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To probe realistically the stability of disks with the MOND dynamics, numerical simulations have been run, solving the N-body problem on a grid, through the equations of Bekenstein & Milgrom (1984). Brada & Milgrom (1999) showed that disks were always more instable in MOND. For the equivalent Newtonian system with a spherical dark halo, the more unstable galaxies are those with massive disks, which are more self-gravitating, while low-mass disks are stabilized by their halo. In MOND, the instability is about the same for massive disks, which are still in the Newtonian regime, however, low-mass disks remain unstable, and their growth rate tend to a constant, instead of vanishing.

3.1. Disk stability in MOND

From detailed comparison of two identical initial disks simulated with Newtonian dynamics+dark matter and MOND, Tiret & Combes (2007) have shown that bars develop quicker with modified gravity (see Figure 4). To have identical starts, the baryonic disk is first computed in equilibrium with its velocity distribution in MOND, and then, the amount of dark matter required to obtain the same derived rotation curve, is added for the Newtonian dynamics run. The evolution of the bar strength in Figure 4 reveals that both bars experience a drop in their strength, and this is due to the vertical resonance, building a peanut-shape feature, evolving in a pseudo-bulge (e.g. Combes & Sanders 1981, Combes et al. 1990, Bureau & Freeman 1999). The peanut occurs later in MOND. The bar remains strong during a longer time-scale, but then weakens, while the Newtonian bar can strengthen again, by exchanging angular momentum with the dark halo (e.g. Athanassoula 2002).

Figure 4

Figure 4. Strength of the bar formed in an Sa-type galaxy purely stellar simulation, measured by its Fourier harmonics m = 2,3,4 and 8 (ratio of tangential to radial force), for the CDM-Newton model (left) and MOND (right). The bar settles earlier in MOND, and stays longer, but after dropping at 4.5 Gyr, it does not develop again as in the CDM (cf Tiret & Combes 2007). The drop at 2.5 Gyr in the DM model as in the MOND model at 4.5 Gyr is due to the formation of a peanut bulge, through the vertical resonance (e.g. Combes et al. 1990).

This different way of growing results also in a different final morphology of the stellar disks: in MOND the disk is more extended, since the bar has grown by angular momentum exchange with the outer disk particules. Figure 4 represents an early-type spiral Sa. When all types are considered, the bar occurs much later in Newtonian models, because later types are more dominated by the dark matter halo, and are less self-gravitating. In MOND it is the contrary, the bar is first stronger in late-types, and then the disk is heated too much and the bar weakens. When the statistics are computed over the whole Hubble sequence, it appears that bars are stronger and more frequent in MOND, when only stellar components are taken into account. The higher MOND bar frequency is more in agreement with observations, where 2/3rds of spiral galaxies are barred (e.g. Laurikainen et al. 2004, 2009).

3.2. Pattern speed evolution

The bar pattern speed evolutions are also different in the two models. As shown in Figure 5 left, Ωbar is almost constant in MOND, while it drops by a factor 3 in 7 Gyr time in the equivalent Newtonian system. This is clearly due to the exchange of angular momentum from the bar to the dark matter halo, through dynamical friction. Indeed, the test run when the Newtonian system is computed with a rigid halo, which cannot deform and produce dynamical friction, has an almost constant Ωbar too.

Figure 5

Figure 5. Left Bar pattern speeds versus time: in MOND, the pattern speed remains constant, as in the Newtonian galaxy with a rigid dark matter halo. When the dark halo particles are taken into account self-consistently, the bar slows down, losing its angular momentum through dynamical friction. Right Frequency curves (from bottom to top, Ω − κ/2, Ω − νz/2, Ω and Ω + κ/2) for the CDM case (top) and MOND (bottom). The thick horizontal line is the pattern speed of the bar in each case (cf Tiret & Combes 2007).

This drop in Ωbar for the Newtonian+dark matter model has several consequences: First the Lindblad resonances in the plane and the vertical resonance move in radius, as shown in Figure 5 right. The pattern speed at the end of the simulation is shown as a thick dash line, and the inner/vertical resonance moves from 2 kpc to 12 kpc. Since the peanut represents stars vertically up-lifted at resonance, this means that the radius of the peanut is moving radially outwards, as shown in Figure 6. In MOND on the contrary, resonances are more long-lived, and can produce more robust effects.

Figure 6

Figure 6. Peanut-shape bulge formation, through vertical resonance with the bar. With CDM (left), the bar slows down with time, and the resonance moves to larger radii. Two peanut features are formed along the evolution, and the last one is rather extended in radius, while with MOND (right), there is only one peanut formed, centrally concentrated (Tiret & Combes 2007). These runs consider only the stellar component. Peanuts are less developed, when the disk is rich in gas.

3.3. Bulges and pseudo-bulges

Until now, the comparison between MOND and the Newtonian equivalent systems has been discussed with purely stellar disks. However, the presence of gas, and its interaction with stars change the picture. Gas as a dissipational component, is subject to a phase shift in its response to the bar pattern. There is a torque from the bar to the gas, that drives it to the center. This changes the potential there, and therefore the Ω frequencies and the resonances. The final result is a weakening of the bar, which can only develop again through gas accretion (e.g. Bournaud & Combes 2002). Gas dissipation and star formation have been taken into account in MOND simulations by Tiret & Combes (2008a). Statistically, bars occur even more rapidly in gas rich disks, and especially in the Newtonian models, which were too stable in the purely stellar disks. This makes the two models more similar, as far as the frequency of bars is concerned. Since the baryonic mass is more concentrated with gas in any model, the vertical resonance and the peanut occur at smaller radii, therefore the pseudo-bulges are smaller and more boxy in appearance.

Finally, the gas is driven by gravity torques inwards inside corotation, and outwards outside. It accumulates in rings at the inner (outer) Lindblad resonances respectively, in star forming rings that reproduce the blue rings observed in barred galaxies (e.g. Buta & Combes, 1996). In MOND, this phenomenon is even more remarkable, since first bars are still stronger and more frequent than in the Newtonian dynamics, but also the exchange of angular momentum between the stellar and the gas components is favored, while in the Newtonian case, there is competition with the dark halo for this exchange.

Summarizing the previous learnings, bars are more frequent in MOND, and consequently the formation of pseudo-bulges is favored. The fraction of classical bulges formed in major or minor mergers is likely to be much less, so that the picture of bulge formation is significantly different in the two regimes. These conclusions are applicable mainly to the local galaxies, at very low redshifts. First bars are less frequent in the past (Sheth et al. 2008), and pseudo-bulges are thought to be the dominant bulge formation at lower redshift (e.g. Kormendy & Kennicutt 2004). Second, it is not well known how the MOND model can be extended at high redshift. It has been remarked that the critical acceleration a0 is of the same order as c H0, with H0 the Hubble constant today, and therefore the critical acceleration could increase with z as H(z). Similarly a0 ∼ c (Λ / 3)1/2 (with Λ being the dark energy parameter), and any kind of variation with time of a0 is possible.

In the standard model, there is another mechanism to form bulges, which is more dominant at high redshift, that we will consider now.

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