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They can thus be used to predict how electrons will behave, and so whether a material is likely to host topological states. This includes both topological insulators and topological semi-metals—materials in which electrons, under certain conditions, act collectively like massless particles. The latter allow for the study of new quantum phenomena, and are being explored for use as catalysts. A third group—including Vishwanath—also found hundreds of topological materials, many of which they deem promising.

Some of these materials might also have useful topological qualities, so Vishwanath and colleagues are exploring ways to apply similar symmetry-based methods to identify topological magnetic materials. Experimentalists now have their work cut out. Researchers will be able to comb the databases to find new topological materials to explore. But until they are probed and measured in experiments, the classifications assigned to each material are still only predictions, Yazyev says.

Not every new topological material will prove interesting. So even more useful, says Judy Cha, an experimental physicist at Yale University in New Haven, Connecticut, would be if theorists could factor into the databases other practical information about the materials, such as how defects in the crystal affect the flow of electrons through it; this would help to whittle the list down to only the most practical. Elizabeth Gibney works for Nature magazine. The generlization of those relations for N -point correlation functions is suspected to hold generally,.

Cosmological N -body simulations approximately support the validity of the above ansatz, but also detect the finite deviation from it [ 82 ]. A complementary approach to characterize the clustering of the Universe beyond the two-point correlation functions is the genus statistics [ 26 ]. This is a mathematical measure of the topology of the isodensity surface.

This may be evaluated, for instance, by taking the ratio of the number of galaxies N x , V f in the volume V f centered at x to its average value :. Genus is one of the topological numbers characterizing the surface defined as. The Gauss-Bonnet theorem implies that the value of g is indeed an integer and equal to the number of holes minus 1. This is qualitatively understood as follows: Expand an arbitrary two-dimensional surface around a point as.

Interestingly the Gaussian density field has an analytic expression for Equation :. In this sense, genus statistics is a complementary measure of the clustering pattern of Universe. Figure taken from [ 54 ]. Even if the primordial density field obeys the Gaussian statistics, the subsequent nonlinear gravitational evolution generates the significant non-Gaussianity.

To distinguish the initial non-Gaussianity from that acquired by the nonlinear gravity is of fundamental importance in inferring the initial condition of the Universe in a standard gravitational instability picture of structure formation. In a weakly nonlinear regime, Matsubara derived an analytic expression for the non-Gaussianity emerging from the primordial Gaussian field [ 49 ]:. This expression plays a key role in understanding if the non-Gaussianity in galaxy distribution is ascribed to the primordial departure from the Gaussian statistics.

All MFs can be expressed as integrals over the excursion set. The general expression is. For a 3-D Gaussian random field, the average MFs per unit volume can be expressed analytically as follows:. The MFs were first introduced to cosmological studies by Mecke et al. Analytic expressions of MFs in weakly non-Gaussian fields are derived in [ 52 ]. As discussed above, luminous objects, such as galaxies and quasars, are not direct tracers of the mass in the Universe. In fact, a difference of the spatial distribution between luminous objects and dark matter, or a bias , has been indicated from a variety of observations.

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Galaxy biasing clearly exists. The fact that galaxies of different types cluster differently see, e.

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In order to confront theoretical model predictions for the mass distribution against observational data, one needs a relation of density fields of mass and luminous objects. The biasing of density peaks in a Gaussian random field is well formulated [ 37 , 4 ], and it provides the first theoretical framework for the origin of galaxy density biasing.

In this scheme, the galaxy-galaxy and mass-mass correlation functions are related in the linear regime via. However, a much more specific linear biasing model is often assumed in common applications, in which the local density fluctuation fields of galaxies and mass are assumed to be deterministically related via the relation.

Note that Equation follows from Equation , but the reverse is not true. The above deterministic linear biasing is not based on a reasonable physical motivation. Even in the simple case of no evolution in comoving galaxy number density, the linear biasing relation is not preserved during the course of fluctuation growth. Indeed, an analytical model for biasing of halos on the basis of the extended Press-Schechter approximation [ 59 ] predicts that the biasing is nonlinear and provides a useful approximation for its behavior as a function of scale, time, and mass threshold.

N -body simulations provide a more accurate description of the nonlinearity of the halo biasing confirming the validity of the Mo and White model [ 35 , ]. Biasing is likely to be stochastic , not deterministic [ 15 ]. An obvious part of this stochasticity can be attributed to the discrete sampling of the density field by galaxies, i. For example, the random variations in the density on smaller scales is likely to be reflected in the efficiency of galaxy formation.

As another example, the local geometry of the background structure, via the deformation tensor, must play a role too. While this relation can be schematically expressed as. For illustrative purposes, we define the biasing factor as the ratio of the density contrasts of luminous objects and mass:. From the above point of view, the local deterministic linear bias is obviously unrealistic, but is still a widely used conventional model for biasing.

In fact, the time- and scale-dependence of the linear bias factor b obj z , R was neglected in many previous studies of biased galaxy formation until very recently. Currently, however, various models beyond the deterministic linear biasing have been seriously considered with particular emphasis on the nonlinear and stochastic aspects of the biasing [ 71 , 15 , 87 , 86 ]. Let us illustrate the biasing from numerical simulations by considering two specific and popular models: primordial density peaks and dark matter halos [ 86 ]. As for the dark matter halos, these are identified using the standard friend-of-friend algorithm with a linking length of 0.

We locate a fiducial observer in the center of the circle.

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Then the comoving position vector r for a particle with a comoving peculiar velocity v at a redshift z is observed at the position s in redshift space:. Figure taken from [ 86 ]. We use two-point correlation functions to quantify stochasticity and nonlinearity in biasing of peaks and halos, and explore the signature of the redshift-space distortion. We also use the superscripts R and S to distinguish quantities defined in real and redshift spaces, respectively. We estimate those correlation functions using the standard pair-count method. The correlation functions of biased objects generally have larger amplitudes than those of mass.

Popular models of the biasing based on the peak or the dark halos are successful in capturing some essential features of biasing. None of the existing models of bias, however, seems to be sophisticated enough for the coming precision cosmology era.

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The development of a more detailed theoretical model of bias is needed. A straightforward next step is to resort to numerical simulations which take account of galaxy formation even if phenomenological at this point. We show an example of such approaches from Yoshikawa et al. Galaxies in their simulations are identified as clumps of cold and dense gas particles which satisfy the Jeans condition and have the SPH density more than times the mean baryon density at each redshift. Dark halos are identified with a standard friend-of-friend algorithm; the linking length is 0.

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  4. In addition, they identify the surviving high-density substructures in dark halos, DM cores see [ ] for further details. The circles in the lower panels indicate the positions of galaxies identified in our simulation. Figure taken from [ ]. Upper left panel: dark matter; upper right panel: gas; lower left panel: DM cores; lower right panel: cold gas.

    The circles in the lower panels indicate the positions of galaxies identified according to our criteria. The comoving size of the box is 6. The conditional mean relation computed directly from the simulation is plotted in solid lines, while dashed lines indicate theoretical predictions of halo biasing by Taruya and Suto [ 87 ]. This is mainly due to the exclusion effect of dark halos due to their finite volume size which is not taken into account in the theoretical model. Since our simulated galaxies have smaller spatial extent than the halos, the exclusion effect is not so serious.

    Solid lines indicate the conditional mean for each object.

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    Dashed lines in each panel depict the theoretical prediction of conditional mean by Taruya and Suto [ 87 ]. We turn next to a more conventional biasing parameter defined through the two-point statistics:. This is due to the fact that the present algorithm of group identification with larger l max tends to pick up lower mass halos which are poorly resolved in our numerical resolution. Two-point correlation functions of dark matter, galaxies, and dark halos from cosmological hydrodynamical simulations. Roughly speaking, z f corresponds to the median formation redshift of stars in the present-day galaxies.

    The old population indeed clusters more strongly than the mass, and the young population is anti-biased.

    The relative bias between the two populations ranges 1. Since the clustering of dark matter halos is well understood now, one can describe the galaxy biasing if the halo model is combined with the relation between the halos and luminous objects. This is another approach to galaxy biasing, halo occupation function HOF , which has become very popular recently. Indeed the basic idea behind HOF has a long history, but the model predictions have been significantly improved with the recent accurate models for the mass function, the biasing and the density profile of dark matter halos.

    Here we briefly outline this approach. We adopt a simple parametric form for the average number of a given galaxy population as a function of the hosting halo mass:. We will put constraints on the three parameters from the observed number density and clustering amplitude for each galaxy population. In short, the number density of galaxies is most sensitive to M 1 which changes the average number of galaxies per halo. The clustering amplitude on large scales is determined by the hosting halos and thus very sensitive to the mass of those halos, M min. The clustering on smaller scales, on the other hand, depends on those three parameters in a fairly complicated fashion; roughly speaking, M min changes the amplitude, while a, and to a lesser extent M 1 as well, change the slope.