Here we discuss the numerical methods presently available for this type of analyses, and we review the main results obtained so far for different classes of alternative cosmologies. These can be grouped into models where structure formation is affected only through a modified expansion history (such as quintessence and early dark-energy models, Section 1.4.1) and models where particles experience modified gravitational forces, either for individual particle species (interacting dark-energy models and growing neutrino models, Section 126.96.36.199) or for all types of particles in the universe (modified gravity models).
In general, in the context of flat FLRW cosmologies, any dynamical evolution of the dark-energy density () determines a modification of the cosmic expansion history with respect to the standard CDM cosmology. In other words, if the dark energy is a dynamical quantity, i.e., if its equation of state parameter exactly, for any given set of cosmological parameters (, , , , ), the redshift evolution of the Hubble function will differ from the standard CDM case .
Quintessence models of dark energy [954, 754] based on a classical scalar field subject to a self-interaction potential have an energy density that evolves in time according to the dynamical evolution of the scalar field, which is governed by the homogeneous Klein–Gordon equation:
For a canonical scalar field, the equation of state parameter , where , will in general be larger than , and the density of dark energy will consequently be larger than at any redshift . Furthermore, for some simple choices of the potential function such as those discussed in Section 1.4.1 (e.g., an exponential potential or an inverse-power potential ), scaling solutions for the evolution of the system can be analytically derived. In particular, for an exponential potential, a scaling solution exists where the dark energy scales as the dominant cosmic component, with a fractional energy density
Early dark energy (EDE) is, therefore, a common prediction of scalar field models of dark energy, and observational constraints put firm bounds on the allowed range of at early times, and consequently on the potential slope .
As we have seen in Section 1.2.1, a completely phenomenological parametrization of EDE, independent from any specific model of dynamical dark energy has been proposed by  as a function of the present dark-energy density , its value at early times , and the present value of the equation of state parameter . From Eq. 1.2.4, the full expansion history of the corresponding EDE model can be derived.
A modification of the expansion history indirectly influences also the growth of density perturbations and ultimately the formation of cosmic structures. While this effect can be investigated analytically for the linear regime, N-body simulations are required to extend the analysis to the nonlinear stages of structure formation. For standard Quintessence and EDE models, the only modification that is necessary to implement into standard -body algorithms is the computation of the correct Hubble function for the specific model under investigation, since this is the only way in which these non standard cosmological models can alter structure formation processes.
This has been done by the independent studies of  and , where a modified expansion history consistent with EDE models described by the parametrization of Eq. 1.2.4 has been implemented in the widely used -body code Gadget-2  and the properties of nonlinear structures forming in these EDE cosmologies have been analyzed. Both studies have shown that the standard formalism for the computation of the halo mass function still holds for EDE models at . In other words, both the standard fitting formulae for the number density of collapsed objects as a function of mass, and their key parameter representing the linear overdensity at collapse for a spherical density perturbation, remain unchanged also for EDE cosmologies.
The work of , however, investigated also the internal properties of collapsed halos in EDE models, finding a slight increase of halo concentrations due to the earlier onset of structure formation and most importantly a significant increment of the line-of-sight velocity dispersion of massive halos. The latter effect could mimic a higher normalization for cluster mass estimates based on galaxy velocity dispersion measurements and, therefore, represents a potentially detectable signature of EDE models.
Another interesting class of non standard dark-energy models, as introduced in Section 1.4.4, is given by coupled dark energy where a direct interaction is present between a Quintessence scalar field and other cosmic components, in the form of a source term in the background continuity equations:
While such direct interaction with baryonic particles () is tightly constrained by observational bounds, and while it is suppressed for relativistic particles () by symmetry reasons (), a selective interaction with cold dark matter (CDM hereafter) or with massive neutrinos is still observationally viable (see Section 1.4.4).
Since the details of interacting dark-energy models have been discussed in Section 1.4.4, here we simply recall the main features of these models that have a direct relevance for nonlinear structure formation studies. For the case of interacting dark energy, in fact, the situation is much more complicated than for the simple EDE scenario discussed above. The mass of a coupled particle changes in time due to the energy exchange with the dark-energy scalar field according to the equation:
As a consequence of these new terms in the Newtonian acceleration equation the growth of density perturbations will be affected, in interacting dark-energy models, not only by the different Hubble expansion due to the dynamical nature of dark energy, but also by a direct modification of the effective gravitational interactions at subhorizon scales. Therefore, linear perturbations of coupled species will grow with a higher rate in these cosmologies In particular, for the case of a coupling to CDM, a different amplitude of the matter power spectrum will be reached at with respect to CDM if a normalization in accordance with CMB measurements at high redshifts is assumed.
Clearly, the new acceleration equation (1.6.6) will have an influence also on the formation and evolution of nonlinear structures, and a consistent implementation of all the above mentioned effects into an -body algorithm is required in order to investigate this regime.
For the case of a coupling to CDM (a coupling with neutrinos will be discussed in the next section) this has been done, e.g., by [604, 870] with 1D or 3D grid-based field solvers, and more recently by means of a suitable modification by  of the TreePM hydrodynamic -body code Gadget-2 .
Nonlinear evolution within coupled quintessence cosmologies has been addressed using various methods of investigation, such as spherical collapse [611, 962, 618, 518, 870, 3, 129] and alternative semi-analytic methods [787, 45]. -body and hydro-simulations have also been done [604, 79, 76, 77, 80, 565, 562, 75, 980]. We list here briefly the main observable features typical of this class of models:
Subsequent studies based on Adaptive Mesh Refinement schemes for the solution of the local scalar field equation  have broadly confirmed these results.
The analysis has been extended to the case of non-constant coupling functions by , and has shown how in the presence of a time evolution of the coupling some of the above mentioned results no longer hold:
All these effects represent characteristic features of interacting dark-energy models and could provide a direct way to observationally test these scenarios. Higher resolution studies would be required in order to quantify the impact of a DE-CDM interaction on the statistical properties of halo substructures and on the redshift evolution of the internal properties of CDM halos.
As discussed in Section 1.6.1, when a variable coupling is active the relative balance of the fifth-force and other dynamical effects depends on the specific time evolution of the coupling strength. Under such conditions, certain cases may also lead to the opposite effect of larger halo inner overdensities and higher concentrations, as in the case of a steeply growing coupling function [see 76]. Alternatively, the coupling can be introduced by choosing directly a covariant stress-energy tensor, treating dark energy as a fluid in the absence of a starting action [619, 916, 193, 794, 915, 613, 387, 192, 388].
In case of a coupling between the dark-energy scalar field and the relic fraction of massive neutrinos, all the above basic equations (1.6.5) – (1.6.8) still hold. However, such models are found to be cosmologically viable only for large negative values of the coupling [as shown by 36], that according to Eq. 1.6.5 determines a neutrino mass that grows in time (from which these models have been dubbed “growing neutrinos”). An exponential growth of the neutrino mass implies that cosmological bounds on the neutrino mass are no longer applicable and that neutrinos remain relativistic much longer than in the standard scenario, which keeps them effectively uncoupled until recent epochs, according to Eqs. (1.6.3 and 1.6.4). However, as soon as neutrinos become non-relativistic at redshift due to the exponential growth of their mass, the pressure terms in Eqs. (1.6.3 and 1.6.4) no longer vanish and the coupling with the DE scalar field becomes active.
Therefore, while before the model behaves as a standard CDM scenario, after the non-relativistic massive neutrinos obey the modified Newtonian equation (1.6.6) and a fast growth of neutrino density perturbation takes place due to the strong fifth force described by Eq. (1.6.8).
The growth of neutrino overdensities in the context of growing neutrinos models has been studied in the linear regime by , predicting the formation of very large neutrino lumps at the scale of superclusters and above (10 – 100 Mpc/h) at redshift .
The analysis has been extended to the nonlinear regime in  by following the spherical collapse of a neutrino lump in the context of growing neutrino cosmologies. This study has witnessed the onset of virialization processes in the nonlinear evolution of the neutrino halo at , and provided a first estimate of the associated gravitational potential at virialization being of the order of for a neutrino lump with radius .
An estimate of the potential impact of such very large nonlinear structures onto the CMB angular power spectrum through the Integrated Sachs–Wolfe effect has been attempted by . This study has shown that the linear approximation fails in predicting the global impact of the model on CMB anisotropies at low multipoles, and that the effects under consideration are very sensitive to the details of the transition between the linear and nonlinear regimes and of the virialization processes of nonlinear neutrino lumps, and that also significantly depend on possible backreaction effects of the evolved neutrino density field onto the local scalar filed evolution.
A full nonlinear treatment by means of specifically designed -body simulations is, therefore, required in order to follow in further detail the evolution of a cosmological sample of neutrino lumps beyond virialization, and to assess the impact of growing neutrinos models onto potentially observable quantities as the low-multipoles CMB power spectrum or the statistical properties of CDM large scale structures.
Modified gravity models, presented in Section 1.4, represent a different perspective to account for the nature of the dark components of the universe. Although most of the viable modifications of GR are constructed in order to provide an identical cosmic expansion history to the standard CDM model, their effects on the growth of density perturbations could lead to observationally testable predictions capable of distinguishing modified gravity models from standard GR plus a cosmological constant.
Since a modification of the theory of gravity would affect all test masses in the universe, i.e., including the standard baryonic matter, an asymptotic recovery of GR for solar system environments, where deviations from GR are tightly constrained, is required for all viable modified gravity models. Such mechanism, often referred to as the “Chameleon effect”, represents the main difference between modified gravity models and the interacting dark-energy scenarios discussed above, by determining a local dependence of the modified gravitational laws in the Newtonian limit.
While the linear growth of density perturbations in the context of modified gravity theories can be studied [see, e.g., 456, 674, 32, 54] by parametrizing the scale dependence of the modified Poisson and Euler equations in Fourier space (see the discussion in Section 1.3), the nonlinear evolution of the “Chameleon effect” makes the implementation of these theories into nonlinear -body algorithms much more challenging. For this reason, very little work has been done so far in this direction. A few attempts to solve the modified gravity interactions in the nonlinear regime by means of mesh-based iterative relaxation schemes have been carried out by [700, 701, 800, 500, 981, 281, 964] and showed an enhancement of the power spectrum amplitude at intermediate and small scales. These studies also showed that this nonlinear enhancement of small scale power cannot be accurately reproduced by applying the linear perturbed equations of each specific modified gravity theory to the standard nonlinear fitting formulae [as, e.g., 844].
Higher resolution simulations and new numerical approaches will be necessary in order to extend these first results to smaller scales and to accurately evaluate the deviations of specific models of modified gravity from the standard GR predictions to a potentially detectable precision level.
A popular analytical approach to study nonlinear clustering of dark matter without recurring to -body simulations is the spherical collapse model, first studied by . In this approach, one studies the collapse of a spherical overdensity and determines its critical overdensity for collapse as a function of redshift. Combining this information with the extended Press–Schechter theory ([743, 147]; see  for a review) one can provide a statistical model for the formation of structures which allows to predict the abundance of virialized objects as a function of their mass. Although it fails to match the details of -body simulations, this simple model works surprisingly well and can give useful insigths into the physics of structure formation. Improved models accounting for the complexity of the collapse exist in the literature and offer a better fit to numerical simulations. For instance,  showed that a significant improvement can be obtained by considering an ellipsoidal collapse model. Furthermore, recent theoretical developments and new improvements in the excursion set theory have been undertaken by  and other authors (see e.g., ).
The spherical collapse model has been generalized to include a cosmological constant by [718, 948].  have used it to study the observational consequences of a cosmological constant on the growth of perturbations. The case of standard quintessence, with speed of sound , have been studied by . In this case, scalar fluctuations propagate at the speed of light and sound waves maintain quintessence homogeneous on scales smaller than the horizon scale. In the spherical collapse pressure gradients maintain the same energy density of quintessence between the inner and outer part of the spherical overdensity, so that the evolution of the overdensity radius is described by
In the following we will discuss the spherical collapse model in the contest of other dark energy and modified gravity models.
In its standard version, quintessence is described by a minimally-coupled canonical field, with speed of sound . As mentioned above, in this case clustering can only take place on scales larger than the horizon, where sound waves have no time to propagate. However, observations on such large scales are strongly limited by cosmic variance and this effect is difficult to observe. A minimally-coupled scalar field with fluctuations characterized by a practically zero speed of sound can cluster on all observable scales. There are several theoretical motivations to consider this case. In the limit of zero sound speed one recovers the Ghost Condensate theory proposed by  in the context of modification of gravity, which is invariant under shift symmetry of the field . Thus, there is no fine tuning in assuming that the speed of sound is very small: quintessence models with vanishing speed of sound should be thought of as deformations of this particular limit where shift symmetry is recovered. Moreover, it has been shown that minimally-coupled quintessence with an equation of state can be free from ghosts and gradient instabilities only if the speed of sound is very tiny, . Stability can be guaranteed by the presence of higher derivative operators, although their effect is absent on cosmologically relevant scales [260, 228, 259].
The fact that the speed of sound of quintessence may vanish opens up new observational consequences. Indeed, the absence of quintessence pressure gradients allows instabilities to develop on all scales, also on scales where dark matter perturbations become nonlinear. Thus, we expect quintessence to modify the growth history of dark matter not only through its different background evolution but also by actively participating to the structure formation mechanism, in the linear and nonlinear regime, and by contributing to the total mass of virialized halos.
Following , in the limit of zero sound speed pressure gradients are negligible and, as long as the fluid approximation is valid, quintessence follows geodesics remaining comoving with the dark matter (see also  for a more recent model with identical phenomenology). In particular, one can study the effect of quintessence with vanishing sound speed on the structure formation in the nonlinear regime, in the context of the spherical collapse model. The zero speed of sound limit represents the natural counterpart of the opposite case . Indeed, in both cases there are no characteristic length scales associated with the quintessence clustering and the spherical collapse remains independent of the size of the object (see [95, 671, 692] for a study of the spherical collapse when of quintessence is small but finite).
Due to the absence of pressure gradients quintessence follows dark matter in the collapse and the evolution of the overdensity radius is described by
Quintessence with zero speed of sound modifies dark matter clustering with respect to the smooth quintessence case through the linear growth function and the linear threshold for collapse. Indeed, for (), it enhances (diminishes) the clustering of dark matter, the effect being proportional to . The modifications to the critical threshold of collapse are small and the effects on the dark matter mass function are dominated by the modification on the linear dark matter growth function. Besides these conventional effects there is a more important and qualitatively new phenomenon: quintessence mass adds to the one of dark matter, contributing to the halo mass by a fraction of order . Importantly, it is possible to show that the mass associated with quintessence stays constant inside the virialized object, independently of the details of virialization. Moreover ,the ratio between the virialization and the turn-around radii is approximately the same as the one for CDM computed by . In Figure 5 we plot the ratio of the mass function including the quintessence mass contribution, for the case to the smooth case. The sum of the two effects is rather large: for values of still compatible with the present data and for large masses the difference between the predictions of the and the cases is of order one.
We now consider spherical collapse within coupled dark-energy cosmologies. The presence of an interaction that couples the cosmon dynamics to another species introduces a new force acting between particles (CDM or neutrinos in the examples mentioned in Section 1.4.4) and mediated by dark-energy fluctuations. Whenever such a coupling is active, spherical collapse, whose concept is intrinsically based on gravitational attraction via the Friedmann equations, has to be suitably modified in order to account for other external forces. As shown in  the inclusion of the fifth force within the spherical collapse picture deserves particular caution. Here we summarize the main results on this topic and we refer to  for a detailed illustration of spherical collapse in presence of a fifth force.
If CDM is coupled to a quintessence scalar field as described in Sections 1.4.4 and 2.11 of the present document, the full nonlinear evolution equations within the Newtonian limit read:spherical collapse picture and can be treated within simulations [77, 565, 76, 562, 75]. We stress that it is crucial to include these additional terms in the equations, as derived from the nonlinear equations, in order to correctly account for the presence of a fifth force. The outlined procedure can easily be generalized to include uncoupled components, for example baryons. In this case, the corresponding evolution equation for , will be fed by . This yields an evolution equation for the uncoupled scale factor that is equivalent to the standard Friedmann equation. In Figure 6 we show the linear density contrast at collapse for three coupled quintessence models with and , , .
An increase of results in an increase of . As shown in , is well described by a simple quadratic fitting formula,
If a coupling between dark energy and neutrinos is present, as described in Sections 1.4.4 and 2.9, bound neutrino structures may form within these models . It was shown in  that their formation will only start after neutrinos become non-relativistic. A nonlinear treatment of the evolution of neutrino densities is thus only required for very late times, and one may safely neglect neutrino pressure as compared to their density. The evolution equations (1.6.16) and (1.6.17) can then also be applied for the nonlinear and linear neutrino density contrast. The extrapolated linear density at collapse for growing neutrino quintessence reflects in all respects the characteristic features of this model and results in a which looks quite different from standard dark-energy cosmologies. We have plotted the dependence of on the collapse redshift in Figure 7 for three values of the coupling. The oscillations seen are the result of the oscillations of the neutrino mass caused by the coupling to the scalar field: the latter has characteristic oscillations as it approaches the minimum of the effective potential in which it rolls, given by a combination of the self-interaction potential and the coupling contribution . Furthermore, due to the strong coupling , the average value of is found to be substantially higher than 1.686, corresponding to the Einstein de Sitter value, shown in black (double-dashed) in Figure 7. Such an effect can have a strong impact on structure formation and on CMB . For the strongly coupled models, corresponding to a low present day neutrino mass , the critical density at collapse is only available for , for , , respectively. This is again a reflection of the late transition to the non-relativistic regime. Nonlinear investigations of single lumps beyond the spherical collapse picture was performed in [963, 179], the latter showing the influence of the gravitational potentials induced by the neutrino inhomogeneities on the acoustic oscillations in the baryonic and dark-matter spectra.
A convenient way to parametrize the presence of a nonnegligible homogeneous dark-energy component at early times was presented in  and has been illustrated in Section 1.2.1 of the present review. If we specify the spherical collapse equations for this case, the nonlinear evolution of the density contrast follows the evolution equations (1.6.16) and (1.6.17) without the terms related to the coupling. As before, we assume relativistic components to remain homogeneous. In Figure 8 we show for two models of early dark energy, namely model I and II, corresponding to the choices () and () respectively. Results show () [368, 962].
Living Rev. Relativity 16, (2013), 6
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