Using the hypothesis that the universe is filled with some form of cold dark matter, it is possible to simultaneously fit observations of the CMB [229] and provide an elegant picture for the growth of large scale structure [444]. Thus, an obvious question is how MOND fares with these subjects. Of course, as we have seen, there is no unique existing MOND theory (Section 7), and the basic theory underlying MOND as a paradigm is probably yet to be found. Nevertheless, we can make a few general considerations about how any MOND theory should behave, and then look in more details at specific predictions from existing relativistic theories. The general picture is that, in some ways MOND does surprisingly well, in others it clearly gives no real unique prediction by now, and in still others it appears to fail outright.

If one alters the force law as envisioned by MOND, the effective long-range force becomes stronger. Though details will, of course, depend on the specific relativistic theory, we can speculate about the consequences of a MOND-like force in cosmology. Note, however, that most of what follows cannot be rigorously justified at the moment for lack of a compelling unique underlying theory. But, obviously, because of the stronger force, dynamical measures of the cosmic mass density will be overestimated, just as in galaxies. Applying MOND to the peculiar motions of galaxies yields [279]. There are large uncertainties in estimating the extragalactic peculiar acceleration field, so this merely shows that MOND might alleviate the need for non-baryonic dark matter inferred conventionally from .

The stronger effective gravitational attraction of MOND would change the growth rate of perturbations. Instead of adding dark mass to speed the growth of structure, we now rely on the modified force law to do the work. While it is obvious that MOND will form structures more rapidly than conventional gravity with the same source perturbation, we immediately encounter a challenge posed by the non-linear nature of the theory, precluding an easy linear perturbation analysis. One can nevertheless sketch a naive overview of how structure might form under the influence of MOND. The following picture emerges from numerical calculations of particles interacting under MOND in an assumed background [386, 341, 226, 250], and is thus obviously slightly (or very) different from the various relativistic MOND theories of Section 7 and from those yet to be found, especially from those MONDian theories involving the existence of some form of dark matter (twin matter, dipolar dark matter, etc.). In the early universe, perturbations cannot grow because the baryons are coupled to the photon fluid. The mass density is lower, so matter domination occurs later than in CDM. Consequently, MOND structure formation initially has to lag behind CDM at very high redshift (). However, as the influence of the photon field declines and perturbations begin to enter the MOND regime, structure formation rapidly speeds up. Large galaxies may form by and clusters by [11, 386], considerably earlier than in CDM. By , the voids have become more empty than in CDM, but otherwise simulations (of collisionless particles, which is, of course, not the best representation of the baryon fluid) show the same qualitative features of the cosmic web [226, 250]. This similarity is not surprising since MOND is a subtle alteration of the force law. The chief difference is in the timing of when structures of a given mass appear, it being easier to assemble a large mass early in MOND. This means that MOND is promising in addressing many of the challenges of Section 4.2, namely the high-z clusters challenge [11] and Local Void challenge, as well as the bulk flow challenge and high collisional velocity of the bullet cluster [16, 251], again due to the much-larger-than-Newtonian MOND force in the structure formation context. What is more, it could allow large massive galaxies to form early () from monolithic dissipationless collapse [393], with well-defined relationships between the mass, radius and velocity dispersion. Consequently, there would be less mergers than in CDM at intermediate redshifts, in accordance with constraints from interacting galaxies (see Section 6.5.3), which could explain the observed abundance of large thin bulgeless disks unaffected by major mergers (see Section 4.2), and in those rare mergers between large spirals, tidal dwarf galaxies would be formed and survive more easily (see Section 6.5.4). This could lead to the intriguing possibility that most dwarf galaxies are not primordial but have been formed tidally in these encounters [239]. These populations of satellite galaxies, associated with globular clusters that formed along with them, would naturally appear in (more than one) closely related planes (because a gas-rich galaxy pair undergoes many close encounters in MOND before merging, see Section 6.5.3), thereby perhaps providing a natural solution to the Milky Way satellites phase-space correlation problem of Section 4.2. What is more, the density-morphology relation for dwarf ellipticals (more dE galaxies in denser environments [239]), observed in the field, in galaxy groups and in galaxy clusters could also find a natural explanation.

Actually, the chief problem seems not to be forming structure in MOND, but the danger of over-producing it [341, 401]. The amplitude of the power spectrum is well measured at in the CMB and at by surveys like the Sloan Digital Sky Survey. Simulations normalized to the CMB overproduce the structure at by a factor of 2. Given the uncertainty in the parent relativistic theory and hence the appropriate form of the expansion history, this seems remarkably close. Given the non-linear nature of the theory, MOND could easily have been wrong by many orders of magnitude in this context. Nevertheless, it may be necessary to somehow damp the growth of structure at late times [401]. In this regard, a laboratory measurement of the ordinary neutrino mass might be relevant. Conventional structure cannot form in CDM if [229]. In contrast, some modest damping from a non-trivial neutrino mass might be desirable in MOND, and is also relevant to the CMB and clusters of galaxies (see Section 6.6.4).

In addition to mapping the growth factor as a function of redshift, one would also like to predict the
power spectrum of mass fluctuations as a function of scale at a given epoch. It is certainly
possible to match the power spectrum of galaxies at [401], but because of MOND’s
non-linearity and the uncertainty in the background cosmology, it is rather harder to know if such a
match faithfully represents a viable theory. Indeed, a natural prediction of baryon-dominated
cosmologies is the presence of strong baryon acoustic oscillations in the matter power spectrum at
[267, 127]. Dodelson [127] portrays this as a problem, but as already pointed out
in [267], the non-linearity of MOND can lead to mode mixing that washes out the initially strong
signal by . A more interesting test would be provided by the galaxy power spectrum at
high redshift (). This is a challenging observation, as one needs both a large survey
volume and high resolution in -space. The latter requirement arises because the predicted
features in the power spectrum are very sharp. The window functions necessarily employed
in the analysis of large scale structure data are typically wider than the predicted features.
Convolution of the predicted power spectrum with the SDSS analysis procedure [326] shows that
essentially all the predicted features wash out, with the possible exception of the strongest
feature on the largest scale. This means that the BAO signal detected by SDSS and consistent
with CDM [135] could also be interpreted as a confirmation of the prediction [267] of such
features^{60}
in MOND. However, there is no definitive requirement that the BAO appears at the same scale as observed,
or that it survives at all. In relativistic theories such as TeVeS (Section 7.3 and 7.4), damping of the
baryonic oscillations can be taken care of by parameters of the theory such as in original TeVeS
(Eq. 84, see Figure 3 of [430]) or the coefficients in generalized TeVeS (Eq. 87). In any
case, as in standard cosmology, the angular power spectrum of the CMB should be a cleaner
probe.

A first attempt to address the CMB was made before the existence of relativistic theories with a simple ansatz [265]: just as MOND returns precisely Newton in high accelerations, so any parent theory should contain GR (almost exactly, although this is not precisely the case for, e.g., TeVeS) in the appropriate strong-field limit. An obvious first assumption is that MOND effects do not yet appear in the very early universe, so that pure GR suffices for calculations concerning the CMB. The chief difference between CDM and a MONDian cosmology is then just the presence or absence of non-baryonic cold dark matter. With this ansatz, we can make one robust prediction: the shape of the acoustic power spectrum should follow pure baryonic diffusion damping. There is no net forcing term, as provided by the extra degree of freedom of non-baryonic cold dark matter. Thus, with nothing but baryons, each acoustic peak should be lower than the previous one [425] as part of a simple damping tail (Figure 45). In contrast, there must be evidence of forcing present in a power spectrum where CDM outweighs the baryons.

The density of both the baryons and the non-baryonic cold dark matter are critical to the shape of the acoustic power spectrum. For a given baryon density, models with CDM will have a larger second peak than models without it. Similarly, the third peak is always lower than the second in purely baryonic models, while it can be either higher or lower in CDM models, depending on the mix of each type of mass. Moreover, both parameters were well constrained prior to observation of the CMB [468]: from BBN [480] and from a variety of methods [116]. Therefore, it seemed like a straightforward exercise to predict the difference one should observe. The most robust prediction that could be made was the ratio of the amplitude of the first to second acoustic peak [265]. For the range of baryon and dark matter densities allowed at the time, CDM predicted a range in this ratio anywhere from 1.5 to 1.9. That is, the first peak should be almost but not quite twice as large as the second, with the precise value containing the information necessary to much better constrain both density parameters. For the same baryon densities allowed by BBN but no dark matter, the models fell in a distinct and much narrower range: 2.2 to 2.6, with the most plausible value being 2.4. The second peak is smaller (so the ratio of first to second higher) because there is no driving term to counteract baryonic damping. In this limit, the small range of relative peak heights follows directly from the narrow range in from BBN.

The BOOMERanG experiment [117] provided the first data capable of testing this prediction, and was in good agreement with the no-CDM prediction [268]. This result was subsequently confirmed by WMAP, which measured a ratio [345]. This is in good quantitative agreement with the prediction of the no-CDM ansatz, and outside the range first expected in CDM. CDM can nevertheless provide a good fit to the CMB power spectrum. The chief parameter adjustment required to obtain a fit is the baryon density, which must be increased: this is the reason for the near doubling of the long-standing value [480] to the more recent [229].

A critical question is whether the baryon density required by CDM is consistent with the
independently-measured abundances of the light isotopes. This question is explored in Figure 46. Historically,
no isotope suggested a value prior to fits to the CMB requiring such a high value. This is an
important fact to bear in mind, since historically cosmology has a long tradition of confirmation
bias^{61}.
More recent measurements of deuterium and helium are consistent with the high baryon density required by
CDM fits to the CMB. Lithium persistently suggests a lower baryon density, consistent with
pre-CMB values. If we are convinced of the correctness of CDM, then it is easy to dismiss
this as some peculiarity of stars – if exposed to the high temperatures in the cores of stars by
turbulent mixing, lithium might be depleted from it primordial value. If we are skeptical of
CDM, then it is no surprise that measurements of the primordial lithium abundance return
the same value now as they did before. From the perspective of the no-dark matter MOND
view, the CMB, lithium, deuterium, and helium all give a consistent baryon density given the
uncertainties.

However, the no-CDM ansatz must fail at some point. It could fail outright if the parent MOND theory deviates substantially from GR in the early universe. However, the more obvious [265] points of failure are rather due to the anticipated early structure formation in MOND discussed above. This should lead, in a true MOND theory, to early re-ionization of the universe and an enhancement of the integrated Sachs–Wolfe effect. Evidence for both these effects are present in the WMAP data [269]. Indeed, it turns out to be rather easy, and perhaps too easy, to enhance the integrated Sachs–Wolfe (ISW) effect in theories like TeVeS or GEA [430, 516]. Nevertheless, early re-inioniaztion is an especially natural consequence of MOND structure formation that was predicted a priori [265]. In contrast, structure is expected to build up more slowly in CDM such that obtaining the observed early re-ionization implies that the earliest objects to collapse were 50 times as efficient at converting mass to ionizing photons as are collapsed objects at the present time [435].

One prediction of the no-CDM anzatz that should not obviously fail is that the third peak should be smaller than the second peak of the acoustic power spectrum of the CMB. In a universe governed by MOND rather than cold dark matter, there is no obvious non-baryonic mass that is decoupled from the photon-baryon fluid. Therefore, it is a strong expectation that we observe only baryonic damping in the power spectrum, and each peak should be smaller in amplitude than the previous one. Contrary to this expectation, WMAP observes the third peak to be nearly equal in amplitude to the second [442, 229]. This approximate equality of the second and third peaks falsifies the simple no-CDM anzatz.

The PLANCK mission should soon report a new and much higher resolution measurement of the CMB acoustic power spectrum.
It is conceivable^{62}
that improved data will reveal a different power spectrum. A third peak as low as that expected
in the no-CDM anzatz would be one of the few observations capable of clearly falsifying the
existence of cosmic non-baryonic dark matter. A more likely result is basic confirmation of existing
observations with only minor tweaks to the exact power spectrum. Such a result would have
little impact on the discussion here as it would simply confirm the need for some degrees of
freedom in relativistic MOND theories that can play a role analogous to CDM. However, the
uncertainties on the best fit cosmological parameters may become negligibly small. Precise as
current data are, cosmology (with the exception of BBN) is still far from being over-constrained.
Hopefully, PLANCK data will be sufficiently accurate that they either agree or clearly do not
agree^{63}
with a host of other observations.

Presuming nothing substantial changes in the CMB data, we must understand the net forcing term in the acoustic oscillations leading to a high third peak. This might be taken in one of three ways:

- Practical falsification of MOND,
- Proof of the existence of some form of non-baryonic matter particles,
- An indication of some necessary additional freedom in relativistic parent theories of MOND,
playing the role of the non-baryonic mass in the CMB
^{64}.

Tempting as the first case (i) is [432], we cannot know whether the CMB falsifies MOND until we have exhaustively explored the predictions of relativistic parent theories (Section 7). The possibility of true non-baryonic mass (ii) seems unelegant, although a modification of gravity and the existence of non-baryonic dark matter are not mutually exclusive concepts. What is more, there is one obviously existing form of non-baryonic mass that may be relevant on cosmic scales: neutrinos. If [434], then the neutrino mass is too small to be of interest in this context. However, as discussed above, a modest neutrino mass may help to prevent MOND from over-predicting the growth of structure. Independently, a mass to 2 eV for the three neutrino species provides a good match to the width of the acoustic peaks of the CMB [269], which are otherwise too wide in a purely baryonic universe. Note that it also provides a match to the missing mass in galaxy clusters of (see Section 6.6.4). However, this neutrino mass is inadequate to explain the relatively high third peak in the no-DM ansatz. Obtaining a match to that instead requires a neutrino mass (for only one species) of 10 eV [9]. Such a large mass violates experimental constraints on the ordinary neutrino mass [234], but it may be possible to have a sterile neutrino with a mass in that ballpark [26]. As strange as this sounds, it provides a good fit to the CMB (Figure 47), and it may provide the unseen mass in all clusters and groups (see Section 6.6.4 [13, 11]). Experiments that can address the existence of such a particle would thus be very interesting [288], although in between it is perhaps best to view it merely as the encapsulation of our ignorance about cosmology in modified gravity theories, much as dark energy currently plays the same role in conventional cosmology. The fit of Figure 47 [9] is at least a proof of concept that cold DM is definitely not required by the CMB alone.

Perhaps the most intriguing possibility is (iii), that the height of the third peak is providing a glimpse of
some new aspect of modified gravity theories. As we have seen, generalizations of GR seeking to incorporate
MONDian phenomenology must, per force, introduce either non-locality (Section 7.10), or new degrees of
freedom in local theories. It is at least conceivable that these new degrees of freedom result in the net
driving of the acoustic oscillations that is implied by the departure from pure baryonic damping. For
instance, Dodelson & Liguori [128] have shown that in TeVeS (Sections 7.3 and 7.4) or GEA (Section 7.7)
theories, based on unit-norm vector fields, the growth of the spatial part of the vector perturbation in
the course of cosmological evolution is acting as an additional seed akin to non-baryonic dark
matter^{65}
(but unlike dark matter, its energy density is subdominant to the baryonic mass). Actually, it has
been shown that, with the help of this effect prior to baryon-photon decoupling, it is actually
possible^{66}
to produce as high a third peak as the second one in TeVeS and GEA theories without non-baryonic dark
matter, but at the cost of leading to unacceptably high temperature anisotropies in the CMB on large
angular scales, due to an over-enhanced ISW effect [430, 516]. Indeed, when making the effect of the
growth of the perturbed vector modes large, one also generates [151, 409, 498] a large gravitational slip
(see Section 8.4) in the perturbed FLRW metric (Eq. 116), which in turn leads to enhanced
ISW^{67}.
For this reason, acceptable fits to the CMB in TeVeS or GEA still need to appeal to non-baryonic
mass [430]. In this case, ordinary neutrinos within their model-independent mass-limit [234] are sufficient,
though^{68}.
However, the gravitational slip could be able to soon exclude at least some of these models
from combined information on the matter overdensity and weak lensing [362, 498]. However,
an important caveat is that all of the above arguments are based on adiabatic initial
conditions^{69}.
While initial isocurvature perturbations are basically ruled out in the GR context, this is not necessarily
true for modified gravity theories, so that correlated mixtures of adiabatic and isocurvature modes could
perhaps lower the ISW effect and/or raise the third peak [429].

Of course, when the additional “dark fields” of relativistic MOND theories are truly massive (as is the case in some theories), they can be thought of as true “dark matter”, whose energy density outweighs the baryonic one in the early universe: this is the case for the second scalar field of BSTV (Section 7.5), the scalar field of Section 7.6, and of course the dipolar dark matter of Section 7.9. In all these cases, reproducing the acoustic peaks of the CMB is, by construction, not a problem at all (nor erasing the baryon acoustic oscillations in the matter power spectrum contrary to [127]), while the MOND phenomenology is still nicely recovered in galaxies. In the case of BIMOND (Section 7.8), the possible appeal to twin matter could also have important consequences on the growth of structure [316] and, of course, on the CMB acoustic peaks too, although the latter analysis is still lacking. In an initially matter-twin matter symmetric universe, if the initial quantum fluctuations are not identical in the two sectors, matter and twin matter would still segregate efficiently, since density differences grow much faster that the sum [316]. The inhomogeneities of the two matter types would then develop, eventually, into mutually avoiding cosmic webs, and the tensors coming from the variation of the interaction term between the two metrics with respect to the matter metric can then act precisely as the energy-momentum tensor of cosmological dark matter [316], besides its contribution to the cosmological constant (see Section 9.1). Finally, the most thought-provoking and interesting possibility would perhaps be to explain all these cosmological observations through non-local effects (Section 7.10). In any case, it is likely that MOND will not be making truly clear predictions regarding cosmology until a more profound theory, based on first principles and underlying the MOND paradigm, is found.

Living Rev. Relativity 15, (2012), 10
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