Spherical Collapse Model Research Articles

Mass scaling relations for dark halos from an analytic universal outer density profile

Context. The average matter density within the turnaround scale, which demarcates where galaxies shift from clustering around a structure to joining the expansion of the Universe, is an important cosmological probe. However, a measurement of the mass enclosed by the turnaround radius is difficult. Analyses of the turnaround scale in simulated galaxy clusters place the turnaround radius at about three times the virial radius in a ΛCDM universe and at a (present-day) density contrast with the background matter density of the Universe of δ ~ 11. Assessing the mass at such extended distances from a cluster’s center is a challenge for current mass measurement techniques. Consequently, there is a need to develop and validate new mass-scaling relations, to connect observable masses at cluster interiors with masses at greater distances. Aims. Our research aims to establish an analytical framework for the most probable mass profile of galaxy clusters, leading to novel mass scaling relations, allowing us to estimate masses at larger scales. We derive such analytical mass profiles and compare them with those from cosmological simulations. Methods. We used excursion set theory, which provides a statistical framework for the density and local environment of dark matter halos, and complement it with the spherical collapse model to follow the non-linear growth of these halos. Results. The profile we developed analytically showed good agreement (better than 30%, and dependent on halo mass) with the mass profiles of simulated galaxy clusters. Mass scaling relations were obtained from the analytical profile with offset better than 15% from the simulated ones. This level of precision highlights the potential of our model for probing structure formation dynamics at the outskirts of galaxy clusters.

Magnetohydrodynamical modeling of star-disk formation: from isolated spherical collapse towards incorporation of external dynamics

The formation of protostars and their disks has been understood as the result of the gravitational collapse phase of an accumulation of dense gas that determines the mass reservoir of the star-disk system. Against this background, the broadly applied scenario of considering the formation of disks has been to model the collapse of a dense core assuming spherical symmetry. Our understanding of the formation of star-disk systems is currently undergoing a reformation though. The picture evolves from interpreting disks as the sole outcome of the collapse of an isolated prestellar core to a more dynamic picture where disks are affected by the molecular cloud environment in which they form. In this review, we provide a status report of the state-of-the-art of spherical collapse models that are highly advanced in terms of the incorporated physics together with constraints from models that account for the possibility of infall onto star-disk systems in simplified test setups, as well as in multi-scale simulations that cover a dynamical range from the Giant Molecular Cloud environment down to the disk. Considering the observational constraints that favor a more dynamical picture of star formation, we finally discuss the challenges and prospects in linking the efforts of tackle the problem of star-disk formation in combined multi-scale, multi-physics simulations.

The density of virialized clusters as a probe of dark energy

ABSTRACT We use the spherical collapse model to demonstrate that the observable average density of virialized clusters depends on the properties of dark energy along with the properties of gravity on cluster scales and can therefore be used as a probe of these properties. As an application of this approach, we derive the predicted virialized densities and radii of cluster mass structures for a wide range of values of the cosmological constant (including negative values) as a function of the turnaround redshift. For the value of ΩΛ,0 = −0.7 (with Ωm,0 = 0.3), we find an amplification of the density of virialized clusters which can be as large as 80 per cent compared to Planck18/lambda Cold Dark Matter (ΛCDM) for a turnaround redshift zmax ≳ 2. Such an amplification may lead to more efficient early galaxy formation in this class of models, in accordance with the recent findings of JWST, which may be partially pertinent to the Λ sign switching models (ΛsCDM), which have been suggested as potential solutions to the observed Hubble and S8 discrepancies.

The cosmic web from perturbation theory

Context. Analysing the large-scale structure (LSS) in the Universe with galaxy surveys demands accurate structure formation models. Such models should ideally be fast and have a clear theoretical framework in order to rapidly scan a variety of cosmological parameter spaces without requiring large training data sets. Aims. This study aims to extend Lagrangian perturbation theory (LPT), including viscosity and vorticity, to reproduce the cosmic evolution from dark matter N-body calculations at the field level. Methods. We extend LPT to a Eulerian framework, which we dub eALPT. An ultraviolet regularisation through the spherical collapse model provided by Augmented LPT turns out to be crucial at low redshifts. This iterative method enables modelling of the stress tensor and introduces vorticity. The eALPT model has two free parameters apart from the choice of cosmology, redshift snapshots, cosmic volume, and the number of particles. Results. We find that compared to N-body solvers, the cross-correlation of the dark matter distribution increases at k = 1 h Mpc−1 and z = 0 from ∼55% with the Zel’dovich approximation (∼70% with ALPT), to ∼95% with the three-timestep eALPT, and the power spectra show percentage accuracy up to k ≃ 0.3 h Mpc−1.

Identification of Galaxy Protoclusters Based on the Spherical Top-hat Collapse Theory

We propose a new method for finding galaxy protoclusters that is motivated by structure formation theory and also directly applicable to observations. We adopt the conventional definition that a protocluster is a galaxy group whose virial mass M vir < M cl at its epoch, where M cl = 1014 M ⊙, but would exceed that limit when it evolves to z = 0. We use the critical overdensity for complete collapse at z = 0 predicted by the spherical top-hat collapse model to find the radius and total mass of the regions that would collapse at z = 0. If the mass of a region centered at a massive galaxy exceeds M cl, the galaxy is at the center of a protocluster. We define the outer boundary of a protocluster as the zero-velocity surface at the turnaround radius so that the member galaxies are those sharing the same protocluster environment and showing some conformity in physical properties. We use the cosmological hydrodynamical simulation Horizon Run 5 (HR5) to calibrate this prescription and demonstrate its performance. We find that the protocluster identification method suggested in this study is quite successful. Its application to the high-redshift HR5 galaxies shows a tight correlation between the mass within the protocluster regions identified according to the spherical collapse model and the final mass to be found within the clusters at z = 0, meaning that the regions can be regarded as the bona fide protoclusters with high reliability. We also confirm that the redshift-space distortion does not significantly affect the performance of the protocluster identification scheme.

Dark matter halo mass functions and density profiles from mass and energy cascade

Halo abundance and structure play a central role for modeling structure formation and evolution. Without relying on a spherical or ellipsoidal collapse model, we analytically derive the halo mass function and cuspy halo density (inner slope of −4/3) based on the mass and energy cascade theory in dark matter flow. The hierarchical halo structure formation leads to halo or particle random walk with a position-dependent waiting time tau _g. First, the inverse mass cascade from small to large scales leads to the halo random walk in mass space with tau _gpropto m_h^{-lambda }, where m_h is the halo mass and lambda is a halo geometry parameter with predicted value of 2/3. The corresponding Fokker-Planck solution for halo random walk in mass space gives rise to the halo mass function with a power-law behavior on small scale and exponential decay on large scale. This can be further improved by considering two different lambda for haloes below and above a critical mass scale m_h^*, i.e. a double-lambda halo mass function. Second, a double-gamma density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time tau _g propto Phi (r)^{-1} propto r^{-gamma }, where Phi is the gravitational potential and r is the particle distance to halo center. Theory predicts gamma =2/3 that leads to a cuspy density profile with an inner slope of −4/3, consistent with the predicted scaling laws from energy cascade. The Press-Schechter mass function and Einasto density profile are just special cases of proposed models. The small scale permanence can be identified due to the scale-independent rate of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the -4/3 scaling law (rho _h propto r^{-4/3}) on small scales. Theory predicts the halo number density scales with halo mass as propto m_h^{-1.9}, while the halo mass density scales as propto m_h^{4/9}. Results were compared against the Illustris simulations. This new perspective provides a theory for nearly universal halo mass functions and density profiles.

Machine learning and structure formation in modified gravity

ABSTRACT In general relativity, approximations based on the spherical collapse model such as Press–Schechter theory and its extensions are able to predict the number of objects of a certain mass in a given volume. In this paper, we use a machine learning algorithm to test whether such approximations hold in screened modified gravity theories. To this end, we train random forest classifiers on data from N-body simulations to study the formation of structures in lambda cold dark matter (ΛCDM) as well as screened modified gravity theories, in particular f(R) and nDGP gravity. The models are taught to distinguish structure membership in the final conditions from spherical aggregations of density field behaviour in the initial conditions. We examine the differences between machine learning models that have learned structure formation from each gravity, as well as the model that has learned from ΛCDM. We also test the generalizability of the ΛCDM model on data from f(R) and nDGP gravities of varying strengths, and therefore the generalizability of extended Press–Schechter spherical collapse to these types of modified gravity.

Voids and haloes-in-voids statistics as a probe of the expansion history of the Universe

ABSTRACT Structures in the Universe are arranged into the cosmic web. Distributions, statistics, and evolutions of the structures can be used as probes for cosmological models. We investigate the number density of voids and dark matter haloes-in-voids in the excursion set theory (EST). We study the Markov and non-Markov frameworks of EST in both spherical and ellipsoidal collapse models. Afterward, we compare the number density of voids and haloes-in-voids in the standard lambda cold dark matter and the reconstructed model. The reconstructed model is a model-independent reconstruction based on background observations. This work explores the effects of the collapse model barrier in the different EST frameworks on the statistics of voids and the statistics of haloes-in-voids. Finally, we find the hint that cosmological models can be distinguished by the number density of haloes-in-voids in the 1.0−2.5 redshift range. The maximum difference is observed in z ∼ 1.9.

Spherical collapse of non-top-hat profiles in the presence of darkenergy with arbitrary sound speed

We study the spherical collapse of non-top-hat matter fluctuations in the presence of dark energy with arbitrary sound speed. The model is described by a system of partial differential equations solved using a pseudo-spectral method with collocation points. This method can reproduce the known analytical solutions in the linear regime with an accuracy better than 10-6% and better than 10-2% for the virialization threshold given by the usual spherical collapse model. We show the impact of nonlinear dark energy fluctuations on matter profiles, matter peculiar velocity and gravitational potential. We also show that phantom dark energy models with low sound speed can develop a pathological behaviour around matter halos, namely negative energy density. The dependence of the virialization threshold density for collapse on the dark energy sound speed is also computed, confirming and extending previous results in the limit for homogeneous and clustering dark energy.

The evolution of high-density cores of the BOSS Great Wall superclusters

Context.High-density cores (HDCs) of galaxy superclusters that embed rich clusters and groups of galaxies are the earliest large objects to form in the cosmic web, and the largest objects that may collapse in the present or future.Aims.We aim to study the dynamical state and possible evolution of the HDCs in the BOSS Great Wall (BGW) superclusters at redshiftz ≈ 0.5 from the CMASS (constant mass) galaxy sample, based on the Baryon Oscillation Spectroscopic Survey (BOSS) in order to understand the growth and evolution of structures in the Universe.Methods.We analysed the luminosity density distribution in the BGW superclusters to determine the HDCs in them. We derived the density contrast values for the spherical collapse model in a wide range of redshifts and used these values to study the dynamical state and possible evolution of the HDCs of the BGW superclusters. The masses of the HDCs were calculated using stellar masses of galaxies in them. We found the masses and radii of the turnaround and future collapse regions in the HDCs of the BGW superclusters and compared them with those of local superclusters.Results.We determined eight HDCs in the BGW superclusters. The masses of their turnaround regions are in the range ofMT ≈ 0.4–3.3 × 1015 h−1 M⊙, and radii are in the range ofRT ≈ 3.5–7 h−1Mpc. The radii of their future collapse regions are in the range ofRFC ≈ 4–8 h−1Mpc. Distances between individual cores in superclusters are much larger: of the order of 25–35 h−1Mpc. The richness and sizes of the HDCs are comparable with those of the HDCs of the richest superclusters in the local Universe.Conclusions.The BGW superclusters will probably evolve to several poorer superclusters with masses similar to those of the local superclusters. This may weaken the tension with the ΛCDM model, which does not predict a large number of very rich and large superclusters in our local cosmic neighbourhood, and explains why there are no superclusters as elongated as those in the BGW in the local Universe.

Analysis of cosmological bias within spherical collapse model

The goal of our research work is to analyze cosmological bias parameter. Parametric equations of spherical collapse model are used to calculate the values of spherical collapse over density and mass variance, which is further used in bias formulae to find the values of cosmological bias. Spherical collapse over density has been calculated in the range of redshift 0 to 1. Also, it is compared with the value according to the spherical collapse model. Bias is one of the parameters which are utilized to infer cosmological parameters. Extracting the cosmological parameters is very much useful to know and understand about the birth and evolution of our universe. As there is no direct probe to get the idea about the existence of dark matter. Bias factor helps to analyze about dark matter. The bias coefficient of higher order terms in Taylor series expansion are found to be in ascending order. Increasing values of bias indicate the large-scale structure formation at current epoch is more and more clustered. Values of bias are discussed in result. Also, bias values have been analyzed for redshift in the range 2 to 0. The graph has been plotted bias versus redshift. Let’s found bias decreases with decrease of redshift. That means bias evolves with redshift. Bias value less than one and negative value of bias implies that structure formation is in linear region and higher values of bias indicates the structure formation occurs in nonlinear region. Negative value of bias is also called as antibias. That means the structure formation has not started yet. It is still in linear region. The bias value nearly equal to one indicates that the structure formation has been transformed from linear region to nonlinear region. So, the result showing bias values greater than one indicates that evolution of structure formation occurs in nonlinear region.

Strong cosmic censorship conjecture at the dust black hole collapse

The cosmic censorship conjecture proposed by Penrose states that the singularity of a black hole is covered by the event horizon and is not naked. The first examples of violation of this conjecture are from the models of the spherical collapse of the perfect fluid black hole in the dust state. In this article, we examine an important class of LTB spherical dust collapse models in which there is a naked singularity. We will first show that the singularity of a black hole will be spacelike. We then show that the event horizon will be outside the singularity. We will also show that if the energy condition is met, the famous category of these models will not have a naked singularity. We will conclude that the strong cosmic censorship conjecture will be held

Tidal virialization of dark matter haloes with clustering dark energy

We extend the analysis of Pace et al. [1] by considering the virialization process in the extended spherical collapse model for clustering dark-energy models, i.e., accounting for dark-energy fluctuations. Differently from the standard approach, here virialization is naturally achieved by properly modelling deviations from sphericity due to shear and rotation induced by tidal interactions. We investigate the time evolution of the virial overdensity Δvir in seven clustering dynamical dark energy models and compare the results to the ΛCDM model and to the corresponding smooth dark-energy models.Taking into account all the appropriate corrections, we deduce the abundance of convergence peaks for Rubin Observatory-LSST and Euclid-like weak-lensing surveys, of Sunyaev-Zel'dovich peaks for a Simon Observatory-like CMB survey, and of X-ray peaks for an eROSITA-like survey. Despite the tiny differences in Δvir between clustering and smooth dark-energy models, owing to the large volumes covered by these surveys, five out of seven clustering dark-energy models can be statistically distinguished from ΛCDM. The contribution of dark-energy fluctuation cannot be neglected, especially for the Chevallier-Polarski-Limber and Albrecht-Skordis models, provided the instrumental configurations provide high signal-to-noise ratio. These results are almost independent of the tidal virialization model.

Bianchi IX gravitational collapse of matter inhomogeneities

We investigate a model of gravitational collapse of matter inhomogeneities where the latter are modelled as Bianchi type IX (BIX) spacetimes. We found that this model contains, as limiting cases, both the standard spherical collapse model and the Zeldovich solution. We study how these models are affected by small anisotropies within the BIX potential. For the spherical collapse case, we found that the model is equivalent to a closed FLRW Universe filled with matter and two perfect fluids representing the anisotropic contributions. From the linear evolution up to the turnaround, the anisotropies effectively shift the value of the FLRW spatial curvature, because the fluids have effective Equation of State (EoS) parameters w ≈ -1/3. Then we estimate the impact of such anisotropies on the number density of haloes using the Press-Schechter formalism.If a fluid description of the anisotropies is still valid after virialization, the averaged over time EoS parameters are w ≈ 1/3. Using this and demanding hydrostatic equilibrium, we find a relation between the mass M, the average radius R and the pressure p of the virialized final structure.When we consider within the BIX ansatz small deviations from the Zeldovich solution, our qualitative analysis suggests that the so called pancakes exhibit oscillatory behavior, as would be expected in the case of a vacuum BIX spacetime.

A Short Review on Clustering Dark Energy

We review dark energy models that can present non-negligible fluctuations on scales smaller than Hubble radius. Both linear and nonlinear evolutions of dark energy fluctuations are discussed. The linear evolution has a well-established framework, based on linear perturbation theory in General Relativity, and is well studied and implemented in numerical codes. We highlight the main results from linear theory to explain how dark energy perturbations become important on the scales of interest for structure formation. Next, we review some attempts to understand the impact of clustering dark energy models in the nonlinear regime, usually based on generalizations of the Spherical Collapse Model. We critically discuss the proposed generalizations of the Spherical Collapse Model that can treat clustering dark energy models and their shortcomings. Proposed implementations of clustering dark energy models in halo mass functions are reviewed. We also discuss some recent numerical simulations capable of treating dark energy fluctuations. Finally, we summarize the observational predictions based on these models.

Evolution of splashback boundaries and gaseous outskirts: insights from mergers of self-similar galaxy clusters

ABSTRACT A self-similar spherical collapse model predicts a dark matter (DM) splashback and accretion shock in the outskirts of galaxy clusters while missing a key ingredient of structure formation – processes associated with mergers. To fill this gap, we perform simulations of merging self-similar clusters and investigate their DM and gas evolution in an idealized cosmological context. Our simulations show that the cluster rapidly contracts during the major merger and the splashback radius rsp decreases, approaching the virial radius rvir. While in the self-similar model rsp depends on a smooth mass accretion rate parameter Γs, our simulations show that in the presence of mergers, rsp responds to the changes in the total mass accretion rate Γvir, which accounts for both mergers and smooth accretion. The scatter of the Γvir − rsp/rvir relation indicates a generally low Γs ∼ 1 in clusters in cosmological simulations. In contrast to the DM, the hot gaseous atmospheres significantly expand by the merger-accelerated (MA-) shocks formed when the runaway merger shocks overtake the outer accretion shock. After a major merger, the MA-shock radius is larger than rsp by a factor of up to ∼1.7 for Γs ≲ 1 and is ∼rsp for Γs ≳ 3. This implies that (1) mergers could easily generate the MA-shock-splashback offset measured in cosmological simulations, and (2) the smooth mass accretion rate is small in regions away from filaments where MA-shocks reside. We further discuss the shapes of the DM haloes, various shocks, and contact discontinuities formed at different epochs of the merger, and the ram-pressure stripping in cluster outskirts.

The Corona Borealis supercluster: connectivity, collapse, and evolution

Context. Rich superclusters of galaxies represent dynamically active environments in which galaxies and their systems form and evolve. Aims. We study the dynamical properties and connectivity of the richest galaxy clusters in the Corona Borealis (CB) supercluster and of the whole supercluster, and analyse star formation of galaxies in them with the aim to understand the evolution of the supercluster and the galaxies within it. We compare it with the supercluster SCl A2142. Methods. We used the luminosity-density field to determine the high-density cores of the CB. We identified the richest galaxy clusters in them and studied the dynamical state of the clusters, analysed their substructure, and studied the star formation properties of galaxies in them using normal mixture modelling and the projected phase space diagram. We determined filaments in the supercluster to analyse the connectivity of clusters. To understand the possible future evolution of the CB, we compared the mass distribution in it with predictions from the spherical collapse model and analysed the gravitational acceleration field in the CB. Results. The richest clusters in the high-density cores of the CB are the Abell clusters A2065, A2061 (together with A2067), A2089, and Gr2064. At a radius R30 around each cluster (corresponding to the density contrast Δρ ≈ 30), the galaxy distribution shows a minimum. The R30 values for individual clusters lie in the range of 3 − 6 h−1 Mpc. The radii of the clusters (splashback radii) lie in the range of Rcl ≈ 2 − 3 Rvir. The projected phase space diagrams and the comparison with the spherical collapse model suggest that R30 regions have passed turnaround and are collapsing, forming infall regions around each cluster. Galaxies in the richest cluster of the CB, A2065, and in its infall region have on average younger stellar populations than other clusters and their environment. The cluster A2061 has the highest fraction of galaxies with very old stellar populations, similar to those in A2142. The number of long filaments that begin near clusters vary from one near A2089 to five near A2061. The total connectivity of these clusters (the number of infalling groups and filaments) varies from two to nine. Conclusions. During the future evolution, the clusters in the main part of the CB may merge and form one of the largest bound systems in the nearby Universe. Another part, with the cluster Gr2064, will form a separate system. Our study suggests that structures with a current characteristic density contrast Δρ ≈ 30 have passed turnaround and started to collapse at redshifts z ≈ 0.3 − 0.4. The comparison of the number and properties of the most massive collapsing supercluster cores from observations and simulations may serve as a test for cosmological models.

Axion miniclusters made easy

We use a modified version of the peak patch excursion set formalism to compute the mass and size distribution of QCD axion miniclusters from a fully non-Gaussian initial density field obtained from numerical simulations of axion string decay. We find strong agreement with $N$-body simulations at significantly lower computational cost. We employ a spherical collapse model, and provide fitting functions for the modified barrier in the radiation era. The halo mass function at $z=629$ has a power-law distribution ${M}^{\ensuremath{-}0.6}$ for masses within the range ${10}^{\ensuremath{-}15}\ensuremath{\lesssim}M\ensuremath{\lesssim}{10}^{\ensuremath{-}10}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$, with all masses scaling as $({m}_{a}/50\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}{)}^{\ensuremath{-}0.5}$. We construct merger trees to estimate the collapse redshift and concentration mass relation, $C(M)$, which is well described using analytical results from the initial power spectrum and linear growth. Using the calibrated analytic results to extrapolate to $z=0$, our method predicts a mean concentration $C\ensuremath{\sim}\mathcal{O}(\mathrm{few})\ifmmode\times\else\texttimes\fi{}{10}^{4}$. The low computational cost of our method makes future investigation of the statistics of rare, dense miniclusters easy to achieve.

Effects of Rastall parameter on perturbation of dark sectors of the Universe

In recent years, Rastall gravity is undergoing a considerable surge in popularity. This theory purports to be a modified gravity theory with a non-conserved energy–momentum tensor (EMT) and an unusual non-minimal coupling between matter and geometry. This work looks for the evolution of homogeneous spherical perturbations within the Universe in the context of Rastall gravity. Using the spherical Top-Hat collapse model, we seek for exact solutions in linear regime for density contrast of dark matter (DM) and dark energy (DE). We find that the Rastall parameter affects crucially the dynamics of density contrasts for DM and DE and the fate of spherical collapse is different in comparison to the case of general relativity (GR). Numerical solutions for perturbation equations in nonlinear regime reveal that DE perturbations could amplify the rate of growth of DM perturbations depending on the values of Rastall parameter.

Mass Inflation and the $$C^2$$-inextendibility of Spherically Symmetric Charged Scalar Field Dynamical Black Holes

It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein-Maxwell-Klein-Gordon equations and \textit{while we do not directly show mass inflation}, we obtain a "mass inflation/ridigity" dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar field on the event horizon, that the Cauchy horizon emanating from time-like infinity is $\mathcal{CH}_{i^+}= \mathcal{D} \cup \mathcal{S}$ for two (possibly empty) disjoint connected sets $\mathcal{D}$ and $\mathcal{S}$ such that: _$\mathcal{D}$ (the dynamical set) is a past set on which the Hawking mass blows up (mass inflation scenario). _$\mathcal{S}$ (the static set) is a future set isometric to a Reissner--Nordstr\"{o}m Cauchy horizon i.e.\ the radiation is zero on $\mathcal{S}$. As a consequence, we establish a novel classification of Cauchy horizons into three types: dynamical ($\mathcal{S}=\emptyset$), static ($\mathcal{D}=\emptyset$) or mixed, and prove that $\mathcal{CH}_{i^+}$ is globally $C^2$-inextendible. Our main motivation is the $C^2$ Strong Cosmic Censorship Conjecture for a realistic model of spherical collapse in which charged matter emulates the repulsive role of angular momentum: in our case the Einstein-Maxwell-Klein-Gordon system on one-ended space-times. As a result, we prove in spherical symmetry that: - two-ended asymptotically flat space-times are $C^2$-future-inextendible i.e. $C^2$ Strong Cosmic Censorship is true for Einstein-Maxwell-Klein-Gordon, assuming the decay of the scalar field on the event horizon at the expected rate. - In the one-ended case, the Cauchy horizon emanating from time-like infinity is $C^2$-inextendible. This result suppresses the main obstruction to $C^2$ Strong Cosmic Censorship in spherical collapse.