9.2 Large-scale structure and Cosmic Microwave Background

Modified gravity theories should, of course, not only produce a reasonable Hubble expansion but also reproduce the observed anisotropies in the CMB, and the matter power spectrum. Taken at face value, these require not only dark matter, but non-baryonic cold dark matter. Any alternative theory must account for these, just as dark matter models need to explain galaxy scale phenomenology.

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 [229Jump To The Next Citation Point] 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 Ω ≈ Ω m b [279Jump To The Next Citation Point]. 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 Ωm > Ωb.

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 [386Jump To The Next Citation Point, 341Jump To The Next Citation Point, 226Jump To The Next Citation Point, 250Jump To The Next Citation Point], 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 ΛCDM. Consequently, MOND structure formation initially has to lag behind ΛCDM at very high redshift (z > 200). 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 z ≈ 10 and clusters by z ≈ 2 [11Jump To The Next Citation Point, 386], considerably earlier than in ΛCDM. By z = 0, the voids have become more empty than in ΛCDM, 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, 250Jump To The Next Citation Point]. 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 [11Jump To The Next Citation Point] and Local Void challenge, as well as the bulk flow challenge and high collisional velocity of the bullet cluster [16Jump To The Next Citation Point, 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 (z ≈ 10) from monolithic dissipationless collapse [393], with well-defined relationships between the mass, radius and velocity dispersion. Consequently, there would be less mergers than in ΛCDM 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 [239Jump To The Next Citation Point]. 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 [341Jump To The Next Citation Point, 401Jump To The Next Citation Point]. The amplitude of the power spectrum is well measured at z = 1091 in the CMB and at z ≈ 0 by surveys like the Sloan Digital Sky Survey. Simulations normalized to the CMB overproduce the structure at z = 0 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 [401Jump To The Next Citation Point]. In this regard, a laboratory measurement of the ordinary neutrino mass might be relevant. Conventional structure cannot form in ΛCDM if m > 0.2 eV ν [229Jump To The Next Citation Point]. 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 z = 0 [401Jump To The Next Citation Point], 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 z = 0 [267Jump To The Next Citation Point, 127Jump To The Next Citation Point]. Dodelson [127Jump To The Next Citation Point] portrays this as a problem, but as already pointed out in [267Jump To The Next Citation Point], the non-linearity of MOND can lead to mode mixing that washes out the initially strong signal by z = 0. A more interesting test would be provided by the galaxy power spectrum at high redshift (z ∼ 5). This is a challenging observation, as one needs both a large survey volume and high resolution in k-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 ΛCDM [135] could also be interpreted as a confirmation of the prediction [267] of such features60 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 K in original TeVeS (Eq. 84View Equation, see Figure 3 of [430Jump To The Next Citation Point]) or the ci coefficients in generalized TeVeS (Eq. 87View Equation). In any case, as in standard cosmology, the angular power spectrum of the CMB should be a cleaner probe.

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Figure 45: The acoustic power spectrum of the cosmic microwave background as observed by WMAP [229Jump To The Next Citation Point] together with the a priori predictions of ΛCDM (red line) and no-CDM (blue line) as they existed in 1999 [265Jump To The Next Citation Point] prior to observation of the acoustic peaks. ΛCDM correctly predicted the position of the first peak (the geometry is very nearly flat) but over-predicted the amplitude of both the second and third peak. The most favorable a priori case is shown; other plausible ΛCDM parameters [468Jump To The Next Citation Point] predicted an even larger second peak. The most important parameter adjustment necessary to obtain an a posteriori fit is an increase in the baryon density Ωb above what had previously been expected from BBN. In contrast, the no-CDM model ansatz made as a proxy for MOND successfully predicted the correct amplitude ratio of the first to second peak with no parameter adjustment [268Jump To The Next Citation Point, 269Jump To The Next Citation Point]. The no-CDM model was subsequently shown to under-predict the amplitude of the third peak [442Jump To The Next Citation Point].

A first attempt to address the CMB was made before the existence of relativistic theories with a simple ansatz [265Jump To The Next Citation Point]: 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 ΛCDM 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 45View Image). 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]: Ωb from BBN [480Jump To The Next Citation Point] and Ωm 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 [265Jump To The Next Citation Point]. For the range of baryon and dark matter densities allowed at the time, ΛCDM 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 Ωb from BBN.

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Figure 46: Estimates of the baryon density Ω h2 b [where h = H ∕(100 km s−1 Mpc −1) 0] over time (updated [273] from [269Jump To The Next Citation Point]). BBN was already a well-established field prior to 1995; earlier contributions are summarized by compilations (green ovals [480Jump To The Next Citation Point, 107]) that gave the long-lived standard value Ωbh2 = 0.0125 [480Jump To The Next Citation Point]. More recent estimates from individual isotopes are shown as triangles (2H), squares (4He), diamonds (3He), and stars (7Li). Estimates of the baryon density based on analyses of the cosmic microwave background are shown by circles (dark blue for ΛCDM; light blue for no-CDM). No measurement of any isotope suggested a value greater than 2 Ωbh = 0.02 prior to observation of the acoustic peaks in the microwave background (dotted lines), which might be seen as a possible illustration of confirmation bias. Fitting the acoustic peaks in ΛCDM requires Ω h2 > 0.02 b. More recent measurements of 2H and 4He have migrated towards the ΛCDM CMB value, while 7Li remains persistently problematic [111]. It has been suggested that turbulent mixing might result in the depletion of primordial lithium necessary to reconcile lithium with the CMB (upward pointing arrow [287]), while others [405] argue that this would merely reconcile some discrepant stars with the bulk of the data defining the Spite plateau, which persists in giving a 7Li abundance discrepant from the ΛCDM CMB value. In contrast, the amplitude of the second peak of the microwave background is consistent with no-CDM and Ωbh2 = 0.014 ± 0.005 [269Jump To The Next Citation Point]. Consequently, from the perspective of MOND, the CMB, lithium, deuterium, and helium all give a consistent baryon density given the uncertainties.

The BOOMERanG experiment [117] provided the first data capable of testing this prediction, and was in good agreement with the no-CDM prediction [268Jump To The Next Citation Point]. This result was subsequently confirmed by WMAP, which measured a ratio 2.34 ± 0.09 [345]. This is in good quantitative agreement with the prediction of the no-CDM ansatz, and outside the range first expected in ΛCDM. ΛCDM 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 Ωbh2 = 0.0125 [480] to the more recent Ω h2 = 0.02249 b [229Jump To The Next Citation Point].

A critical question is whether the baryon density required by ΛCDM is consistent with the independently-measured abundances of the light isotopes. This question is explored in Figure 46View Image. Historically, no isotope suggested a value Ωbh2 > 0.02 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 bias61. More recent measurements of deuterium and helium are consistent with the high baryon density required by ΛCDM fits to the CMB. Lithium persistently suggests a lower baryon density, consistent with pre-CMB values. If we are convinced of the correctness of ΛCDM, 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 ΛCDM, 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 [265Jump To The Next Citation Point] 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 [269Jump To The Next Citation Point]. 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 [430Jump To The Next Citation Point, 516Jump To The Next Citation Point]. Nevertheless, early re-inioniaztion is an especially natural consequence of MOND structure formation that was predicted a priori [265Jump To The Next Citation Point]. In contrast, structure is expected to build up more slowly in ΛCDM 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 [442Jump To The Next Citation Point, 229]. This approximate equality of the second and third peaks falsifies the simple no-CDM anzatz.

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Figure 47: CMB data as measured by the WMAP satellite year five data release (filled circles) and the ACBAR 2008 data release (triangles). Dashed line: ΛCDM fit. Solid line: HDM fit with a sterile neutrino of mass 11 eV. Image courtesy of Angus, reproduced by permission from [9Jump To The Next Citation Point].

The PLANCK mission should soon report a new and much higher resolution measurement of the CMB acoustic power spectrum. It is conceivable62 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 agree63 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:

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 √ ----- m ν ≈ Δm2 [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 m ν ≈ 1 eV to 2 eV for the three neutrino species provides a good match to the width of the acoustic peaks of the CMB [269Jump To The Next Citation Point], 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 T > 4 keV (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 [9Jump To The Next Citation Point]. Such a large mass violates experimental constraints on the ordinary neutrino mass [234Jump To The Next Citation Point], 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 47View Image), and it may provide the unseen mass in all clusters and groups (see Section 6.6.4 [13Jump To The Next Citation Point, 11Jump To The Next Citation Point]). 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 47View Image [9Jump To The Next Citation Point] 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 matter65 (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 possible66 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 [430Jump To The Next Citation Point, 516]. Indeed, when making the effect of the growth of the perturbed vector modes large, one also generates [151, 409, 498Jump To The Next Citation Point] a large gravitational slip (see Section 8.4) in the perturbed FLRW metric (Eq. 116View Equation), which in turn leads to enhanced ISW67. For this reason, acceptable fits to the CMB in TeVeS or GEA still need to appeal to non-baryonic mass [430Jump To The Next Citation Point]. In this case, ordinary neutrinos within their model-independent mass-limit [234] are sufficient, though68. 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 conditions69. 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 [127Jump To The Next Citation Point]), 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 [316Jump To The Next Citation Point] 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 [316Jump To The Next Citation Point]. 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.

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