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 1.4.4.4) 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 [956] 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 [406] and [367], 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 [857
] 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 [406], 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 [79
] of the TreePM hydrodynamic
-body code Gadget-2
[857].
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 [561] have broadly confirmed these results.
The analysis has been extended to the case of non-constant coupling functions by [76
], 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 [668], 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 [963] 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 [727].
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 [413]. 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 [976
] 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, [823] 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 [609
] and other authors (see e.g.,
[821]).
The spherical collapse model has been generalized to include a cosmological constant by [718, 948]. [540]
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 [937
]. 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 [56
] 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 [258], 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 [574] 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
where the energy density of quintessence 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 [540
]. 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 [962] 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 [962
] 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:
These equations can be derived from the non-relativistic Navier–Stokes equations and from the Bianchi identities written in presence of an external source of the type: where An increase of results in an increase of
. As shown in [962
],
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 [180]. It was shown in [668] 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 [727
]. 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 [956] 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
].
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