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Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are obtained, in the range of exponents $p>1$, $\sigma\ge-2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as $t\to\infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case $\sigma=-2$ we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher-KPP equation is derived and employed in order to deduce these properties.

The following Fisher-KPP type equation $$ u_t=Ku_{xx}-Bu^q+Au^p, \quad (x,t)\in\real\times(0,\infty), $$ with $p>q>0$ and $A$, $B$, $K$ positive coefficients, is considered. For both $p>q>1$ and $p>1$, $q=1$, we construct stationary solutions, establish their behavior as $|x|\to\infty$ and prove that they are separatrices between solutions decreasing to zero in infinite time and solutions presenting blow-up in finite time. We also establish decay rates for the solutions that decay to zero as $t\to\infty$.

We investigate the effect of primordial non-Gaussianities on halo number counts using N-body simulations with different values of $f_{\rm NL}^{\rm loc}$. We show how current theoretical models fail to adequately describe the non-Gaussian mass function of halos identified with different overdensity thresholds, $\Delta_{\rm b}$. We explain how these discrepancies are related to a variation in the density profile of dark matter halos, finding that the internal steepness (i.e. the compactness) of halos depends on the value of $f_{\rm NL}^{\rm loc}$. We then parametrize these deviations in halo number counts with a factor $\kappa(\Delta_{\rm b})$ that modifies the linear density threshold for collapse according to the halo identification threshold used, defined with respect to the Universe background density. We rely on a second-degree polynomial to describe $\kappa$ and employ a Bayesian analysis to determine the coefficients of this polynomial. In addition, we verify the independence of the latter on the sign and absolute value of $f_{\rm NL}^{\rm loc}$. Finally, we show how this re-parametrization prevents the extraction of biased constraints on $f_{\rm NL}^{\rm loc}$, correcting for large systematic errors especially in the case of halos identified with high density thresholds. This improvement is crucial in the perspective of deriving cosmological constraints with the non-Gaussian mass function from real data, as different mass definitions can be employed depending on the properties of the survey., Comment: 28 pages, 8 figures, to be submitted to JCAP

Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $\sigma=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-\alpha t}f(|x|e^{\beta t}), \qquad \alpha=\frac{2}{m-1}\beta,$$ it is shown that, for $m+q>2$, there exists a unique exponent $\beta_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.

We extend the evolution mapping approach, originally proposed by Sanchez (2022) to describe non-linear matter density fluctuations, to statistics of the cosmic velocity field. This framework classifies cosmological parameters into shape parameters, which determine the shape of the linear matter power spectrum, $P_L(k, z)$, and evolution parameters, which control its amplitude at any given redshift. Evolution mapping leverages the fact that density fluctuations in cosmologies with identical shape parameters but different evolution parameters exhibit remarkably similar non-linear evolutions when expressed as a function of the clustering amplitude. We use a suite of N-body simulations sharing identical shape parameters but spanning a wide range of evolution parameters. Using an efficient method for estimating the volume-weighted velocity field based on the Voronoi tesselation of the simulation particles, we study the non-linear evolution of the power spectra of the velocity divergence, $P_{\theta\theta}(k)$, and its cross-power spectrum with the density field, $P_{\delta\theta}(k)$. By analysing snapshots at redshifts where the linear matter perturbations have the same amplitude, we demonstrate that evolution mapping accurately applies to $P_{\theta\theta}(k)$ and $P_{\delta\theta}(k)$. Deviations at small scales can be modelled in terms of differences in the suppression factor, $g(a) = D(a)/a$, akin to those observed for the density field. Evolution mapping simplifies the description of the cosmological dependence of non-linear density and velocity statistics, streamlining the sampling of large cosmological parameter spaces for the analysis of cosmological observables., Comment: 10 pages, 4 figures, including an appendix. v2 has minor changes from peer review suggestions

We classify radially symmetric self-similar profiles presenting finite time blow-up to the quasilinear diffusion equation with weighted source $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, $T>0$, in dimension $N\geq1$ and in the range of exponents $-2<\sigma<\infty$, $1

We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=\Delta u^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always at least one solution in backward self-similar form if $p>p_s=m(N+2)/(N-2)$. In particular, this establishes \emph{non-optimality of the Lepin critical exponent} introduced in \cite{Le90} in the semilinear case $m=1$ and extended for $m>1$ in \cite{GV97, GV02}, for the existence of self-similar blow-up solutions. We also prove that there are multiple solutions in the same range, provided $N$ is sufficiently large. This is in strong contrast with the semilinear case, where the Lepin critical exponent has been proved to be optimal.

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,T)$, with exponents $10$, are established. More precisely, we identify the \emph{optimal class of initial conditions} $u_0$ for which (local in time) existence is ensured and prove \emph{non-existence of solutions} for the complementary set of data. We establish then (local in time) \emph{uniqueness and a comparison principle} for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem \emph{blows up in a finite time} $T\in(0,\infty)$ and \emph{finite speed of propagation} holds true for $t\in(0,T)$: if $u_0\in L^{\infty}(\mathbb{R}^N)$ is an initial condition with compact support and blow-up time $T>0$, then $u(t)$ is compactly supported for $t\in(0,T)$. We also establish in this work the \emph{absence of localization at the blow-up time} $T$ for solutions stemming from compactly supported data.

Combining galaxy clustering information from regions of different environmental densities can help break cosmological parameter degeneracies and access non-Gaussian information from the density field that is not readily captured by the standard two-point correlation function (2PCF) analyses. However, modelling these density-dependent statistics down to the non-linear regime has so far remained challenging. We present a simulation-based model that is able to capture the cosmological dependence of the full shape of the density-split clustering (DSC) statistics down to intra-halo scales. Our models are based on neural-network emulators that are trained on high-fidelity mock galaxy catalogues within an extended-$\Lambda$CDM framework, incorporating the effects of redshift-space, Alcock-Paczynski distortions and models of the halo-galaxy connection. Our models reach sub-percent level accuracy down to $1\,h^{-1}{\rm Mpc}$ and are robust against different choices of galaxy-halo connection modelling. When combined with the galaxy 2PCF, DSC can tighten the constraints on $\omega_{\rm cdm}$, $\sigma_8$, and $n_s$ by factors of 2.9, 1.9, and 2.1, respectively, compared to a 2PCF-only analysis. DSC additionally puts strong constraints on environment-based assembly bias parameters. Our code is made publicly available on Github., Comment: Submitted to MNRAS. Source code to generate the figures available in the captions. Updated to add missing references

We present a clustering analysis of the BOSS DR12 CMASS galaxy sample, combining measurements of the galaxy two-point correlation function and density-split clustering down to a scale of $1\,h^{-1}{\rm Mpc}$. Our theoretical framework is based on emulators trained on high-fidelity mock galaxy catalogues that forward model the cosmological dependence of the clustering statistics within an extended-$\Lambda$CDM framework, including redshift-space and Alcock-Paczynski distortions. Our base-$\Lambda$CDM analysis finds $\omega_{\rm cdm} = 0.1201\pm 0.0022$, $\sigma_8 = 0.792\pm 0.034$, and $n_s = 0.970\pm 0.018$, corresponding to $f\sigma_8 = 0.462\pm 0.020$ at $z \approx 0.525$, which is in agreement with Planck 2018 predictions and various clustering studies in the literature. We test single-parameter extensions to base-$\Lambda$CDM, varying the running of the spectral index, the dark energy equation of state, and the density of massless relic neutrinos, finding no compelling evidence for deviations from the base model. We model the galaxy-halo connection using a halo occupation distribution framework, finding signatures of environment-based assembly bias in the data. We validate our pipeline against mock catalogues that match the clustering and selection properties of CMASS, showing that we can recover unbiased cosmological constraints even with a volume 84 times larger than the one used in this study., Comment: Submitted to MNRAS. Source code to generate the figures available in the captions. Updated to add missing references and fix legend of Fig. 6

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$, $N\geq1$, are classified with respect to the exponents $m\in(0,1)$, $\sigma\in(-2,\infty)$ and $p>\max\{1,p_L(\sigma)\}$, where $$ p_L(\sigma)=1+\frac{\sigma(1-m)}{2}. $$ In the supercritical range $m_c=(N-2)_{+}/Np_L(\sigma)$, but do not exist for any $\sigma\geq0$ and $p\in(p_F(\sigma),p_s(\sigma))$. In the subcritical range $0

We present the Texas Euclid Survey for Lyman-Alpha (TESLA), a spectroscopic survey in the 10 square degree of the Euclid North Ecliptic Pole (NEP) field. Using TESLA, we study how the physical properties of Lyman-alpha emitters (LAEs) correlate with Lyman-alpha emission to understand the escape of Lyman alpha from galaxies at redshifts 2 -- 3.5. We present an analysis of 43 LAEs performed in the NEP field using early data from the TESLA survey. We use Subaru Hyper Suprime-Cam imaging in the grizy-bands, Spitzer/IRAC channels 1 and 2 from the Hawaii 20 square degree (H20) survey and spectra acquired by the Visible Integral-Field Replicable Unit Spectrograph (VIRUS) on the Hobby-Eberly Telescope. We perform spectral energy distribution (SED) fitting to compute the galaxy properties of 43 LAEs, and study correlations between stellar mass, star formation rate (SFR), and dust, to the Lyman-alpha rest-frame equivalent widths (EW). We uncover marginal (1 sigma significance) correlations between stellar mass and Lyman-alpha EW, and star formation rate (SFR) and Lyman-alpha EW, with a Spearman correlation coefficient of -0.$34_{-.14}^{+.17}$ and -0.$37_{-.14}^{+.16}$ respectively. We show that the Lyman-alpha distribution of the 43 LAEs is consistent with being drawn from an exponential distribution with an e-folding scale of 150 Angstrom. Once complete the TESLA survey will enable the study of ~ thousands of LAEs to explore correlations between galaxy properties and Lyman-alpha EW. The large sample size will allow the construction of a predictive model for the Lyman-alpha EW as a function of SED-derived galaxy properties, which could be used to improve Lyman-alpha based constraints on reionization.

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, where $m>1$, $p\in(0,1)$ and $\sigma>0$. Initial data are taken to be bounded, non-negative and compactly supported. In the range when $m+p\geq2$, we prove \emph{local existence of solutions} together with a \emph{finite speed of propagation} of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist \emph{infinitely many solutions} to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range $m+p<2$, we establish new \emph{Aronson-B\'enilan estimates} satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish \emph{infinite speed of propagation} of the supports of solutions if $m+p<2$, that is, $u(x,t)>0$ for any $x\in\real^N$, $t>0$, even in the case when the initial condition $u_0$ was compactly supported.

We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$, in the sub-critical range of the fast diffusion equation $00$ and $\max\{p_c(\sigma),1\}0$, in the form $$ u(x,t)=t^{\alpha}f(|x|t^{\beta}), \qquad f(\xi)\sim C\xi^{-(N-2)/m}, \qquad \alpha>0, \ \beta>0 $$ exist, provided $0

Some transformations acting on radially symmetric solutions to the following class of non-homogeneous reaction-diffusion equations $$ |x|^{\sigma_1}\partial_tu=\Delta u^m+|x|^{\sigma_2}u^p, \qquad (x,t)\in\real^N\times(0,\infty), $$ which has been proposed in a number of previous mathematical works as well as in several physical models, are introduced. We consider here $m\geq1$, $p\geq1$, $N\geq1$ and $\sigma_1$, $\sigma_2$ real exponents. We apply these transformations in connection to previous results on the one hand to deduce general qualitative properties of radially symmetric solutions and on the other hand to construct self-similar solutions which are expected to be patterns for the dynamics of the equations, strongly improving the existing theory. We also introduce mappings between solutions which work in the semilinear case $m=1$.

We analyse the full shape of anisotropic clustering measurements from the extended Baryon Oscillation Spectroscopic survey (eBOSS) quasar sample together with the combined galaxy sample from the Baryon Oscillation Spectroscopic Survey (BOSS). We obtain constraints on the cosmological parameters independent of the Hubble parameter $h$ for the extensions of the $\Lambda$CDM models, focusing on cosmologies with free dark energy equation of state parameter $w$. We combine the clustering constraints with those from the latest CMB data from Planck to obtain joint constraints for these cosmologies for $w$ and the additional extension parameters - its time evolution $w_{\rm{a}}$, the physical curvature density $\omega_{K}$ and the neutrino mass sum $\sum m_{\nu}$. Our joint constraints are consistent with flat $\Lambda$CDM cosmological model within 68\% confidence limits. We demonstrate that the Planck data are able to place tight constraints on the clustering amplitude today, $\sigma_{12}$, in cosmologies with varying $w$ and present the first constraints for the clustering amplitude for such cosmologies, which is found to be slightly higher than the $\Lambda$CDM value. Additionally, we show that when we vary $w$ and allow for non-flat cosmologies and the physical curvature density is used, Planck prefers a curved universe at $4\sigma$ significance, which is $\sim2\sigma$ higher than when using the relative curvature density $\Omega_{\rm{K}}$. Finally, when $w$ is varied freely, clustering provides only a modest improvement (of 0.021 eV) on the upper limit of $\sum m_{\nu}$., Comment: 12 pages, 6 figures, submitted to MNRAS

We prove existence and uniqueness of self-similar solutions with exponential form $$ u(x,t)=e^{\alpha t}f(|x|e^{-\beta t}), \qquad \alpha, \ \beta>0 $$ to the following quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $10$ if $u_0$ is compactly supported.

We present the results of a stellar-population analysis of Lyman-alpha emitting galaxies (LAES) in GOODS-N at 1.9 < z < 3.5 spectroscopically identified by the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). We provide a method for connecting emission-line detections from the blind spectroscopic survey to imaging counterparts, a crucial tool needed as HETDEX builds a massive database of ~1 million Lyman-alpha detections. Using photometric data spanning as many as 11 filters covering 0.4-4.5 microns from the Hubble and Spitzer Space Telescopes, we study the objects' global properties and explore which properties impact the strength of Lyman-alpha emission. We measure a median stellar mass of 0.8 (^+2.9_-0.5) x 10^9 Msol and conclude that the physical properties of HETDEX spectroscopically-selected LAEs are comparable to LAEs selected by previous deep narrow band studies. We find that stellar mass and star formation rate correlate strongly with the Lyman-alpha equivalent width. We then use a known sample of z>7 LAEs to perform a proto-study of predicting Lyman-alpha emission from galaxies in the Epoch of Reionization, finding agreement at the 1-sigma level between prediction and observation for the majority of strong emitters., Comment: 29 pages, 22 figures, Accepted to the Astrophysical Journal

In this paper we present COMET, a Gaussian process emulator of the galaxy power spectrum multipoles in redshift-space. The model predictions are based on one-loop perturbation theory and we consider two alternative descriptions of redshift-space distortions: one that performs a full expansion of the real- to redshift-space mapping, as in recent effective field theory models, and another that preserves the non-perturbative impact of small-scale velocities by means of an effective damping function. The outputs of COMET can be obtained at arbitrary redshifts (up to $z \sim 3$), for arbitrary fiducial background cosmologies, and for a large parameter space that covers the shape parameters $\omega_c$, $\omega_b$, and $n_s$, as well as the evolution parameters $h$, $A_s$, $\Omega_K$, $w_0$, and $w_a$. This flexibility does not impair COMET's accuracy, since we exploit an exact degeneracy between the evolution parameters that allows us to train the emulator on a significantly reduced parameter space. While the predictions are sped up by at least two orders of magnitude, validation tests reveal an accuracy of $0.1\,\%$ for the monopole and quadrupole ($0.3\,\%$ for the hexadecapole), or alternatively, better than $0.25\,\sigma$ for all three multipoles in comparison to statistical uncertainties expected for the Euclid survey with a tenfold increase in volume. We show that these differences translate into shifts in mean posterior values that are at most of the same size, meaning that COMET can be used with the same confidence as the exact underlying models. COMET is a publicly available Python package that also provides the tree-level bispectrum multipoles in redshift-space and Gaussian covariance matrices., Comment: 19 pages, 11 figures; for the COMET Python package, see https://gitlab.com/aegge/comet-emu; v2: added discussion on binning and discreteness effects, minor improvements to match published version

We report and analyse the presence of foregrounds in the cosmic microwave background (CMB) radiation associated to extended galactic halos. Using the cross correlation of Planck and WMAP maps and the 2MRS galaxy catalogue, we find that the mean temperature radial profiles around nearby galaxies at $cz\le 4500~\rm{km~s^{-1}}$ show a statistically significant systematic decrease of $\sim 15~\mu \rm{K}$ extending up to several galaxy radii. This effect strongly depends on the galaxy morphological type at scales within several tens of times the galaxy size, becoming nearly independent of galaxy morphology at larger scales. The effect is significantly stronger for the more extended galaxies, with galaxy clustering having a large impact on the results. Our findings indicate the presence of statistically relevant foregrounds in the CMB maps that should be considered in detailed cosmological studies. Besides, we argue that these can be used to explore the intergalactic medium surrounding bright late-type galaxies and allow for diverse astrophysical analyses., Comment: 10 pages, 9 figures, submitted to MNRAS

We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p0$.

Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=\Delta u^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p0$ and $p\in[1,\infty)$. As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition $u_0$, contrasting with previous results in literature for the critical limit $p=m$.

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma > 2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t > 0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed in dimension $N\geq2$, with $m>1$, $\sigma\in(-2,0)$ and $11$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^{\sigma}$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison.

We present the analysis of the full shape of anisotropic clustering measurement from the extended Baryon Oscillation Spectroscopic Survey (eBOSS) quasar sample together with the combined galaxy sample from the Baryon Oscillation Spectroscopic Survey (BOSS), re-analysed using an updated recipe for the non-linear matter power spectrum and the non-local bias parameters. We obtain constraints for flat $\Lambda$CDM cosmologies, focusing on the cosmological parameters that are independent of the Hubble parameter $h$. Our recovered value for the RMS linear perturbation theory variance as measured on the scale of $12\,{\rm Mpc}$ is $\sigma_{12}=0.805\pm 0.049$, while using the traditional reference scale of $8\,h^{-1}{\rm Mpc}$ gives $\sigma_{8}=0.815\pm 0.044$. We quantify the agreement between our measurements and the latest CMB data from Planck using the suspiciousness metric, and find them to be consistent within $0.64 \pm 0.03\sigma$. Combining our clustering constraints with the $3\times2$pt data sample from the Dark Energy Survey (DES) Year 1 release slightly degrades this agreement to the level of $1.54 \pm 0.08\sigma$, while still showing an overall consistency with Planck. We furthermore study the effect of imposing a Planck - like prior on the parameters that define the shape of the linear matter power spectrum, and find significantly tighter constraints on the parameters that control the evolution of density fluctuations. In particular, the combination of low-redshift data sets prefers a value of the physical dark energy density $\omega_{\rm DE}=0.335 \pm 0.011$, which is 1.7$\sigma$ higher than the one preferred by Planck., Comment: 14 pages, 10 figures, matches the version accepted by MNRAS

We present a new approach to describe statistics of the non-linear matter density field that exploits a degeneracy in the impact of different cosmological parameters on the linear dimensionless matter power spectrum, $\Delta^2_{\rm L}(k)$. We classify all cosmological parameters into two groups, shape parameters, which determine the shape of $\Delta^2_{\rm L}(k)$, and evolution parameters, which only affect its amplitude at any given redshift. With this definition, the time evolution of $\Delta^2_{\rm L}(k)$ in models with identical shape parameters but different evolution parameters can be mapped from one to the other by relabelling the redshifts that correspond to the same clustering amplitude, which we characterize by the linear mass fluctuation in spheres of radius $12\,{\rm Mpc}$, $\sigma_{12}(z)$. We use N-body simulations to show that the same evolution mapping relation gives a good description of the non-linear power spectrum, the halo mass function, or the full density field. The deviations from the exact degeneracy are the result of the different structure formation histories experienced by each model to reach the same clustering amplitude and can be accurately described in terms of differences in the suppression factor $g(a) = D(a)/a$. These relations can be used to drastically reduce the number of parameters required to describe the cosmology dependence of the power spectrum. We show how this can help to speed up the inference of parameter constraints from cosmological observations. We also present a new design of an emulator of the non-linear power spectrum whose predictions can be adapted to an arbitrary choice of evolution parameters and redshift., Comment: 13 pages, 11 figures, replaced to match version accepted by MNRAS

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0,1)$ and $\sigma>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are \emph{compactly supported} and might present two different types of interface behavior and three different possible \emph{good behaviors} near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of $\sigma$. This paper generalizes in dimension $N>1$ previous results by the authors in dimension $N=1$ and also includes some finer classification of the profiles for $\sigma$ large that is new even in dimension $N=1$.

This is the second part of a thorough investigation of the redshift-space effects that affect void properties and the impact they have on cosmological tests. Here, we focus on the void-galaxy cross-correlation function, specifically, on the projected versions that we developed in a previous work. The pillar of the analysis is the one-to-one relationship between real and redshift-space voids above the shot-noise level identified with a spherical void finder. Under this mapping, void properties are affected by three effects: (i) a systematic expansion as a consequence of the distortions induced by galaxy dynamics, (ii) the Alcock-Paczynski volume effect, which manifests as an overall expansion or contraction depending on the fiducial cosmology, and (iii) a systematic off-centring along the line of sight as a consequence of the distortions induced by void dynamics. We found that correlations are also affected by an additional source of distortions: the ellipticity of voids. This is the first time that distortions due to the off-centring and ellipticity effects are detected and quantified. With a simplified test, we verified that the Gaussian streaming model is still robust provided all these effects are taken into account, laying the foundations for improvements in current models in order to obtain unbiased cosmological constraints from spectroscopic surveys. Besides this practical importance, this analysis also encodes key information about the structure and dynamics of the Universe at the largest scales. Furthermore, some of the effects constitute cosmological probes by themselves, as is the case of the void ellipticity., Comment: 14 pages, 9 figures, published in MNRAS, accepted after minor comments

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 01, $$ and the critical value for the weight $$ \sigma=\frac{2(p-1)}{1-m}. $$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a \emph{change of sign of both self-similar exponents} at $m=m_s=(N-2)/(N+2)$, leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.

The construction of catalogues of a particular type of galaxy can be complicated by interlopers contaminating the sample. In spectroscopic galaxy surveys this can be due to the misclassification of an emission line; for example in the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX) low redshift [OII] emitters may make up a few percent of the observed Ly${\alpha}$ emitter (LAE) sample. The presence of contaminants affects the measured correlation functions and power spectra. Previous attempts to deal with this using the cross-correlation function have assumed sources at a fixed redshift, or not modelled evolution within the adopted redshift bins. However, in spectroscopic surveys like HETDEX, where the contamination fraction is likely to be redshift dependent, the observed clustering of misclassified sources will appear to evolve strongly due to projection effects, even if their true clustering does not. We present a practical method for accounting for the presence of contaminants with redshift-dependent contamination fractions and projected clustering. We show using mock catalogues that our method, unlike existing approaches, yields unbiased clustering measurements from the upcoming HETDEX survey in scenarios with redshift-dependent contamination fractions within the redshift bins used. We show our method returns auto-correlation functions with systematic biases much smaller than the statistical noise for samples with at least as high as 7 per cent contamination. We also present and test a method for fitting for the redshift-dependent interloper fraction using the LAE-[OII] galaxy cross-correlation function, which gives less biased results than assuming a single interloper fraction for the whole sample., Comment: This is a pre-copy edited, author-produced PDF of an article accepted for publication in Monthly Notices of the Royal Astronomical Society following peer review. The version of record Farrow et al, MNRAS, 2021 is available online at: https://academic.oup.com/mnras/advance-article-abstract/doi/10.1093/mnras/stab1986/6322852

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: $$ \partial_tu=\Delta u^m + |x|^{\sigma}u^m, $$ posed for $x\in\mathbb{R}^N$, $t\geq0$, where $m>1$, dimension $N\geq2$ and $\sigma>0$. A new and explicit critical exponent $$ \sigma_c=\frac{2(m-1)(N-1)}{3m+1} $$ is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent $\sigma_c$ and also depends strongly on whether the dimension $N\geq4$ or $N\in\{2,3\}$. These results extend previous works of the authors in dimension $N=1$.

We investigate the impact of different assumptions in the modeling of one-loop galaxy bias on the recovery of cosmological parameters, as a follow up of the analysis done in the first paper of the series at fixed cosmology. We use three different synthetic galaxy samples whose clustering properties match the ones of the CMASS and LOWZ catalogues of BOSS and the SDSS Main Galaxy Sample. We investigate the relevance of allowing for either short range non-locality or scale-dependent stochasticity by fitting the real-space galaxy auto power spectrum or the combination of galaxy-galaxy and galaxy-matter power spectrum. From a comparison among the goodness-of-fit ($\chi^2$), unbiasedness of cosmological parameters (FoB), and figure-of-merit (FoM), we find that a four-parameter model (linear, quadratic, cubic non-local bias, and constant shot-noise) with fixed quadratic tidal bias provides a robust modelling choice for the auto power spectrum of the three samples, up to $k_{\rm max}=0.3\,h\,\mathrm{Mpc}^{-1}$ and for an effective volume of $6\,h^{-3}\,\mathrm{Gpc}^3$. Instead, a joint analysis of the two observables fails at larger scales, and a model extension with either higher derivatives or scale-dependent shot-noise is necessary to reach a similar $k_{\rm max}$, with the latter providing the most stable results. These findings are obtained with three, either hybrid or perturbative, prescriptions for the matter power spectrum, \texttt{RESPRESSO}, gRPT and EFT. In all cases, the inclusion of scale-dependent shot-noise increases the range of validity of the model in terms of FoB and $\chi^2$. Interestingly, these model extensions with additional free parameters do not necessarily lead to an increase in the maximally achievable FoM for the cosmological parameters $\left(h,\,\Omega_ch^2,\,A_s\right)$, which are generally consistent to those of the simpler model at smaller $k_{\rm max}$., Comment: 24 pages, 8 figures

We present a joint likelihood analysis of the real-space power spectrum and bispectrum measured from a variety of halo and galaxy mock catalogs. A novel aspect of this work is the inclusion of nonlinear triangle configurations for the bispectrum, made possible by a complete next-to-leading order ("one-loop") description of galaxy bias, as is already common practice for the power spectrum. Based on the goodness-of-fit and the unbiasedness of the parameter posteriors, we accomplish a stringent validation of this model compared to the leading order ("tree-level") bispectrum. Using measurement uncertainties that correspond to an effective survey volume of $6\,(\mathrm{Gpc}/h)^3$, we determine that the one-loop corrections roughly double the applicable range of scales, from $\sim 0.17\,h/\mathrm{Mpc}$ (tree-level) to $\sim 0.3\,h/\mathrm{Mpc}$. This converts into a $1.5 - 2$x improvement on constraints of the linear bias parameter at fixed cosmology, and a $1.5 - 2.4$x shrinkage of uncertainties on the amplitude of fluctuations $A_s$, which clearly demonstrates the benefit of extracting information from nonlinear scales despite having to marginalize over a larger number of bias parameters. Besides, our precise measurements of galaxy bias parameters up to fourth order allow for thorough comparisons to coevolution relations, showing excellent agreement for all contributions generated by the nonlocal action of gravity. Using these relations in the likelihood analysis does not compromise the model validity and is crucial for obtaining the quoted improvements on $A_s$. We also analyzed the impact of higher-derivative and scale-dependent stochastic terms, finding that for a subset of our tracers the former can boost the performance of the tree-level model with constraints on $A_s$ that are only slightly degraded compared to the one-loop model., Comment: 33 pages, 13 figures, supporting Mathematica notebook available at https://doi.org/10.5281/zenodo.4529886, Python package for Delaunay binning correction available at https://github.com/aegge/BispTools

We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$, with $m>1$, $02$. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.

Accurate modelling of redshift-space distortions (RSD) is challenging in the non-linear regime for two-point statistics e.g. the two-point correlation function (2PCF). We take a different perspective to split the galaxy density field according to the local density, and cross-correlate those densities with the entire galaxy field. Using mock galaxies, we demonstrate that combining a series of cross-correlation functions (CCFs) offers improvements over the 2PCF as follows: 1. The distribution of peculiar velocities in each split density is nearly Gaussian. This allows the Gaussian streaming model for RSD to perform accurately within the statistical errors of a ($1.5\,h^{-1}$Gpc)$^3$ volume for almost all scales and all split densities. 2. The PDF of the density field at small scales is non-Gaussian, but the CCFs of split densities capture the non-Gaussianity, leading to improved cosmological constraints over the 2PCF. We can obtain unbiased constraints on the growth parameter $f\sigma_{12}$ at the per-cent level, and Alcock-Paczynski (AP) parameters at the sub-per-cent level with the minimal scale of $15\,h^{-1}{\rm Mpc}$. This is a $\sim$30 per cent and $\sim$6 times improvement over the 2PCF, respectively. The diverse and steep slopes of the CCFs at small scales are likely to be responsible for the improved constraints of AP parameters. 3. Baryon acoustic oscillations (BAO) are contained in all CCFs of split densities. Including BAO scales helps to break the degeneracy between the line-of-sight and transverse AP parameters, allowing independent constraints on them. We discuss and compare models for RSD around spherical densities., Comment: 23 pages, 16 figures, MNRAS accepted version after peer review, minor comments

Minkowski functionals (MFs) are a set of statistics that characterise the geometry and topology of the cosmic density field and contain complementary information to the standard two-point analyses. We present MEDUSA, an implementation of an accurate method for estimating the MFs of three-dimensional point distributions. These estimates are inferred from triangulated isodensity surfaces that are constructed from the Delaunay tessellation of the input point sample. Contrary to previous methods, MEDUSA can account for periodic boundary conditions, which is crucial for the analysis of N-body simulations. We validate our code against several test samples with known MFs, including Gaussian random fields with a ${\Lambda}$CDM power spectrum, and find excellent agreement with the theory predictions. We use MEDUSA to measure the MFs of synthetic galaxy catalogues constructed from N-body simulations. Our results show clearly non-Gaussian signatures that arise from the non-linear gravitational evolution of the density field. We find that, although redshift-space distortions significantly change our MFs estimates, their impact is considerably reduced if these measurements are expressed as a function of the volume-filling fraction. We also show that the effect of Alcock-Paczynski (AP) distortions on the MFs can be described by scaling them with different powers of the isotropic AP parameter $q$ defined in terms of the volume-averaged distance $D_{\rm V}(z)$. Thus the MFs estimates by MEDUSA are useful probes of non-linearities in the density field, and the expansion and growth of structure histories of the Universe., Comment: 14 pages, 16 figures, submitted to MNRAS

All estimators of the two-point correlation function are based on a random catalogue, a set of points with no intrinsic clustering following the selection function of a survey. High-accuracy estimates require the use of large random catalogues, which imply a high computational cost. We propose to replace the standard random catalogues by glass-like point distributions or glass catalogues, which are characterized by a power spectrum $P(k)\propto k^4$ and exhibit significantly less power than a Poisson distribution with the same number of points on scales larger than the mean inter-particle separation. We show that these distributions can be obtained by iteratively applying the technique of Zeldovich reconstruction commonly used in studies of baryon acoustic oscillations (BAO). We provide a modified version of the widely used Landy-Szalay estimator of the correlation function adapted to the use of glass catalogues and compare its performance with the results obtained using random samples. Our results show that glass-like samples do not add any bias with respect to the results obtained using Poisson distributions. On scales larger than the mean inter-particle separation of the glass catalogues, the modified estimator leads to a significant reduction of the variance of the Legendre multipoles $\xi_\ell(s)$ with respect to the standard Landy-Szalay results with the same number of points. The size of the glass catalogue required to achieve a given accuracy in the correlation function is significantly smaller than when using random samples. Even considering the small additional cost of constructing the glass catalogues, their use could help to drastically reduce the computational cost of configuration-space clustering analysis of future surveys while maintaining high-accuracy requirements., Comment: 9 pages, 7 figures, minor changes to match version accepted by MNRAS

We present constraints on extensions to the flat $\Lambda$CDM cosmological model by varying the spatial curvature $\Omega_K$, the sum of the neutrino masses $\sum m_\nu$, the dark energy equation of state parameter $w$, and the Hu-Sawicki $f(R)$ gravity $f_{R0}$ parameter. With the combined $3\times2$pt measurements of cosmic shear from the Kilo-Degree Survey (KiDS-1000), galaxy clustering from the Baryon Oscillation Spectroscopic Survey (BOSS), and galaxy-galaxy lensing from the overlap between KiDS-1000, BOSS, and the spectroscopic 2-degree Field Lensing Survey (2dFLenS), we find results that are fully consistent with a flat $\Lambda$CDM model with $\Omega_K=0.011^{+0.054}_{-0.057}$, $\sum m_\nu<1.76$ eV (95% CL), and $w=-0.99^{+0.11}_{-0.13}$. The $f_{R0}$ parameter is unconstrained in our fully non-linear $f(R)$ cosmic shear analysis. Considering three different model selection criteria, we find no clear preference for either the fiducial flat $\Lambda$CDM model or any of the considered extensions. Besides extensions to the flat $\Lambda$CDM parameter space, we also explore restrictions to common subsets of the flat $\Lambda$CDM parameter space by fixing the amplitude of the primordial power spectrum to the Planck best-fit value, as well as adding external data from supernovae and lensing of the CMB. Neither the beyond-$\Lambda$CDM models nor the imposed restrictions explored in this analysis are able to resolve the $\sim 3\sigma$ tension in $S_8$ between the $3\times2$pt constraints and Planck, with the exception of $w$CDM, where the $S_8$ tension is resolved. The tension in the $w$CDM case persists, however, when considering the joint $S_8$-$w$ parameter space. The joint flat $\Lambda$CDM CMB lensing and $3\times2$pt analysis is found to yield tight constraints on $\Omega_{\rm m}=0.307^{+0.008}_{-0.013}$, $\sigma_8=0.769^{+0.022}_{-0.010}$, and $S_8=0.779^{+0.013}_{-0.013}$., Comment: 15 pages, 9 figures, 3 tables, accepted in A&A. This paper concludes the KiDS-1000 series of papers: Heymans, Tr\"oster et al. (arXiv:2007.15632), Asgari et al. (arXiv:2007.15633), Hildebrandt et al. (arXiv:2007.15635), Joachimi et al. (arXiv:2007.01844), and Giblin et al. (arXiv:2007.01845). Data products, likelihoods, and posteriors can be found http://kids.strw.leidenuniv.nl/DR4/lensing.php

We present a joint cosmological analysis of weak gravitational lensing observations from the Kilo-Degree Survey (KiDS-1000), with redshift-space galaxy clustering observations from the Baryon Oscillation Spectroscopic Survey (BOSS), and galaxy-galaxy lensing observations from the overlap between KiDS-1000, BOSS and the spectroscopic 2-degree Field Lensing Survey (2dFLenS). This combination of large-scale structure probes breaks the degeneracies between cosmological parameters for individual observables, resulting in a constraint on the structure growth parameter $S_8=\sigma_8 \sqrt{\Omega_{\rm m}/0.3} = 0.766^{+0.020}_{-0.014}$, that has the same overall precision as that reported by the full-sky cosmic microwave background observations from Planck. The recovered $S_8$ amplitude is low, however, by $8.3 \pm 2.6$ % relative to Planck. This result builds from a series of KiDS-1000 analyses where we validate our methodology with variable depth mock galaxy surveys, our lensing calibration with image simulations and null-tests, and our optical-to-near-infrared redshift calibration with multi-band mock catalogues and a spectroscopic-photometric clustering analysis. The systematic uncertainties identified by these analyses are folded through as nuisance parameters in our cosmological analysis. Inspecting the offset between the marginalised posterior distributions, we find that the $S_8$-difference with Planck is driven by a tension in the matter fluctuation amplitude parameter, $\sigma_8$. We quantify the level of agreement between the CMB and our large-scale structure constraints using a series of different metrics, finding differences with a significance ranging between $\sim\! 3\,\sigma$, when considering the offset in $S_{8}$, and $\sim\! 2\,\sigma$, when considering the full multi-dimensional parameter space., Comment: 25 pages, 11 figures, 5 tables, A&A accepted, including a new appendix on Intrinsic Alignments. The KiDS-1000 data products are available for download at http://kids.strw.leidenuniv.nl/DR4/lensing.php. This data release includes open source software, the shear-photo-z catalogue, the cosmic shear and 3x2pt data vectors and covariances, and posteriors in the form of Multinest chains

Voids are promising cosmological probes. Nevertheless, every cosmological test based on voids must necessarily employ methods to identify them in redshift space. Therefore, redshift-space distortions (RSD) and the Alcock-Paczynski effect (AP) have an impact on the void identification process itself generating distortion patterns in observations. Using a spherical void finder, we developed a statistical and theoretical framework to describe physically the connection between the identification in real and redshift space. We found that redshift-space voids above the shot noise level have a unique real-space counterpart spanning the same region of space, they are systematically bigger and their centres are preferentially shifted along the line of sight. The expansion effect is a by-product of RSD induced by tracer dynamics at scales around the void radius, whereas the off-centring effect constitutes a different class of RSD induced at larger scales by the global dynamics of the whole region containing the void. The volume of voids is also altered by the fiducial cosmology assumed to measure distances, this is the AP change of volume. These three systematics have an impact on cosmological statistics. In this work, we focus on the void size function. We developed a theoretical framework to model these effects and tested it with a numerical simulation, recovering the statistical properties of the abundance of voids in real space. This description depends strongly on cosmology. Hence, we lay the foundations for improvements in current models of the abundance of voids in order to obtain unbiased cosmological constraints from redshift surveys., Comment: 16 pages, 12 figures, accepted in MNRAS

We measure the anisotropic clustering of the quasar sample from Data Release 16 (DR16) of the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey (eBOSS). A sample of $343,708$ spectroscopically confirmed quasars between redshift $0.8

We present the cosmological implications from final measurements of clustering using galaxies, quasars, and Ly$\alpha$ forests from the completed Sloan Digital Sky Survey (SDSS) lineage of experiments in large-scale structure. These experiments, composed of data from SDSS, SDSS-II, BOSS, and eBOSS, offer independent measurements of baryon acoustic oscillation (BAO) measurements of angular-diameter distances and Hubble distances relative to the sound horizon, $r_d$, from eight different samples and six measurements of the growth rate parameter, $f\sigma_8$, from redshift-space distortions (RSD). This composite sample is the most constraining of its kind and allows us to perform a comprehensive assessment of the cosmological model after two decades of dedicated spectroscopic observation. We show that the BAO data alone are able to rule out dark-energy-free models at more than eight standard deviations in an extension to the flat, $\Lambda$CDM model that allows for curvature. When combined with Planck Cosmic Microwave Background (CMB) measurements of temperature and polarization the BAO data provide nearly an order of magnitude improvement on curvature constraints. The RSD measurements indicate a growth rate that is consistent with predictions from Planck primary data and with General Relativity. When combining the results of SDSS BAO and RSD with external data, all multiple-parameter extensions remain consistent with a $\Lambda$CDM model. Regardless of cosmological model, the precision on $\Omega_\Lambda$, $H_0$, and $\sigma_8$, remains at roughly 1\%, showing changes of less than 0.6\% in the central values between models. The inverse distance ladder measurement under a o$w_0w_a$CDM yields $H_0= 68.20 \pm 0.81 \, \rm km\, s^{-1} Mpc^{-1}$, remaining in tension with several direct determination methods. (abridged), Comment: A summary of all SDSS BAO and RSD measurements with accompanying legacy figures can be found here: https://www.sdss.org/science/final-bao-and-rsd-measurements/ . The full cosmological interpretation of these measurements can be found here: https://www.sdss.org/science/cosmology-results-from-eboss/

We test the regime of validity of one-loop galaxy bias for a wide variety of biased tracers. Our most stringent test asks the bias model to simultaneously match the galaxy-galaxy and galaxy-mass spectrum, using the measured nonlinear matter spectrum from the simulations to test one-loop effects from the bias expansion alone. In addition, we investigate the relevance of short-range nonlocality and halo exclusion through higher-derivative and scale-dependent noise terms, as well as the impact of using co-evolution relations to reduce the number of free fitting parameters. From comparing validity and merit of these assumptions we find that a four-parameter model (linear, quadratic, cubic nonlocal bias, and constant shot noise) with fixed quadratic tidal bias provides a robust modeling choice for the auto power spectrum of the less massive halos in our set of samples and their galaxy populations (up to $k_{\mathrm{max}} = 0.35\,h/\mathrm{Mpc}$ for a sample volume of $6\,(\mathrm{Gpc}/h)^3$). For the more biased tracers it is most beneficial to include scale-dependent noise. This is also the preferred option when considering combinations of the auto and cross power spectrum, which might be relevant in joint studies of galaxy clustering and weak lensing. We also test the use of perturbation theory to account for matter loops through gRPT, EFT and the hybrid approach RESPRESSO. While all these have similar performance, we find the latter to be the best in terms of validity and recovered mean posterior values, in accordance with it being based partially on simulations., Comment: 30 pages, 12 figures; v2: includes gRPT expressions and some minor edits, matches published version

We classify the self-similar blow-up profiles for the following reaction-diffusion equation with critical strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $02$ completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for $\sigma>2$. Moreover all the blow-up profiles have compact support and their supports are \emph{localized}: there exists an explicit $\eta>0$ such that any blow-up profile satisfies ${\rm supp}\,f\subseteq[0,\eta]$. This property is unexpected and contrasting with the range $m+p>2$. We also classify the possible behaviors of the profiles near the origin., Comment: 38 pages, 7 figures

We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for $x\in\real$, $t\geq0$, where $m>1$, $02(1-p)/(m-1)$. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for $m+p>2$ and $\sigma$ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when $p<1$. We moreover prove that, if the condition $m+p>2$ is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist \emph{two different interface behaviors} for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases \emph{global blow-up} occurs, and in some other cases finite time blow-up occurs \emph{only at space infinity}. We also show that there is no self-similar solution if $m+p<2$, while the critical range $m+p=2$ with $\sigma>2$ is postponed to a different work due to significant technical differences.

It is common to express cosmological measurements in units of $h^{-1}{\rm Mpc}$. Here, we review some of the complications that originate from this practice. A crucial problem caused by these units is related to the normalization of the matter power spectrum, which is commonly characterized in terms of the linear-theory rms mass fluctuation in spheres of radius $8\,h^{-1}{\rm Mpc}$, $\sigma_8$. This parameter does not correctly capture the impact of $h$ on the amplitude of density fluctuations. We show that the use of $\sigma_8$ has caused critical misconceptions for both the so-called $\sigma_8$ tension regarding the consistency between low-redshift probes and cosmic microwave background data, and the way in which growth-rate estimates inferred from redshift-space distortions are commonly expressed. We propose to abandon the use of $h^{-1}{\rm Mpc}$ units in cosmology and to characterize the amplitude of the matter power spectrum in terms of $\sigma_{12}$, defined as the mass fluctuation in spheres of radius $12\,{\rm Mpc}$, whose value is similar to the standard $\sigma_8$ for $h\sim 0.67$., Comment: 6 pages, 4 figures. Minor changes. Matches version published in PRD