Emergent Gravity and the Dark Universe

Emergent Gravity and the Dark Universe

Erik Verlinde, Delta-Institute for Theoretical Physics, Institute of Physics, University of Amsterdam.

SciPost Physics 2, 016 (2017).

Abstract

Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional "dark" gravitational force describing the "elastic" response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton's constant and the Hubble acceleration scale, and provide evidence for the fact that this additional "dark gravity force" explains the observed phenomena in galaxies and clusters currently attributed to dark matter.

1. Introduction and summary

According to Einstein's theory of general relativity spacetime has no intrinsic properties other than its curved geometry: it is merely a stage, albeit a dynamical one, on which matter moves under the influence of forces. There are well motivated reasons, coming from theory as well as observations, to challenge this conventional point of view. From the observational side, the fact that 95 percent of our Universe consists of mysterious forms of energy or matter gives sufficient motivation to reconsider this basic starting point. And from a theoretical perspective, insights from black hole physics and string theory indicate that our "macroscopic" notions of spacetime and gravity are emergent from an underlying microscopic description in which they have no a priori meaning.

1.1 Emergent spacetime and gravity from quantum information

The first indication of the emergent nature of spacetime and gravity comes from the laws of black hole thermodynamics. A central role herein is played by the Bekenstein-Hawking entropy and Hawking temperature: the entropy equals the horizon area divided by four times Newton's constant and the reduced Planck constant, and the temperature equals the reduced Planck constant times the surface acceleration divided by two pi (Equation 1). Here the area denotes the area of the horizon and the surface acceleration is the horizon's surface gravity.

In the past decades the theoretical understanding of the Bekenstein-Hawking formula has advanced significantly, starting with the explanation of its microscopic origin in string theory and the subsequent development of the AdS/CFT correspondence. In the latter context it was realized that this same formula also determines the amount of quantum entanglement in the vacuum. It was subsequently argued that quantum entanglement plays a central role in explaining the connectivity of the classical spacetime. These important insights formed the starting point of the recent theoretical advances that have revealed a deep connection between key concepts of quantum information theory and the emergence of spacetime and gravity.

Currently the first steps are being taken towards a new theoretical framework in which spacetime geometry is viewed as representing the entanglement structure of the microscopic quantum state. Gravity emerges from this quantum information theoretic viewpoint as describing the change in entanglement caused by matter. These novel ideas are best understood in Anti-de Sitter space, where the description in terms of a dual conformal field theory allows one to compute the microscopic entanglement in a well defined setting. In this way it was proven that the entanglement entropy indeed obeys the Bekenstein-Hawking formula, when the vacuum state is divided into two parts separated by a Killing horizon. This fact was afterwards used to extend earlier work on the emergence of gravity by deriving the linearized Einstein equations from general quantum information theoretic principles.

The fact that the entanglement entropy of the spacetime vacuum obeys an area law has motivated various proposals that represent spacetime as a network of entangled units of quantum information, called "tensors". The first proposal of this kind is the multi-scale entanglement renormalization approach, in which the boundary quantum state is constructed, or deconstructed, by a multi-scale entanglement renormalization procedure. More recently it was proposed that the bulk spacetime operates as a holographic error correcting code. In this approach the tensor network representing the emergent spacetime produces a unitary bulk to boundary map defined by entanglement.

The language of quantum error correcting codes and tensor networks gives useful insights into the entanglement structure of spacetime. In particular, it suggests that the microscopic constituents from which spacetime emerges should be thought of as basic units of quantum information whose short range entanglement gives rise to the Bekenstein-Hawking area law and provides the microscopic "bonds" or "glue" responsible for the connectivity of spacetime.

1.2 Emergent gravity in de Sitter space

The conceptual ideas behind the emergence of spacetime and gravity appear to be general and are in principle applicable to other geometries than Anti-de Sitter space. Our goal is to identify these general principles and apply them to a universe closer to our own, namely de Sitter space. Here we have less theoretical control, since at present there is no completely satisfactory microscopic description of spacetimes with a positive cosmological constant. Our strategy will be to apply the same general logic as in AdS, but to make appropriate adjustments to take into account the differences that occur in dS spacetimes. The most important aspect we have to deal with is that de Sitter space has a cosmological horizon. Hence, it carries a finite entropy and temperature, where the surface acceleration is given in terms of the Hubble parameter and the Hubble scale by the speed of light times the Hubble parameter, equivalently the speed of light squared divided by the Hubble scale — the acceleration scale that will play a particularly important role in this paper (Equation 2).

The fact that de Sitter space has no boundary at spatial infinity casts doubt on the possible existence of a holographic description. One may try to overcome this difficulty by viewing dS as an analytic continuation of AdS and use a temporal version of the holographic correspondence. We will not adopt such a holographic approach, since we interpret the presence of the cosmological horizon and the absence of spatial (or null) infinity as signs that the entanglement structure of de Sitter space differs in an essential way from that of AdS (or flat space). The horizon entropy and temperature indicate that microscopically de Sitter space corresponds to a thermal state in which part of the microscopic degrees of freedom are being "thermalized".

An important lesson that has come out of the complementarity, firewall and ER equals EPR discussions is that the quantum information counted by the Bekenstein-Hawking entropy is not localized on the horizon itself, but either represents the entanglement entropy across the horizon (two-sided case) or is interpreted as a thermodynamic entropy (one-sided case) which is stored non-locally. In the latter situation, the quantum states associated with the horizon entropy are maximally entangled with bulk excitations carrying a typical energy set by the temperature. In this paper we will argue that this one-sided perspective also applies to de Sitter space. Furthermore, we propose that the thermal excitations responsible for the de Sitter entropy constitute the positive dark energy. In this physical picture the positive dark energy and accelerated expansion are caused by the slow thermalization of the emergent spacetime.

We propose that microscopically de Sitter space corresponds to an ensemble of metastable quantum states that together carry the Bekenstein-Hawking entropy associated with the cosmological horizon. The metastability has purely an entropic origin: the high degeneracy together with the ultra-slow dynamics prevent the microscopic system from relaxing to the true ground state. At long timescales the microscopic de Sitter states satisfy the eigenstate thermalization hypothesis, which implies that they contain a thermal volume law contribution to the entanglement entropy.

To derive the Einstein equations one requires a strict area law for the entanglement entropy. In condensed matter systems a strict area law arises almost exclusively in ground states of gapped systems with strong short range correlations. A small but non-zero volume law entropy, for instance due to thermalization, would compete with and at large distances overwhelm the area law. We propose that precisely this phenomenon occurs in de Sitter space and is responsible for the presence of a cosmological horizon. Our aim is to study the emergent laws of gravity in de Sitter space while taking into account its thermal volume law entropy.

1.3 Hints from observations: the missing mass problem

In this paper we provide evidence for the fact that the observed dark energy and the phenomena currently attributed to dark matter have a common origin and are connected to the emergent nature of spacetime and gravity. The observed flattening of rotation curves, as well as many other observations of dark matter phenomena, indicate that they are controlled by the Hubble acceleration scale, as first pointed out by Milgrom. It is an empirical fact that the "missing mass problem", usually interpreted as observational evidence for dark matter, only occurs when the gravitational acceleration falls below a certain critical value that is of the order of that acceleration scale. This criterion can be alternatively formulated in terms of the surface mass density.

Consider a spherical region with boundary area four pi times the radius squared, containing matter with total mass near its center. We define the surface mass density as the ratio of the mass and the area. Empirically the directly observed gravitational phenomena attributed to dark matter, such as the flattening of rotation curves in spiral galaxies and the evidence from weak lensing data, occur when the surface mass density falls below a universal value determined by the acceleration scale: the surface mass density stays under the acceleration scale divided by eight pi times Newton's constant (Equation 3).

The appearance of the cosmological acceleration scale in galactic dynamics is striking and gives a strong hint towards an explanation in terms of emergent gravity. To make this point more clear let us rewrite the above inequality as a statement about entropies (Equation 4): the quantity two pi times the mass divided by the reduced Planck constant and the acceleration scale stays under the area divided by four times Newton's constant and the reduced Planck constant. The quantity on the left hand side represents the change in the de Sitter entropy caused by adding the mass, while the right hand side is the entropy of a black hole that would fit inside the region bounded by the area.

Our goal is to give a theoretical explanation for why the emergent laws of gravity differ from those of general relativity precisely when this inequality is obeyed. We will find that this criterion is directly related to the presence of the volume law contribution to the entanglement entropy. At scales much smaller than the Hubble radius gravity is in most situations well described by general relativity, because the entanglement entropy is still dominated by the area law of the vacuum. But at large distances and long time scales the enormous de Sitter entropy in combination with the extremely slow thermal dynamics lead to modifications to these familiar laws. We will determine these modifications and show that precisely when the surface mass density falls below the critical value, the reaction force due to the thermal contribution takes over from the "normal" gravity force caused by the area law.

1.4 Outline: from emergent gravity to apparent dark matter

The central idea of this paper is that the volume law contribution to the entanglement entropy, associated with the positive dark energy, turns the otherwise "stiff" geometry of spacetime into an elastic medium. We find that the elastic response of this "dark energy" medium takes the form of an extra "dark" gravitational force that appears to be due to "dark matter". For spherical situations and under the right circumstances it is shown that the surface mass densities of the baryonic and apparent dark matter obey a scaling relation in a given number of spacetime dimensions (Equation 5): the squared apparent dark matter surface density equals the acceleration scale divided by eight pi times Newton's constant, times the baryonic surface density, divided by the number of spacetime dimensions minus one.

The first form of this relation connects to the entropic criterion above and makes the thermal and entropic origin manifest. The second relation can alternatively be written in terms of the gravitational accelerations due to the apparent dark matter and to the baryons, which are related to the surface densities by dimension-dependent factors (Equation 6). Hence these accelerations obey the scaling relation that the apparent dark acceleration equals the square root of the baryonic acceleration times a characteristic acceleration, itself the Hubble acceleration scale times a dimension-dependent factor (Equation 7).

In four dimensions these equations are equivalent to the baryonic Tully-Fisher relation that relates the velocity of the flattening galaxy rotation curves and the baryonic mass. In this case one finds that the characteristic acceleration is the Hubble acceleration scale divided by six, which is indeed the acceleration scale that appears in Milgrom's phenomenological fitting formula. We like to emphasise that these scaling relations are not new laws of gravity or inertia, but appear as estimates of the strength of the extra dark gravitational force. From our derivation it will become clear in which circumstances these relations hold, and when they are expected to fail. This point will be further clarified in the concluding section.

This paper is organized as follows. In section 2 we present our main hypothesis regarding the entropy content of de Sitter space. In section 3 we discuss several conceptual issues related to the glassy dynamics and memory effects that occur in emergent de Sitter gravity. We determine the effect of matter on the entropy content in section 4, and explain the origin of the entropic criterion. We also give a heuristic derivation of the scaling relation. In section 5 we relate the definition of mass in de Sitter to the reduction of the total entropy using the Wald formalism. This serves as a preparation for section 6, where we give a detailed correspondence between the gravitational and elastic equations. Section 7 contains the main result of our paper: here we derive the apparent dark matter density in terms of the baryonic mass distribution and compare our findings with known observational facts. Finally, the discussion and conclusions are presented in section 8.

(Sections 2 to 6 — the entropy content of de Sitter space, the glassy dynamics and memory effects of emergent gravity, the effect of matter on the entropy and dark energy, the first law of horizons and the definition of mass, and the elastic phase of emergent gravity — are omitted for length; the complete text is at the source.)

7. Apparent dark matter from emergent gravity

7.1 From an elastic memory effect to apparent dark matter (closing argument)

We are finally in a position to combine all ingredients and obtain the main result of our analysis. First we express the largest principal strain in terms of the apparent dark matter surface density. Next we assume that the conditions for the equality sign hold, and that the boundary integral of the displacement field can be evaluated by replacing the displacement field by the corresponding expression in terms of Newton's potential of the ordinary "baryonic" matter — a natural assumption in the case that the boundary coincides with an equipotential surface of that potential. Combined with the elastic energy relation this leads to an integral relation for the apparent dark matter surface density in terms of the Newtonian potential for the baryonic matter (Equation 95): the volume integral of the squared surface density, divided by the squared acceleration scale, equals eight pi times Newton's constant, times a dimensional factor, times the boundary integral of the baryonic potential divided by the acceleration scale.

Since the integration region can be chosen arbitrarily, we can also derive a local relation by first converting the right hand side into a volume integral by applying Stokes' theorem and then equating the integrands (Equation 96). In the next subsection we will use this relation for a spherically symmetric situation to derive the mass density for the apparent dark matter from a given distribution of baryonic matter.

7.2 A formula for apparent dark matter density in galaxies and clusters

To be able to compare our results with observations it will be useful to re-express our results directly as a relation between the densities of the baryonic matter and the apparent dark matter. It is not a straightforward task to do this for a general mass distribution, so we will specialize to the case that the baryonic matter is spherically symmetric. We will also put the number of spacetime dimensions equal to four.

Let us begin by reminding ourselves of the relation between the surface mass density and Newton's potential. For a spherically symmetric situation this relation can be written as the surface density equalling minus the potential divided by four pi times Newton's constant and the radius, which equals the enclosed mass divided by the area (Equation 97), where the enclosed mass is the radial integral of the density times the area (Equation 98).

With the help of these equations it is straightforward to re-express the integral relation in terms of the apparent dark matter mass and the baryonic mass. This leads to the statement that the radial integral of Newton's constant times the squared apparent dark matter mass divided by the squared radius equals the baryonic mass times the acceleration scale times the radius, divided by six (Equation 99).

This is the main formula and central result of our paper, since it allows one to make a direct comparison with observations. It describes the amount of apparent dark matter in terms of the amount of baryonic matter for approximately spherically symmetric and isolated astronomical systems in non-dynamical situations. After having determined the apparent dark matter mass one can then compute the total acceleration as the sum of the baryonic and apparent dark accelerations (Equation 100), where both are given by their usual Newtonian expressions in terms of the enclosed masses (Equation 101).

We will now discuss the consequences of this equation and present it in different forms so that the comparison with observations becomes more straightforward. First we note that the same relation can also be obtained from the simple heuristic derivation presented in section 4.4. By inserting all relevant definitions for the strain and the volume of the inclusion, one precisely recovers our main formula.

In the special case that the baryonic mass is entirely centered in the origin it is easy to derive the well-known form of the baryonic Tully-Fisher relation. In this case one can simply differentiate with respect to the radius while keeping the baryonic mass constant. It is easily verified that this leads to the relation that the apparent dark acceleration equals the square root of the characteristic acceleration times the baryonic acceleration, with the characteristic acceleration equal to the Hubble acceleration scale divided by six (Equation 102). That parameter is the famous acceleration scale introduced by Milgrom in his phenomenological fitting formula for galaxy rotation curves. We have thus given an explanation for the phenomenological success of Milgrom's fitting formula, in particular in reproducing the flattening of rotation curves. An alternative way to express this is as a result for the asymptotic velocity of the flattened galaxy rotation curve: the fourth power of that velocity equals the characteristic acceleration times Newton's constant times the baryonic mass (Equation 103). This is known as the baryonic Tully-Fisher relation and has been well tested by observations of a very large number of spiral galaxies.

We like to emphasize that we have not derived the theory of modified Newtonian dynamics as proposed by Milgrom. In our description there is no modification of the law of inertia, nor is our result to be interpreted as a modified gravitational field equation. It is derived from an estimate of an effect induced by the displacement of the free energy of the underlying microscopic state of de Sitter space due to matter. This elastic response is then reformulated as an estimate of the gravitational self-energy due to the apparent dark matter in the form of the integral relation. Hence, although we derived the same relation as modified Newtonian dynamics, the physics is very different. For this reason we referred to the relation as a fitting formula, since it is important to make a clear separation between an empirical relation and a proposed law of nature. There is little dispute about the observed scaling relation, but the disagreement in the scientific community has mainly been about whether it represents a new law of physics. In our description it does not.

The validity of the central formula depends on a number of assumptions and holds only when certain conditions are being satisfied. These conditions include that one is dealing with a centralized, spherically symmetric mass distribution, which has been in dynamical equilibrium during its evolution. Dynamical situations as those that occur in the Bullet cluster are not described by these same equations. The system should also be sufficiently isolated so that it does not experience significant effects of nearby mass distributions. Finally, in the previous subsection we actually derived an inequality, which means that to get to the central equation we have made an assumption about the largest principal strain. While this assumption is presumably true in quite general circumstances, in particular sufficiently near the main mass distribution where the apparent dark matter first becomes noticeable, as one gets further out, or when other mass distributions come into play, we are left only with an inequality.

It is known that the Milgrom relation fails to explain the observed gravitational acceleration in clusters, since it underestimates the amount of apparent dark matter. To get the right amount one would need to multiply the characteristic acceleration by about a factor of 3. That relation also cannot account for the observed strong gravitational lensing due to dark matter in the central parts of the cluster, since the projected surface mass densities required for strong lensing are larger than the expected value by about a factor of 6. For these reasons proponents of modified Newtonian dynamics still have to assume a form of particle dark matter at the cluster scale.

These discrepancies can be significantly reduced and perhaps completely explained away in our theoretical description. To go from the central formula to the Milgrom relation we assumed that the matter is entirely located in the origin, since in taking the derivative with respect to radius we kept the baryonic mass constant. In most galaxies this is indeed a good approximation, but this assumption is not justified in clusters. Most of the baryonic mass in clusters is contained in X-ray emitting gas, which extends all the way to the outer parts of the cluster. In fact, even for galaxies a more precise treatment requires the use of the mass density profile instead of a point mass approximation.

So let us go back to the central formula and take its derivative while taking into account the radial dependence of the baryonic mass. We introduce the averaged mass densities inside a sphere of radius by writing the integrated masses as four pi thirds of the cubed radius times those averaged densities (Equation 104). We also introduce the slope parameters as minus the logarithmic derivative of each averaged density with respect to the logarithm of radius (Equation 105). When these slope parameters are approximately constant they give us the power law behavior of the averaged mass densities. By differentiating the integral relation with respect to radius and rewriting the result, one finds that the squared average apparent dark matter density equals four minus the baryonic slope parameter, times the acceleration scale times the averaged baryonic density, divided by eight pi times Newton's constant and the radius (Equation 106).

For a central point mass the baryonic slope parameter is equal to 3, hence the prefactor would be equal to one. The apparent dark matter has in that case a distribution with a slope of 2, which means that it falls off like one over the radius squared. A similar formula holds in modified Newtonian dynamics, except without the prefactor.

In the central parts of a cluster the slope parameter of the mass distribution is generally observed to be smaller than 1 or even close to 0, while in the outer parts the slope can still be significantly smaller than 3. We thus find that in our description we gain a factor in between 1.5 and 3.5 depending on the region of the cluster compared to modified Newtonian dynamics. This means that the "missing mass problem" in clusters is significantly reduced and, given the uncertainty about the amount of baryons, possibly entirely removed. In fact, at this point it is good to mention that also other matter particles, whatever they are, would in our description have to be counted in the baryonic mass density. This means that they would also lead to an increase in the apparent dark matter component.

Given the averaged apparent dark matter density one can find the actual mass density for the apparent dark matter via a relation involving its own slope parameter (Equation 107).

We now like to illustrate that these equations can, in contrast to modified Newtonian dynamics, lead to strong lensing phenomena in the cores of clusters in cases where there is a significant dark matter contribution. For this purpose let us consider an idealized situation in which the dark matter and baryonic matter in the core region inside a core radius have exactly the same density profile with both slope parameters equal to 1. This corresponds to the case where both surface mass densities are equal to the maximal value, the acceleration scale divided by eight pi times Newton's constant, corresponding to unit strain. The total mass density profile inside the core is then the acceleration scale divided by four pi times Newton's constant and the radius (Equation 108). One then finds that the projected surface mass density of the entire core region, as astrophysicists would define it by integrating along the line of sight, is equal to the speed of light times the Hubble parameter divided by pi times Newton's constant, which should be sufficient to cause strong gravitational lensing, especially in the inner parts of the core region. This strong lensing effect would in this case be equally due to baryonic and dark matter.

As a final fun comment let us, just out of curiosity, take the density formula and apply it to the entire universe. By this we mean the following: we assume a constant baryonic mass density, so we set the baryonic slope parameter to zero, and in addition we take the radius to be equal to the Hubble radius. Now we note that the critical mass density of the universe equals three times the squared Hubble parameter divided by eight pi times Newton's constant, equivalently three times the acceleration scale divided by eight pi times Newton's constant and the Hubble scale (Equation 109). Hence, when we put the radius equal to the Hubble scale in the density formula we obtain a relation between the standard cosmological density parameters of the baryonic and dark matter. We find that the squared dark matter density parameter equals four thirds of the baryonic density parameter (Equation 110).

This relation holds remarkably well for the values obtained by the WMAP and Planck collaborations. We ask the reader not to read too much into this striking and somewhat surprising fact, because it is far from clear that our derivation of the density formula would be applicable to the entire universe. For instance, an immediate question that comes to mind is whether this relation continues to hold throughout the cosmological evolution of the universe. We have worked exclusively in a static situation near the center of the static patch of a dark energy dominated universe. Any questions regarding the cosmological evolution of the universe are beyond the scope of this paper, and will hopefully be addressed in future work. This point will be reiterated in our conclusion.

8. Discussion and outlook

8.1 Particle dark matter versus emergent gravity

The observational evidence for the presence of dark matter appears to be overwhelming. The first known indications came from the observed velocity profiles in galaxies and clusters of galaxies. Other strong evidence comes from strong and weak gravitational lensing data, which show signs of what appears to be additional clumpy matter in clusters and around galaxies and groups of galaxies. Dark matter also plays a crucial role in the explanation of the spectrum of fluctuations in the cosmic microwave background and the theory of structure formation.

Since up to now there appeared to be no evidence that general relativity or Newtonian gravity could be wrong at the scales in question, the most generally accepted point of view is that these observations indicate that our universe contains an enormous amount of a yet unknown form of dark matter particle. However, the discrepancy between the observed gravitational force and the one caused by the visible baryonic matter is so enormous that it is hard to claim that these observations provide evidence for the validity of general relativity or Newtonian gravity in these situations. Purely based on the observations it is more appropriate to say that these familiar gravitational theories can only be saved by assuming the presence of dark matter. Therefore, without further knowledge, the evidence in favour of dark matter is just as much evidence for the possible breakdown of the currently known laws of gravity.

The real reason why most physicists believe in the existence of particle dark matter is not the observations, but because there was no theoretical evidence nor a conceptual argument for the breakdown of these laws at the scales where the new phenomena are being observed. It has been the aim of this paper to provide a theoretical and conceptual basis for the claim that this situation changes when one regards gravity as an emergent phenomenon. We have shown that the emergent laws of gravity, when one takes into account the volume law contribution to the entropy, start to deviate from the familiar gravitational laws precisely in those situations where the observations tell us they do. We have only made use of the natural constants of nature, and provided reasonably straightforward arguments and calculations to derive the scales and the behavior of the observed phenomena. Especially the natural appearance of the acceleration scale should in our view be seen as a particularly convincing aspect of our approach.

In our view this undercuts the common assumption that the laws of gravity should stay as they are, and hence it removes the rationale of the dark matter hypothesis. Once there is a conceptual reason for a new phase of the gravitational force, which is governed by different laws, and this is combined with a confirmation of its quantitative behavior, the weight of the evidence tips in the other direction. Admittedly, the observed scaling relations have played a role in developing the theoretical description, and motivated our hypothesis that the entropy of de Sitter space is distributed over the bulk of spacetime. But the theoretical arguments that support this hypothesis together with the successful derivation of the observed scaling relations are in our view sufficient proof of hypothesis. Our main conclusion therefore is:

The observed phenomena that are currently attributed to dark matter are the consequence of the emergent nature of gravity and are caused by an elastic response due to the volume law contribution to the entanglement entropy in our universe.

In order to explain the observed phenomena we did not postulate the existence of a dark matter particle, nor did we modify the gravitational laws in an ad hoc way. Instead we have tried to understand their origin and their mutual relation by taking seriously the theoretical indications coming from string theory and black hole physics that spacetime and gravity are emergent. We believe this approach and the results we obtained tell us that the phenomena associated with dark matter are an unavoidable and logical consequence of the emergent nature of spacetime itself. The net effect should be that in our conventional framework one has to add a dark component to the stress energy tensor, which behaves very much like the cold dark matter needed to explain structure formation, but which in its true origin is an intrinsic property of spacetime rather than being caused by some unknown particle. Indeed, we have argued that the observed dark matter phenomena are a remnant, a memory effect, of the emergence of spacetime together with the ordinary matter in it.

In particular, we have made clear why the apparent dark matter behaves exactly in the right way to explain the phenomenological success of modified Newtonian dynamics, as well as its failures, without the introduction of any freely adjustable parameters. We have found that in many, but not all, aspects the apparent dark matter behaves similar to what one would expect from particle dark matter. In particular, the excess gravity and the gravitational potential wells that play a role in these scenarios also appear in our description.

Perhaps superficially our approach is similar in spirit to some earlier works on the relationship between dark matter and the thermodynamics of spacetime. But the details of our derivations and especially the conceptual argumentation differ significantly from these papers. Our theoretical framework incorporates and has been motivated by the recent developments on emergent gravity from quantum information, and is in our view a logical extension of this promising research direction.

8.2 Emergent gravity and apparent dark matter in cosmological scenarios

In this paper we have focussed on the explanation of the observed gravitational phenomena attributed to dark matter. By this we mean the excess in the gravitational force or the missing mass that is observed in spiral or elliptical galaxies and in galaxy clusters. Of course, dark matter plays a central role in many other aspects of the current cosmological paradigm, in particular in structure formation and the explanation of the acoustic peaks in the cosmic microwave background. In none of these scenarios is it required that dark matter is a particle: all that is needed is that its cosmological evolution and dynamics is consistent with a pressureless fluid. In our description we eventually end up with an estimate of the apparent dark matter density that in many respects behaves as required for structure formation and perhaps even for the explanation of the CMB spectrum. Namely, effectively the apparent dark matter that comes out of our emergent gravity description also leads to a gravitational potential that attracts the baryonic matter as cold dark matter would do.

However, the arguments and calculations that we presented in this paper are not yet sufficient to answer the questions regarding the cosmological evolution of our equations. In particular, we made use of the value of the present-day Hubble parameter in our equations, which immediately raises the question whether one should use another value for the Hubble parameter at other cosmological times. In our calculations the Hubble parameter was assumed to be constant, since we made the approximation that our universe is entirely dominated by dark energy and that ordinary matter only leads to a small perturbation. This suggests that the Hubble parameter, or rather the acceleration scale, should actually be defined in terms of the dark energy density, or the value of the cosmological constant. This would imply that the acceleration scale is indeed constant, even though it takes a slightly different value.

A related issue is that in our analysis we assumed that dark energy is the dominant contribution to the energy density of our universe. According to our standard cosmological scenarios this is no longer true in the early times of our universe, in particular at the time of decoupling. This poses again the question whether a theory in which apparent dark matter is explained via emergent gravity would be able to reproduce the successful description of the CMB spectrum, the large scale structure and galaxy formation. These questions need to be understood before we can make any claim that our description of dark matter phenomena is as successful as the Lambda-CDM paradigm in describing the early universe and cosmology at large scales.

By changing the way we view gravity, namely as an emergent phenomenon in which the Einstein equations need be derived from the thermodynamics of quantum entanglement, one also has to change the way we view the evolution of the universe. In particular, one should be able to derive the cosmological evolution equations from emergent gravity. For this one needs to first properly understand the role of quantum entanglement and the evolution of the total entropy of our universe. So it is still an open question if and how the standard cosmological picture is incorporated in a theory of emergent gravity. How does one interpret the expansion of the universe from this perspective? Or does inflation still play a role in an emergent cosmological scenario?

All these questions are beyond the scope of the present paper. So we will not make an attempt to answer all or even a part of these questions. This also means that before these questions are investigated it is too early to make a judgement on whether our emergent gravity description of dark matter will also be able to replace the current particle dark matter paradigm in early cosmological scenarios.

Acknowledgements

This work has been performed during the past 6 years and I have benefitted from discussions and received encouragement from many colleagues. I like to begin by thanking Herman Verlinde for sharing his insights, encouragement and for collaboration on projects that are closely related to the ideas presented in this paper and which have influenced my thinking. I have benefitted from discussions at different stages and on different aspects of this work, with Jan de Boer, Kyriakos Papadodimas, Bartek Czech, David Berenstein, Irfan Ilgin, Ted Jacobson, Justin Khoury, Tom Banks, Gerard 't Hooft, Lenny Susskind and Juan Maldacena. A special thanks goes to Sander Mooij for stimulating discussions and enthusiastic support, and to Manus Visser for discussions, verifying the calculations and his invaluable help in correcting this manuscript. I am also grateful for encouragement from Stanley Deser, Robbert Dijkgraaf, Eliezer Rabinovici, Neil Turok, Paul Steinhardt, Coby Sonnenschein, John Preskill, Steve Shenker, and especially David Gross, whose critical and sharp questions have helped me to stay focussed on the main issues.

I thank the participants of the Bits and Branes program at KITP and the summer workshop at the Aspen center in 2014 for many discussions (among others with Joe Polchinski and Don Marolf on the Firewall Paradox) that helped sharpen my ideas for this work as well. Also I am grateful for the hospitality of the Caltech theory group in the past years.

I like to thank various astronomers and cosmologists whose knowledge benefitted this work and whose support I have appreciated greatly. These include Moti Milgrom, Pavel Kroupa, Hongsheng Zhao and Bob Sanders. Finally, I regret that Jacob Bekenstein is no longer around to be able to read this finished work, since his ideas and interests are playing a central role.

This research has been made possible by the EMERGRAV advanced grant from the European Research Council (ERC), the Spinoza Grant of the Dutch Science Organisation (NWO), and the NWO Gravitation Program for the Delta Institute for Theoretical Physics.

(Sections 2 to 6 and the numbered reference list are omitted for length; the complete text, with its equations set as mathematics, is at the source.)