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The Future of the Local Large Scale Structure: the roles of Dark Matter and Dark Energy

We study the distinct effects of Dark Matter and Dark Energy on the future evolution of nearby large scale structures using constrained N-body simulations. We contrast a model of Cold Dark Matter and a Cosmological Constant (LCDM) with an Open CDM (OCDM) model with the same matter density Omega_m =0.3 and the same Hubble constant h=0.7. Already by the time the scale factor increased by a factor of 6 (29 Gyr from now in LCDM; 78 Gyr from now in OCDM) the comoving position of the Local Group is frozen. Well before that epoch the two most massive members of the Local Group, the Milky Way and Andromeda, will merge. However, as the expansion rates of the scale factor in the two models are different, the Local Group will be receding in physical coordinates from Virgo exponentially in a LCDM model and at a roughly constant velocity in an OCDM model. More generally, in comoving coordinates the future large scale structure will look like a sharpened image of the present structure: the skeleton of the cosmic web will remain the same, but clusters will be more `isolated' and the filaments will become thinner. This implies that the long-term fate of large scale structure as seen in comoving coordinates is determined primarily by the matter density. We conclude that although the LCDM model is accelerating at present due to its Dark Energy component while the OCDM model is non accelerating, their large scale structure in the future will look very similar in comoving coordinates.


Abstract.
We study the distinct effects of Dark Matter and Dark Energy on the future evolution of nearby large scale structures using constrained N-body simulations. We contrast a model of Cold Dark Matter and a Cosmological Constant (ΛCDM) with an Open CDM (OCDM) model with the same matter density Ω m = 0.3 and the same Hubble constant h = 0.7. Already by the time the scale factor increased by a factor of 6 (29 Gyr from now in ΛCDM; 78 Gyr from now in OCDM) the comoving position of the Local Group is frozen. Well before that epoch the two most massive members of the Local Group, the Milky Way and Andromeda, will merge. However, as the expansion rates of the scale factor in the two models are different, the Local Group will be receding in physical coordinates from Virgo exponentially in a ΛCDM model and at a roughly constant velocity in an OCDM model. More generally, in comoving coordinates the future large scale structure will look like a sharpened image of the present structure: the skeleton of the cosmic web will remain the same, but clusters will be more 'isolated' and the filaments will become thinner. This implies that the long-term fate of large scale structure as seen in comoving coordinates is determined primarily by the matter density. We conclude that although the ΛCDM model is accelerating at present due to its Dark Energy component while the OCDM model is non accelerating, their large scale structure in the future will look very similar in comoving coordinates.
Submitted to: Acepted for publication in JCAP

Introduction
Cosmologists usually study the past evolution of the universe. Indeed, the Local Universe has been used before as a laboratory for studying dynamics and cosmological parameters, for example the timing argument (Raychaudhury & Lynden-Bell 1989, Karachentsev 2005, or N-body methods (Peebles et al., 1989). Similarly, we can gain insight into the dynamics and the role of the Dark Matter and Dark Energy components of the universe by considering the future evolution of nearby large scale structure. The background cosmology is now reasonably well established based on the measurements of the cosmic microwave background, the large redshift surveys, Supernovae Ia data and other probes (e.g. Spergel et al. 2006 and references therein). These measurements support a 'concordance' model in which the universe is flat and contains approximately 4% baryons, 21% Cold Dark Matter and 75% Dark Energy, with a possible small contribution of massive neutrinos. The nature of the Dark Matter and the Dark Energy are still to be understood, in particular the possibility of a more general equation of state of the Dark Energy component, w = P/ρ, which might also be evolving with cosmic epoch. The case w = −1 corresponds to Einstein's Cosmological Constant Λ, which we shall consider in this paper, as it is consistent with the current observations. We shall refer to this model as Λ Cold Dark Matter (ΛCDM).
The fate of a ΛCDM universe as a whole has been discussed before (e.g. Loeb, 2002;Krauss & Scherrer, 2007). The goal of the present study is to contrast the ΛCDM model with an OCDM model with the same matter density parameter, to investigate the roles of the Dark Matter and the Dark Energy in the evolution of clustering. When the universe will be completely dominated by the Λ term the scale factor will expand exponentially, a(t) ∝ exp( Λ/3t) (a 'de-Sitter phase'). It is less clear what will happen to the growth of structure and to individual objects in the universe. e.g. will they manage to survive as bound objects in an exponentially expanding universe? Papers by Nagamine & Loeb (2003, hereafter NL03) and Busha et al. (2005) have already considered some of these questions.
In particular NL03 used a constrained N-body simulation for a ΛCDM universe to predict the future evolution of nearby large scale structure. They found that structures will freeze in comoving coordinates. In particular they found that "the Local Group will get somewhat closer to the Virgo cluster in comoving coordinates, but will be pulled away from Virgo in physical coordinates due to the accelerated expansion of the universe." However, one should recall that the freeze out of the growth of structure is not the signature of the cosmological constant but rather of the fact that the density of matter is less than the critical. Namely, in a universe dominated either by the negative curvature or by the Λ term gravitational instability stalls. The linear theory growth of density perturbations in a universe with and without a cosmological constant is illustrated in numerous papers (e.g. Figure 2 of ( Lahav & Suto , 2004)). This has motivated us to explore the fate of the universe in the general case where the mean matter density is subcritical and compare the cases of a Λ dominated universe with an open one.
Here we extend the NL03 study to explore to what extent this fate of the local structure depends on the Dark Matter and Dark Energy contents of the universe. We use the tool of Constrained Realizations of Gaussian random fields (Hoffman & Ribak, 1991) to constrain the initial conditions for the N-body simulations by observational data. The resulting simulations reproduce quite faithfully the observed LSS out to a few tens of Mpc from the LG. We apply the simulations to both ΛCDM model and an Open Cold Dark Matter (OCDM) model with the same Ω m = 0.3 and the same Hubble constant h = 0.7. We look in particular at the universe at present epoch a = 1 ( where the age of the universe is 13.5 Gyr and 11.3 Gyr in ΛCDM and OCDM, respectively) and at a = 6 (where the universe will be 42.4 Gyr old for ΛCDM and 89.2 Gyr old in OCDM). We find that in comoving coordinates the evolution of large scale structure is quite similar. The main difference is in physical coordinates, as the evolution of the scale factor is obviously different for the two models.
The outline of the paper is as follows. In section 2 we summarize briefly the properties of the Local Group and the Local Supercluster. In Section 3 and in the Appendix we present analytic considerations for the evolution of structure and in Section 4 we give details of the constrained simulations. The results and plots from the simulations are shown in Section 5 and we discuss the results in Section 6.

Nearby Structures in the Local Universe
The Local Universe has been mapped in great detail since the 1980s, with the aid of whole sky galaxy surveys like IRAS, 2MASS and peculiar velocity surveys (e.g. Strauss & Willick 1995 for a review). Roughly speaking by Local Universe we mean the volume of a sphere of radius ≈ 200h −1 Mpc centred at the Milky Way (MW). Here we focus on the Local Group (LG) and the Local Supercluster (LSC). For definitions of the local structure see e.g. van den Bergh (1999) and Tully & Fisher (1987). Here we only summarize briefly the terminology of local structures relevant for our study. The Milky Way, Andromeda (M31) and other 30 other small galaxies within a few Mpc form the LG. At the present epoch the two major galaxies of the LG are approaching one another at an infall velocity of about 120 km/s and thus they constitute a bound dynamical system. The distance between the Milky Way and M31 is 740 ± 40 kpc. The total mass of the LG is estimated to be 2.3 ± 0.6 × 10 12 M ⊙ (van den Bergh, 1999).
Until the early 1980s the Virgo cluster was regarded as the centre of an overdense region called the LSC (e.g. , Davis & Huchra 1982, Lilje, Yahil, & Jones 1986 and it was assumed to be the major supercluster in the local universe. However, in the late 1980's it was recognized in whole-sky surveys that much larger superclusters, such as the Great Attractor and Perseus-Pisces, dominate the local universe. These and other superclusters and voids generate tidal forces which affect the motion of the LG towards the Virgo cluster. Despite this complexity of the local structure we find it useful below to consider the time evolution of the distance between the LG and the centre of Virgo. Obviously in principle we can define other distances or statistics to quantify the local structure. We also note the Supergalactic coordinate system, defined based on a planar structure in the galaxy distribution (de Vaucouleurs, 1975), which we shall use for convenience in some of our plots.
The Virgo-LG system is much less dynamically evolved than the LG itself. The observed mean overdensity in the number count of galaxies within the Virgocentric sphere is ≈ 2 (Davis & Huchra, 1982) and the (line of sight) peculiar velocity of the Virgo cluster relative to the LG is 932 km/s at a distance of 16±2 Mpc (van den Bergh, 1999).
Both the LG and the LSC constitute departure from the homogeneous and isotropic expanding universe. An unperturbed open universe or a flat Λ dominated universe expand forever. Bound objects that have collapsed and virialized by the present epoch will remain so in spite of the future expansion of the universe. The question is what is the fate of the objects in the nearby universe when the universe will be freely or exponentially expanding. In particular we shall follow the evolution of the two objects that dominate local dynamics, namely the LG and LSC.

Theoretical expectations
A rough estimate on the dynamical evolution is provided by the spherical top-hat model (Gunn & Gott, 1972). The evolution of spherical density perturbations in a general Friedmann universe dominated by non-relativistic matter and a Cosmological Constant was considered e.g. by Lahav et al. (1991), Wang & Steinhardt (1998) and Lokas & Hoffman (2001, hereafter LH01), Maor & Lahav (2005) and Percival (2005). A brief summary of the LH01 results and an extension to include velocity perturbations as well are presented in the Appendix.
Using the formalism presented in the Appendix the following values for the critical over-density to future collapse are obtained. For the case of a vanishing velocity perturbation the critical over-density is 17.6 (ΛCDM; in agreement with NL03) and 2.33 (OCDM). Adopting a Virgocentric infall velocity of 16H 0 − 932 = 188km/s (for H 0 = 70km/sec/Mpc) with the critical value is 14.6 (ΛCDM) and 1.3 (OCDM) (cf Appendix). Given the observed Virgocentric overdensity (in galaxy count) of ≈ 2 one can safely assume that in the ΛCDM model the Virgocentric infall is expected not to proceed to a collapse and virialization but rather to reach a freeze out. The case of the OCDM is not clear as the current estimation of the Virgocentric infall and overdensity are only marginally consistent with a freeze out of the infall. Numerical simulations are needed here to resolve the issue of the future of the LSC in the OCDM model.
The LG has already passed its turn-around phase and the MW and M31 are heading at a (line of sight) velocity of 120 km/s towards a merger to become one object. The mean overdensity in the LG is ≈ 260 (by averaging the mass of the MW and M31 over a sphere with a radius which is half the distance MW-M31) and it is much larger than ∆ crit for both the ΛCDM and OCDM models. It follows that we expect the LG to collapse and merge into one object in both cosmologies.

N-body simulations: methodology
The goal of the present paper is to study the future evolution of the nearby LSS in both OCDM and a flat ΛCDM cosmologies. The nearby universe seems to constitute a very typical realization of the flat-ΛCDM and the OCDM models. The LG, in particular, is not a unique or an unusual object in the universe, yet not being a relaxed object in virial equilibrium it has its own characteristics that affect the outcome of any dynamical test that would be applied to it. The key to a successful numerical study of the LG and the nearby universe is the ability to reproduce the characteristic dynamical properties of the LG and its surrounding. Namely a quasi-linear object located at about (10−15) h −1 Mpc from a Virgo-like cluster, within a supercluster that extends as filament connecting two major structures, the Perseus-Pices supercluster and the Great Attractor that are located some 90 h −1 Mpc apart. The LG is caught in a 'tug of war' in between these two major structures, which affects the local dynamics. The key for a successful numerical study of the local universe is the ability to design numerical simulations which reproduce the main dynamical features of the local universe. The optimal way of achieving that goal is by the use of constrained simulations (CSs; Kravtsov et al. 2002, Mathis et al. 2002, Klypin et al. 2003, namely simulations based on initial conditions set by means of constrained realizations of Gaussian fields (Hoffman & Ribak, 1991).
The data used to constrain the initial conditions of the simulations is made of two kinds. The first data set is made of radial velocities of galaxies drawn from the MARK III (Willick et al., 1997), SBF (Tonry et al., 2001) and the Karachentsev (2005) catalogs. Peculiar velocities are less affected by non-linear effects and are used as constraints as if they were linear quantities (Zaroubi et al., 1999). This follows the CSs performed by Kravtsov et al. (2002) and Klypin et al. (2003). The other constraints are obtained from the catalog of nearby X-ray selected clusters of galaxies (Reiprich & Böhringer, 2002). Given the virial parameters of a cluster and assuming the spherical top-hat model one can derive the linear overdensity of the cluster. The estimated linear overdensity is imposed on the mass scale of the cluster as a constraint. It should be noted that neither the MW and M31 nor the LG have been imposed directly on the simulations by the constraints. The constraints used here constrain quite closely the structure on scale larger than ≈ 5 h −1 Mpc (Klypin et al., 2003). Different CSs with different random realizations have been calculated and they all exhibit a clear and unambiguous LSClike structure that dominates the entire simulation, much in the same way as in the actual universe in which the LSC dominates the nearby LSS. The lack of explicit LG constraints causes the simulations to vary with respect to the particular details of the LG-like object. Yet, the fact that the large scale structure of the local universe is faithfully reproduced implies that the CSs have a high probability of producing quite reasonable LG-like objects. This is definitely confirmed by the present simulations.
We assume a flat ΛCDM model with Ω m = 0.3, σ 8 = 0.9 and h = 0.7 (where σ 8 is the power spectrum normalization factor). A somewhat refined set of cosmological parameters has been adopted after WMAP third year data release (Spergel et al., 2006), but the change in the values of the cosmological parameters would not affect the results presented here in any significant way. The other model used here is the OCDM, which is obviously inconsistent with recent observations of the CMB (Spergel et al., 2006). The model is considered here so as to show that the main affect that drives the freeze out of structure is the fact that Ω m is less than unity. For the sake of concreteness we assume that the cosmological parameters of the OCDM model coincide with that of the ΛCDM model apart from setting Ω Λ = 0, and thereby implying an open universe.
The simulations corresponds to a periodic cubic box of 64 h −1 Mpc on a side, spanned by a 256 3 grid. This translates into a mass per particle of 1.3 × 10 9 h −1 M ⊙ . The softening of the gravitational force corresponds to an equivalent Plummer softening of comoving 5 kpc/h, which reproduce the exact Newtonian force at 2.8 times this scale, implying a force resolution of 15 kpc/h. The mass resolution is adequate for resolving a LG-like object made of two ≈ 10 12 h −1 M ⊙ objects and following its internal dynamics. Yet, the internal dynamics of the main halos cannot be resolved. This to be compared with the mass resolution of 3.6 × 10 11 h −1 M ⊙ of NL03, which does not resolve the LG. The current simulations are among the ones used by Martinez-Vaquero et al. (2007). The parallel TREEPM N-body code GADGET2 (Springel (2005)) has been used to run the simulation. A more detailed description of the simulation is presented in Martinez-Vaquero et al. (2007). The halos are found by the AMIGA (Gill et al., 2004) halo finder, and the mass M corresponds to the mass enclosed within the virial radius.

Constrained simulation of the local universe
The CS clearly manifests the main characteristics of the local universe. Fig. 1 shows the projected dark matter density of a 12 h −1 Mpc thick slice centered on the Supergalactic Plain in both the ΛCDM and OCDM models. The local structure is dominated by the LSC filaments which crosses horizontally the Supergalactic Plain at roughly SGY ≈ 15 h −1 Mpc. The simulated LG is located in a filaments that run perpendicularly to the LSC at SGX ≈ −7 h −1 Mpc. Apart from a general shift by a few Mpc in the −SGX direction of the of the whole cosmic web, the simulated structure recovers the observed one. It should be noted at the outset that the present (and future) LSS exhibited by the two models looks very similar. The main dynamical parameters that characterize the simulated local universe at the present and future epochs for both the ΛCDM and OCDM models are presented in Table 1. Both models reproduce the structure of the LSC at the present epoch with a simulated LG located 13.0 (ΛCDM) and 11.6 Mpc/h (OCDM) away from the LG, compared with the actual value of 11.2Mpc/h. The mass of the simulated Virgo is 1.7 10 14 h −1 M ⊙ (ΛCDM) and 0.7 10 14 h −1 M ⊙ (OCDM), compared with the observed ≈ 10 14 h −1 M ⊙ . The mass of the simulated LG is very close to the observationally inferred value of 1.6 10 12 h −1 M ⊙ (van den Bergh, 1999). The OCDM simulated MW-M31 distance is close to the observed value of 0.7 Mpc, but the ΛCDM simulated distance is more than twice larger. That last discrepancy does not affect in any way the conclusions drawn here on the future evolution of the local universe.

The evolution of the LSC
The ΛCDM simulated local universe is presented at the present epoch of a = 1 and future epoch of a = 6 ( Fig. 1), where a is the expansion factor. The projected density maps are presented in term of co-moving Supergalactic coordinates. An inspections of Fig. 1 reveals that the cosmic web at the two epochs has hardly changed. Yet, the late epoch web is more 'skinny' and is more dominated by massive halos and less by the smaller mass halos. It should be noted here that the similarity between the present and future structure exists only upon using co-moving coordinates. In physical coordinates the cosmic web gets more empty and the typical length scale of the web expands exponentially. This is clearly seen in Figs. 2 which shows the evolution of the the Virgo-LG distance (in physical and co-moving coordinates). The first 30 Gyrs reflect the Virgocentric infall of the LG towards the Virgo. The infall is manifested in the co-moving sense only and it levels off in about 35 Gyrs. In physical coordinates the LG is receding away from the Virgo cluster almost exponentially. Note that after t 0 + t H the comoving distance remains constant with time, where t H = H −1 = Λ/3 ∼ 1.7H −1 0 , in agreement with Nagamine & Loeb (2003).
The present (a = 1) and future (a = 6) simulated local structure in the OCDM model are displayed in Fig. 1. The LSS, viewed in comoving coordinates exhibited by the ΛCDM and OCDM models at the two epochs shows a great resemblance. The general appearance of the cosmic web of the two models is almost indistinguishable. Yet, a close inspection of the LSC shows some differences. The LSC almost freezes out in the ΛCDM model. In the OCDM case the center of the LSC keeps on evolving. In particular the Virgo cluster moves along the filament that constitutes the LSC, from right to left in the Supergalactic projection, and merges with other clusters. The LG, on the other hand, is essentially frozen in comoving coordinates. As a result the Virgocentric distance of the LG keeps on growing in comoving coordinates. Figure 2. The evolution of the distance of the LG from the Virgo cluster in the ΛCDM (blue curves) and OCDM (red curves) models is presented in comoving (solid line) and physical (dashed lines) coordinates. The upper panel presents the distance as a function of the scale factor a and the lower one as a function of time. In the ΛCDM case the Virgocentric infall extends to about twice the present Hubble time before it converges to its asymptotic value, where the inflow freezes. In physical coordinates the distance increases at all time and eventually it grows exponentially. In the OCDM the time evolution of the distance does not show the freezing of the flow. The position of the LG indeed becomes frozen in comoving coordinates but the Virgo cluster moves away from it because of the internal dynamics within the LSC.

The evolution of the LG
As the observed distance between the Milky Way and Andromeda is 740 kpc and they are approaching one another at 120 km/sec, we can estimate naively that they will collide in about 6 Gyr if relative transverse velocity is ignored (Cox & Loeb, 2007). The simulated ΛCDM LG is shown at the present epoch in Fig. 3 exhibits two DM halos with masses of about 10 12 h −1 M ⊙ , at a distance 1.1 h −1 Mpc and located in a slightly overdense filament that connects the LG group with the Virgo cluster of a mass of 10 14 h −1 M ⊙ located 10 h −1 Mpc away. The relative infall velocity of the two halos is about 140 km/sec and the transversal relative velocity is 50 km/s. It follows that the CS has reproduced a LG-like object that resembles the actual LG apart from the MW-M31 distance which is larger than the true distance of 0.74Mpc. From the ΛCDM simulation we find that the simulated MW and M31 merge into one object, at the epoch of a = 3.0, corresponding for h = 0.7 to 17.3 Gyr from now (we note this is longer than that expected from just the two-body radial motion based on the the simulation values: with infall of 140 km/sec starting at relative distance of 1570 kpc, we expect 11.0 Gyr).
Our large simulated present epoch MW-M31 distance obviously implies that the estimated time till the LG merger is overestimated. Indeed, Cox & Loeb (2007) have recently estimated that the merger will take place in 5Gyr.
The simulated OCDM LG is very similar to the ΛCDM one, both in structure and evolution (Fig. 3). The present epoch MW-M31 distance of the OCDM LG is somewhat smaller than in the ΛCDM case, ≈ 0.5 h −1 Mpc and therefore the MW-M31 merger is taking place earlier, at a ∼ 1.1, corresponding to (12.7-11.3) Gyr = 1.4 Gyr from now. Fig. 3 shows the final merged object much later (at a = 6) in both simulations.
It is clear that in the present CSs the internal structure of the LG is very weakly constrained and different realizations exhibit a large scatter with respect to the structure of the LG. This leads to a considerable scatter in the collapse time of the LG. Yet, all simulated LG-like objects will collapse much before a = 1.

General remarks on the future evolution
The cumulative mass function n(> M) of DM halos serves as a good indicator of the growth of structure of a cosmological model, where n(> M) is the number of DM halos more massive than M per unit co-moving volume. We have calculated the mass function of both the ΛCDM and OCDM models in the present a = 1 and future a = 6 epochs. Generally speaking the mass function of the two models are very similar at both epochs. This similarity follows from the very similar dynamical nature of the two models, when analyzed in co-moving coordinates. The cumulative mass function hardly evolves in both models, yet at a = 6 there are somewhat fewer small, ≈ 10 12 h −1 M ⊙ , DM halos and roughly twice as many rich cluster -like halos. A detailed analysis shows that the depletion in the number of small halos and the appearance of more massive clusters is somewhat more pronounced in the OCDM model compared with the ΛCDM one.
A close inspection of Fig 2 shows that the LG-Virgo distance is already frozen by a = 6 in the ΛCDM model, while it is still slowly increasing in the OCDM case at that epoch. This is consistent with the evolution displayed by the mass function, namely by a = 6 the OCDM model is somewhat more evolved than the ΛCDM model. The slight differences between the models is clearly understood.
A simple analytical reasoning leads to the understanding that once the dynamics of the universe is no longer dominated by the cold matter density, gravitational instability comes to an end and the growth of structure freezes out. This property is shared by both the ΛCDM and OCDM models. Yet, the rate of transition to the asymptotic state is different. In the ΛCDM case the asymptotic state is achieved when the (constant for w = −1) dark energy term dominates the cold matter term (which decays like a −3 ) Figure 3. The mass distribution around the LG in the Supergalactic plane in the ΛCDM (left column) and OCDM (right column) models presented at the present epoch (a = 1; upper row) and the future (a = 6; lower row). The plot shows the projection of the particles in a slab of thickness 3 h −1 Mpc. The color coding of the particles indicate the density field. At the present epoch the LG is dominated by two massive halos of mass ≈ 10 12 h −1 M ⊙ at a distance of ≈ 1.1 h −1 Mpc and 0.5 h −1 Mpc in ΛCDM and OCDM respectively. There is very little difference between the dynamics of the simulated LG in the ΛCDM and OCDM models.
in the Friedmann equation. In the OCDM case, on the other hand, the freeze out is achieved when the curvature term (which decays as a −2 ) dominates. It follows that the ΛCDM model converged much faster, in time and scale factor as ell, to the freeze out state than the OCDM model.

Discussion
Exploring the future of evolution of the large scale structure of the local universe is more than just a curiosity. Busha et al. (2005) have demonstrated that by running simulations into the future it is possible to identify more efficiently which objects will form bound objects. It also allows us to get further insight into the effect of Dark Matter and Dark Energy in the evolution of large scale structure. In particular we extended here the study of NL03 who analyzed with the aid of constrained N-body simulations the evolution of nearby large scale structure only in a flat ΛCDM model. By contrasting the ΛCDM simulation with an OCDM with the same matter density Ω m = 0.3 and the same Hubble constant h = 0.7 we can learn what is the role of the Dark Matter compared with the role of the Dark Energy.
Our main conclusion is that the long-term fate of large scale structure as seen in comoving coordinates is determined primarily by the matter density. This generalizes the result of NL03, and clarifies the distinct role of the matter density in defining the freezing out of structure, and the combined effects of both the Dark Matter and Dark Energy in the evolution of the scale factor and hence in the physical coordinates.
In more detail our main results are: (i) The Milky Way and Andromeda will merge already at a = 3 and at a = 1.1 according to our ΛCDM and OCDM simulations (where the present epoch distances between the two galaxies in our simulations are somewhat different from the true observed value of 740 kpc).
(ii) Already by a = 6 the comoving position of the LG becomes frozen at its asymptotic value. In the ΛCDM case the comoving LG-Virgo distance reaches its asymptotic value. In the OCDM case the Virgo cluster still moves within the LSC and hence that distance is still slightly growing. However, in physical coordinates the LG will be receding from Virgo exponentially in a ΛCDM model and at an almost constant velocity in an OCDM model. This is just a manifestation of the different evolution of the scale factor a(t) in the two models.
(iii) When we consider the overall cosmic web we find that qualitatively the future large scale structure will look like a sharpened image of the present structure: the skeleton of the cosmic web will remain the same, but clusters will be more 'isolated' and the filaments will become thinner.
The discussion has focused so far on the dynamical evolution of our cosmological neighborhood, which hardly distinguishes between the comoving structure of the ΛCDM and OCDM models. Yet, the two models appear fundamentally very different when viewed in physical coordinates and in particular when the notion of the horizon is considered. The comoving event horizon in the ΛCDM model of the present epoch at the present time is 3.4h −1 Gpc and at a = 6 it is 0.6h −1 Gpc. As was pointed by Loeb (2002) the future LG observer in the ΛCDM model will be living in an Island Universe. This will happen at a ≈ 350 (namely, at t ≈ 8H −1 0 ), when the Virgo cluster will be at the event horizon of the LG observer. The cosmic web will not be accessible to these observers and the whole notion of comoving coordinates will be meaningless (Krauss & Scherrer, 2007).
The work presented here can be extended further in a number of ways. The evolution of the skeleton structure with time in different models could be quantified by advanced statistics (e.g. the probability distribution function and Minkowski functionals). The models could be extended to study the much discussed equation of state parameter w, and to allow it to be epoch dependent, e.g. w(a) ≈ w 0 + w a (1 − a). Major surveys of Dark energy are underway to quantify w(a) (e.g. the DETF report (Albrecht et al., 2006) and ESO/ESA report (Peacock & Schneider, 2006), and it would be interesting to understand the implications for the growth of structure in the universe if w(a) turns out to be different than the w = −1 assumed in this paper. In addition, one may explore the impact of changes in the value of Ω m , the effect of massive neutrinos (which suppress structure on large scales) and the variations of the normalization of power spectrum σ 8 .
Finally, we remark on the connection between our simulation results and the Anthropic principle. The basic Anthropic argument is that if the matter density was too high the universe would have collapsed by the present epoch without providing enough time for life to emerge and evolve. On the other hand, if the matter density was too small, structure would have not collapsed and again life would not have evolved. It has been argued further that the observed value of the cosmological constant can also be supported by Anthropic arguments, (e.g. Weinberg 1987, Efstathiou 1995, Tegmark & Rees 1998, Weinberg 2005 and references therein). In a nutshell, Weinberg's original argument was that the Cosmological Constant (in the form of a vacuum energy) cannot be too large and positive, because then galaxies would not form. The vacuum density should be less than the matter density of the universe at the epoch when galaxies formed. The probability of a random observer seeing the value of the vacuum energy as small as observed is about 15%, under certain assumptions (Weinberg, 2005). We note in particular the sensitivity of the argument to the assumed amplitude of density fluctuations (Tegmark & Rees, 1998). More recently, the possibility of a ΛCDM universe has been considered in the context of a 'string landscape', where the universe we live in is only one of many in the 'multiverse' ( (Weinberg, 2005) and references therein). It has been argued in the above references that a ΛCDM universe is suitable for the emergence of life, while other universes with other extreme values of matter and vacuum densities would be hostile to life.
While the observed value of the Cosmological Constant is consistent with this argument, we argue that as the large scale structures expected in ΛCDM and the OCDM scenarios look so similar at the present epoch (a = 1), there is no preference from anthropic arguments alone for one or the other. The dominant effect of the formation of structure (as seen in comoving coordinates) is the matter density rather than the vacuum energy. and effective local density parameters by where x stands form m or Λ. Eq. 3 can now be easily solved by replacing the global cosmological parameters H, Ω m and Ω Λ by the local values of h l , ω m and ω Λ evaluated at t i .
Eq. 4 implies that in the case of no velocity perturbation the critical overdensity is 17.6 (ΛCDM) and 2.33 (OCDM). In the case of velocity perturbations Eq. 4 is still valid, but with a modified local Hubble constant and density parameter (Eqs. 8 and 9). Adopting for the Virgo cluster a distance of 16Mpc, a Hubble constant of H 0 = 70km s −1 Mpc −1 and a radial recession velocity of 932km/s (cf §3), one finds (1 + v p /v H ) = 0.83. It follows that the critical overdensity for the Virgocentric infall is 14.6 (ΛCDM) and 1.3 (OCDM).