] [INSPIRE].

[66] M. Milgrom, *MOND as manifestation of modified inertia*, arXiv:2310.14334 [INSPIRE].

[67] L. Tan, N.C. Tsamis and R.P. Woodard, *How inflationary gravitons affect the force of gravity*, Universe **8** (2022) 376 [arXiv:2206.11467] [INSPIRE].

[68] S.P. Miao and R.P. Woodard, *Gravitons enhance fermions during inflation*, Phys. Rev. D **74** (2006) 024021 [gr-qc/0603135] [INSPIRE].

[69] C.L. Wang and R.P. Woodard, *Excitation of photons by inflationary gravitons*, Phys. Rev. D **91** (2015) 124054 [arXiv:1408.1448] [INSPIRE].

[70] S.P. Miao, *Quantum gravitational effects on massive fermions during inflation I*, Phys. Rev. D **86** (2012) 104051 [arXiv:1207.5241] [INSPIRE].

[71] A.H. Chamseddine and V. Mukhanov, *Mimetic dark matter*, JHEP **11** (2013) 135 [arXiv:1308.5410] [INSPIRE].

[72] A.H. Chamseddine, V. Mukhanov and A. Vikman, *Cosmology with mimetic matter*, JCAP **06** (2014) 017 [arXiv:1403.3961] [INSPIRE].