The cosmic web is one of the most striking features of the distribution of galaxies and dark matter on the largest scales in the Universe. It is composed of dense regions packed full of galaxies, long filamentary bridges, flattened sheets and vast low density voids. The study of the cosmic web has focused primarily on the identification of such features, and on understanding the environmental effects on galaxy formation and halo assembly. As such, a variety of different methods have been devised to classify the cosmic web -depending on the data at hand, be it numerical simulations, large sky surveys or other. In this paper we bring twelve of these methods together and apply them to the same data set in order to understand how they compare. In general these cosmic web classifiers have been designed with different cosmological goals in mind, and to study different questions. Therefore one would not a priori expect agreement between different techniques however, many of these methods do converge on the identification of specific features. In this paper we study the agreements and disparities of the different methods. For example, each method finds that knots inhabit higher density regions than filaments, etc. and that voids have the lowest densities. For a given web environment, we find substantial overlap in the density range assigned by each web classification scheme. We also compare classifications on a halo-by-halo basis; for example, we find that 9 of 12 methods classify around a third of group-mass haloes (i.e. M halo ∼ 10 13.5 h −1 M ⊙ ) as being in filaments. Lastly, so that any future cosmic web classification scheme can be compared to the 12 methods used here, we have made all the data used in this paper public.

We use recent proper motion measurements of the tangential velocity of M31, along with its radial velocity and distance, to derive the likelihood of the sum of halo masses of the Milky Way and M31. This is done using a sample halo pairs in the Bolshoi cosmological simulation of ΛCDM cosmology selected to match properties and environment of the Local Group. The resulting likelihood gives estimate of the sum of masses of M MW,200c + M M31,200c = 2.40 +1.95 −1.05 × 10 12 M ⊙ (90% confidence interval). This estimate is consistent with individual mass estimates for the Milky Way and M31 and is consistent, albeit somewhat on the low side, with the mass estimated using the timing argument. We show that although the timing argument is unbiased on average for all pairs, for pairs constrained to have radial and tangential velocities similar to that of the Local Group the argument overestimates the sum of masses by a factor of 1.6. Using similar technique we estimate the total dark matter mass enclosed within 1 Mpc from the Local Group barycenter to be M LG (r < 1 Mpc) = 4.2 +3.4 −2.0 × 10 12 M ⊙ (90% confidence interval).

We present a new method to identify large-scale filaments and apply it to a cosmological simulation. Using positions of haloes above a given mass as node tracers, we look for filaments between them using the positions and masses of all the remaining dark matter (DM) haloes. In order to detect a filament, the first step consists in the construction of a backbone linking two nodes, which is given by a skeleton-like path connecting the highest local DM density traced by non-node haloes. The filament quality is defined by a density and gap parameters characterizing its skeleton, and filament members are selected by their binding energy in the plane perpendicular to the filament. This membership condition is associated to characteristic orbital times; however if one assumes a fixed orbital time-scale for all the filaments, the resulting filament properties show only marginal changes, indicating that the use of dynamical information is not critical for the method. We test the method in the simulation using massive haloes (M > 10 14 h −1 M ) as filament nodes. The main properties of the resulting highquality filaments (which corresponds to 33 per cent of the detected filaments) are (i) their lengths cover a wide range of values of up to 150 h −1 Mpc, but are mostly concentrated below 50 h −1 Mpc; (ii) their distribution of thickness peaks at d = 3.0 h −1 Mpc and increases slightly with the filament length; (iii) their nodes are connected on average to 1.87 ± 0.18 filaments for 10 14.1 M nodes; this number increases with the node mass to 2.49 ± 0.28 filaments for 10 14.9 M nodes; (iv) on average, the central density along the filaments starts at almost a hundred times the average density in the regions surrounding the nodes and then drops to about a few times the mean density at larger distances, where it remains roughly constant over 20-80 per cent of the filament length (this result may depend on the filament length); (v) there is a strong relation between length, quality and how straight a filament is, where shorter filaments are those characterized by higher qualities and more straight-line-like geometries.

Previous studies showed that an estimate of the likelihood distribution of the Milky Way halo mass can be derived using the properties of the satellites similar to the Large and Small Magellanic Clouds (LMC and SMC). However, it would be straightforward to interpret such an estimate only if the properties of the Magellanic Clouds (MCs) are fairly typical and are not biased by the environment. In this study we explore whether the environment of the Milky Way affects the properties of the SMC and LMC such as their velocities. To test for the effect of the environment, we compare velocity distributions for MC-sized subhalos around Milky Way hosts in a sample selected simply by mass and in the second sample of such halos selected with additional restrictions on the distance to the nearest cluster and the local galaxy density, designed to mimic the environment of the Local Group (LG). We find that satellites in halos in the LG-like environments do have somewhat larger velocities, as compared to the halos of similar mass in the sample without environmental constraints. For example, the fraction of subhalos matching the velocity of the LMC is 23±2% larger in the LG-like environments. We derive the host halo likelihood distribution for the samples in the LG-like envirionment and in the control sample and find that the environment does not significantly affect the derived likelihood. We use the updated properties of the SMC and LMC to derive the constraint on the MW halo mass log (M 200 /M ⊙ ) = 12.06 +0.31 −0.19 (90% confidence interval). We also explore the incidence of close pairs with relative velocities and separations similar to those of the LMC and SMC and find that such pairs are quite rare among ΛCDM halos. Only 2% of halos in the MW mass range have a relatively close pair (∆r < 40kpc and ∆s < 160 km s −1 ) of subhalos with circular velocities v circ > 50 km s −1 . Pairs with masses and separations similar to those of the LMC and SMC (∆r MC = 23.4 ± 10 kpc, and ∆s MC = 128 ± 32 km s −1 ) are found only in one out of ≈ 30000 MW-sized halos. Interestingly, the halo mass likelihood distribution for host halos constrained to have MC-like close pairs of subhalos is quite different from the global likelihood from which the MW halo mass constraint discussed above was derived. Taking into account the close separation of the MCs in the Busha et al. 2011 method results in the shift of the MW halo mass estimate to smaller masses, with the peak shifting approximately by a factor of two.