Pairwise Velocity Research Articles

Modelling the redshift-space cluster–galaxy correlation function on Mpc scales with emulation of the pairwise velocity distribution

ABSTRACT We present a method for modelling the cluster–galaxy correlation function in redshift space, down to $\sim$ Mpc scales. The method builds upon the so-called galaxy infall kinematics (GIK) model, a parametric model for the pairwise velocities of galaxies with respect to nearby galaxy clusters. We fit the parameters of the GIK model to a suite of simulations run with different cosmologies, and use Gaussian processes to emulate how the GIK parameters depend upon cosmology. This emulator can then be combined with knowledge of the real-space clustering of clusters and galaxies, to predict the cluster–galaxy correlation function in redshift space, $\xi _\mathrm{cg}^s$. Fitting this model to an observed $\xi _\mathrm{cg}^s$ enables the extraction of cosmological parameter constraints, and we present forecasts for a survey like that currently being done by the Dark Energy Spectroscopic Instrument (DESI). We also perform tests of the robustness of our constraints from fitting to mock data extracted from N-body simulations, finding that fitting to scales $\lesssim 3 \, h^{-1}\, \mathrm{Mpc}$ leads to a biased inference on cosmology, due to model mis-specification on these scales. Finally, we discuss what steps will need to be taken in order to apply our method to real data.

On the statistical theory of self-gravitating collisionless dark matter flow: High order kinematic and dynamic relations

Dark matter, if it exists, accounts for five times as much as ordinary baryonic matter. To better understand the self-gravitating collisionless dark matter flow on different scales, a statistical theory involving kinematic and dynamic relations must be developed for different types of flow, e.g., incompressible, constant divergence, and irrotational flow. This is mathematically challenging because of the intrinsic complexity of dark matter flow and the lack of a self-closed description of flow velocity. This paper extends our previous work on second-order statistics Xu [Phys. Fluids 35, 077105 (2023)] to kinematic relations of any order for any type of flow. Dynamic relations were also developed to relate statistical measures of different orders. The results were validated by N-body simulations. On large scales, we found that (i) third-order velocity correlations can be related to density correlation or pairwise velocity; (ii) the pth-order velocity correlations follow ∝a(p+2)/2 for odd p and ∝ap/2 for even p, where a is the scale factor; (iii) the overdensity δ is proportional to density correlation on the same scale, ⟨δ⟩∝⟨δδ′⟩; (iv) velocity dispersion on a given scale r is proportional to the overdensity on the same scale. On small scales, (i) a self-closed velocity evolution is developed by decomposing the velocity into motion in haloes and motion of haloes; (ii) the evolution of vorticity and enstrophy are derived from the evolution of velocity; (iii) dynamic relations are derived to relate second- and third-order correlations; (iv) while the first moment of pairwise velocity follows ⟨ΔuL⟩=−Har (H is the Hubble parameter), the third moment follows ⟨(ΔuL)3⟩∝εuar that can be directly compared with simulations and observations, where εu≈10−7 m2/s3 is the constant rate for energy cascade; (v) the pth order velocity correlations follow ∝a(3p−5)/4 for odd p and ∝a3p/4 for even p. Finally, the combined kinematic and dynamic relations lead to exponential and one-fourth power-law velocity correlations on large and small scales, respectively.

Kinematic Sunyaev-Zel′dovich pairwise velocity reconstruction with machine learning

Kinematic Sunyaev-Zel′dovich pairwise velocity reconstruction with machine learning

Measuring neutrino mass and asymmetry with matter pairwise velocities

ABSTRACT Neutrinos are believed to be the most abundant fermions in the Universe, but their masses are unknown, except for being non-zero but much smaller than other fermions. Cosmological relic neutrinos could also have non-zero chemical potentials (or asymmetries). Using neutrino-involved N-body simulations, we investigate the neutrino effects on the matter pairwise velocity, which itself is an interesting probe of cosmology. We find that for light-halo ([1011, 1013] M⊙) mean pairwise velocity, in the transition range ([4, 15] Mpc), the effects of neutrino masses overwhelm the effects of neutrino asymmetries, while in the two-halo-group range ([25, 50] Mpc), for both light and heavy haloes ([1013, 1015] M⊙), the effects of neutrino asymmetries dominate, making it possible to disentangle the two effects. We provide fitting formulae to quantify the effects of neutrino mass and asymmetry on halo–halo pairwise velocities.

Convergence of halo statistics: code comparison between rockstar and compaso using scale-free simulations

ABSTRACT In this study, we perform a halo-finder code comparison between rockstar and compaso. Based on our previous analysis aiming at quantifying resolution of N-body simulations by exploiting large (up to N = 40963) simulations of scale-free cosmologies run using abacus, we focus on convergence of the halo mass function, two-point correlation function, and mean radial pairwise velocities of halo centres selected with the aforementioned two algorithms. We establish convergence, for both rockstar and compaso, of mass functions at the 1 per cent precision level and of the mean pairwise velocities (and also two-point correlation function) at the 2 per cent level. At small scales and masses, we find that rockstar exhibits greater self-similarity. We also highlight the role played by the merger-tree post-processing of compaso haloes on their convergence. Finally, we give resolution limits expressed as a minimum particle number per halo in a form that can be directly extrapolated to Lambda cold dark matter.

Constraining accuracy of the pairwise velocities in N-body simulations using scale-free models

ABSTRACT We present a continuation of an analysis that aims to quantify resolution of N-body simulations by exploiting large (up to N = 40963) simulations of scale-free cosmologies run using abacus. Here, we focus on radial pairwise velocities of the matter field, both by direct estimation and through the cumulative two-point correlation function (using the pair conservation equation). We find that convergence at the 1 per cent level of the mean relative pairwise velocity can be demonstrated over a range of scales, evolving from a few times the grid spacing at early times to slightly below this scale at late times. We show the analysis of two different box sizes as well as from averaging results from the smaller boxes, and compare the power of the two aforementioned estimators in constraining accuracy at each scale. Down to scales of the order of the smoothing parameter, convergence is obtained at ∼$5~{{\rm per\, cent}}$ precision, and shows a behaviour indicating asymptotic stable clustering. We also infer for LCDM simulations conservative estimates on the evolution of the lower cut-off to resolution (at 1 and 5 per cent precision) as a function of redshift.

Velocity-dependent J-factors for Milky Way dwarf spheroidal analogues in cosmological simulations

We study the impact of the dark matter velocity distribution modelling on signals from velocity-dependent dark matter annihilation in Milky Way dwarf spheroidal galaxies. Using the high resolution APOSTLE simulations, we identify analogues corresponding to Milky Way dwarf spheroidal galaxies, and from these directly determine the dark matter pair-wise relative velocity distribution, and compare to best-fitting Maxwell-Boltzmann distribution models. For three velocity-dependent annihilation models, p-wave, d-wave, and the Sommerfeld model, we quantify the errors introduced when using the Maxwell-Boltzmann parameterization. We extract a simple power-law relation between the maximum circular velocity of the dwarf spheroidal analogue and the peak speed of the Maxwell-Boltzmann distribution. We show that this relation can be used to accurately calculate the dark matter relative velocity distribution, and find that it allows us to estimate the dark matter annihilation signal without the need to directly calculate the relative velocity distribution for each galaxy. The scatter in the J-factors calculated from the analogues dominates the uncertainty obtained when compared to the J-factor as determined from the observational data for each dwarf spheroidal, with the largest scatter from d-wave models and the smallest from Sommerfeld models.

CellListMap.jl: Efficient and customizable cell list implementation for calculation of pairwise particle properties within a cutoff

N-body simulations and trajectory analysis rely on the calculation of attributes that depend on pairwise particle distances within a cutoff. Interparticle potential energies, forces, distribution functions, neighbor lists, and distance-dependent distributions, for example, must be calculated. Cell lists are widely used to avoid computing distances outside the cutoff. However, efficient cell list implementations are difficult to customize. Here, we provide a fast and parallel implementation of cell lists in Julia that allows the mapping of custom functions dependent on particle positions in 2 or 3 dimensions. Arbitrary periodic boundary conditions are supported. Automatic differentiation and unit propagation can be used. The implementation provides a framework for the development of new analysis tools and simulations with custom potentials. The performance of resulting computations is comparable to state-of-the-art implementations of neighbor list algorithms and cell lists, available in specialized software. Examples are provided for the computation of potential energies, forces, distribution of pairwise velocities, neighbor lists and other typical calculations in molecular and astrophysical simulations. The Julia package is freely available at http://m3g.github.io/CellListMap.jl. Interfacing with Python and R with minimal overhead is possible. Program summaryProgram Title: CellListMap.jlCPC Library link to program files:https://doi.org/10.17632/5kf5zh9zms.1Developer's repository link:http://github.com/m3g/CellListMap.jlLicensing provisions: MITProgramming language: JuliaNature of problem: Particle simulations and trajectory analysis, in chemistry, atomic and molecular physics, and astrophysics, require the computation of pairwise properties within a cutoff, for example potential energies, forces, distance distributions, among others. Cell lists can be used to accelerate these computations and are used in major simulation software, but these implementations cannot be easily adapted for novel custom calculations, simulations, and novel research.Solution method: This package provides a fast and customizable implementation of cell lists for the development of new simulation and analysis software for particle simulations. To provide a customizable implementation, the Julia language was used, allowing the user to write high-level yet performant custom functions to be mapped into pairs of particles satisfying the distance constraint desired. State-of-the art performance is obtained in serial and parallel executions.Additional comments including restrictions and unusual features: The implementation is type-generic, allowing systems in 2 or 3 dimensions to be studied, and the computations can be written in such a way to allow the propagation of units, uncertainties, and automatic differentiation of the computed properties.

Possible Ongoing Merger Discovered by Photometry and Spectroscopy in the Field of the Galaxy Cluster PLCK G165.7+67.0

We present a detailed study of the Planck-selected binary galaxy cluster PLCK G165.7+67.0 (G165; z = 0.348). A multiband photometric catalog is generated incorporating new imaging from the Large Binocular Telescope/Large Binocular Camera and Spitzer/IRAC to existing imaging. To cope with the different image characteristics, robust methods are applied in the extraction of the matched-aperture photometry. Photometric redshifts are estimated for 143 galaxies in the 4 arcmin2 field of overlap covered by these data. We confirm that strong-lensing effects yield 30 images of 11 background galaxies, of which we contribute new photometric redshift estimates for three image multiplicities. These constraints enable the construction of a revised lens model with a total mass of M 600 kpc = (2.36 ± 0.23) × 1014 M ⊙. In parallel, new spectroscopy using MMT/Binospec and archival data contributes thirteen galaxies that meet our velocity and transverse radius criteria for cluster membership. The two cluster components have a pair-wise velocity of ≲100 km s−1, favoring an orientation in the plane of the sky with a transverse velocity of 100–1700 km s−1. At the same time, the brightest cluster galaxy (BCG) is offset in velocity from the systemic mean value, suggesting dynamical disturbance. New LOFAR and Very Large Array data uncover head-tail radio galaxies in the BCG and a large red galaxy in the northeast component. From the orientation and alignment of the four radio trails, we infer that the two cluster components have already traversed each other, and are now exiting the cluster.

Towards an accurate model of small-scale redshift-space distortions in modified gravity

ABSTRACT The coming generation of galaxy surveys will provide measurements of galaxy clustering with unprecedented accuracy and data size, which will allow us to test cosmological models at much higher precision than achievable previously. This means that we must have more accurate theoretical predictions to compare with future observational data. As a first step towards more accurate modelling of the redshift space distortions (RSD) of small-scale galaxy clustering in modified gravity (MG) cosmologies, we investigate the validity of the so-called Skew-T (ST) probability distribution function (PDF) of halo pairwise peculiar velocities in these models. We show that, combined with the streaming model of RSD, the ST PDF substantially improves the small-scale predictions by incorporating skewness and kurtosis, for both Λ cold dark matter (ΛCDM) and two leading MG models: f(R) gravity and the DGP braneworld model. The ST model reproduces the velocity PDF and redshift-space halo clustering measured from MG N-body simulations very well down to $\sim 5 \, h^{-1}\, \mathrm{Mpc}$. In particular, we investigate the enhancements of halo pairwise velocity moments with respect to ΛCDM for a larger range of MG variants than previous works, and present simple explanations to the behaviours observed. By performing a simple Fisher analysis, we find a significant increase in constraining power to detect modifications of General Relativity by introducing small-scale information in the RSD analyses.

Cosmology with the kinetic Sunyaev–Zeldovich effect: Independent of the optical depth and σ8

Cosmological constraints obtained by the kinetic Sunyaev–Zeldovich experiments are degenerate with the optical depth measurement – an effect that is commonly known as the optical-depth degeneracy. In this work, we introduce a new statistic based on the first moment of relative velocity between pairs in a triplet, which is capable of constraining cosmological parameters independently of the optical depth and of σ8. Using 22 000 N-body simulations from the Quijote suite, we quantified the information content in the new statistic using Fisher matrix forecast. We find that it is able to obtain strong constraints on the cosmological parameters, particularly on the summed neutrino mass. The constraints bring an improvement on all cosmological model parameters by a factor of 6.2–12.9 and 2.3–5.7 when compared to those obtained from the mean pairwise velocity and from the redshift-space halo power spectrum, respectively. Thus, this new statistic paves a way forward in constraining cosmological parameters independent of the optical depth and σ8 when using data from future kinetic Sunyaev–Zeldovich experiments alone.

Ultralight axions and the kinetic Sunyaev-Zel’dovich effect

Measurements of secondary cosmic microwave background (CMB) anisotropies, such as the Sunyaev-Zel'dovich (SZ) effect, will enable new tests of neutrino and dark sector properties. The kinetic SZ (kSZ) effect is produced by cosmological flows, probing structure growth. Ultra-light axions (ULAs) are a well-motivated dark-matter candidate. Here the impact of ULA dark matter (with mass $10^{-27}~{\rm eV}$ to $10^{-23}~{\rm eV}$) on kSZ observables is determined, applying new analytic expressions for pairwise cluster velocities and Ostriker-Vishniac signatures in structure-suppressing models. For the future CMB-S4 and ongoing DESI galaxy surveys, the kSZ effect (along with primary anisotropies) will probe ULA fractions $\eta_a = \Omega_{\rm{axion}}/\Omega_{\rm DM}$ as low as $\sim 5\%$ if $m_{a}\simeq 10^{-27}~{\rm eV}$ (at 95\% C.L.), with sensitivity extending up to $m_{a}\simeq 10^{-25}~{\rm eV}$. If reionization and the primary CMB can be adequately modeled, Ostriker-Vishniac measurements could probe values $\eta_{a}\simeq 10^{-3}$ if $10^{-27}~{\rm eV}\lesssim m_{a}\lesssim 10^{-24}~{\rm eV}$, or $\eta_{a}\simeq 1$ if $m_{a}\simeq 10^{-22}~{\rm eV}$, within the fuzzy dark matter window.

Information content in mean pairwise velocity and mean relative velocity between pairs in a triplet

Velocity fields provide a complementary avenue to constrain cosmological information, either through the peculiar velocity surveys or the kinetic Sunyaev Zel’dovich effect. One of the commonly used statistics is the mean radial pairwise velocity. Here, we consider the three-point mean relative velocity (i.e. the mean relative velocities between pairs in a triplet). Using halo catalogs from the Quijote suite ofN-body simulations, we first showcase how the analytical prediction for the mean relative velocities between pairs in a triplet achieve better than 4−5% accuracy using standard perturbation theory at leading order for triangular configurations with a minimum separation ofr ≥ 50 h−1Mpc. Furthermore, we present the mean relative velocity between pairs in a triplet as a novel probe of neutrino mass estimation. We explored the full cosmological information content of the halo mean pairwise velocities and the mean relative velocities between halo pairs in a triplet. We did this through the Fisher-matrix formalism using 22 000 simulations from the Quijote suite and by considering all triangular configurations with a minimum and a maximum separation of 20 h−1Mpc and 120 h−1Mpc, respectively. We find that the mean relative velocities in a triplet allows a 1σneutrino mass (Mν) constraint of 0.065 eV, which is roughly 13 times better than the mean pairwise velocity constraint (0.877 eV). This information gain is not limited to neutrino mass, but it extends to other cosmological parameters: Ωm, Ωb,h,ns, andσ8, achieving an information gain of 8.9, 11.8, 15.5, 20.9, and 10.9 times, respectively. These results illustrate the possibility of exploiting the mean three-point relative velocities to constrain the cosmological parameters accurately from future cosmic microwave background experiments and peculiar velocity surveys.

The Atacama Cosmology Telescope: Detection of the pairwise kinematic Sunyaev-Zel’dovich effect with SDSS DR15 galaxies

We present a 5.4σ detection of the pairwise kinematic Sunyaev-Zeldovich (kSZ) effect using Atacama Cosmology Telescope (ACT) and Planck CMB observations in combination with Luminous Red Galaxy samples from the Sloan Digital Sky Survey (SDSS) DR15 catalog. Results are obtained using three ACT CMB maps: co-added 150 and 98 GHz maps, combining observations from 2008–2018 (ACT DR5), which overlap with SDSS DR15 over 3,700 sq. deg., and a component-separated map using night-time only observations from 2014–2015 (ACT DR4), overlapping with SDSS DR15 over 2,089 sq. deg. Comparisons of the results from these three maps provide consistency checks in relation to potential frequency-dependent foreground contamination. A total of 343,647 galaxies are used as tracers to identify and locate galaxy groups and clusters from which the kSZ signal is extracted using aperture photometry. We consider the impact of various aperture photometry assumptions and covariance estimation methods on the signal extraction. Theoretical predictions of the pairwise velocities are used to obtain best-fit, mass-averaged, optical depth estimates for each of five luminosity-selected tracer samples. A comparison of the kSZ-derived optical depth measurements obtained here to those derived from the thermal SZ effect for the same sample is presented in a companion paper.3 MoreReceived 4 February 2021Accepted 27 May 2021DOI:https://doi.org/10.1103/PhysRevD.104.043502© 2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCosmic microwave backgroundCosmological parametersLarge scale structure of the UniverseGravitation, Cosmology & Astrophysics

Expansion series of the pairwise velocity generating function and its implications on redshift space distortion modeling

The pairwise velocity generating function G has a deep connection with both the pairwise velocity probability distribution function and modeling of redshift space distortion (RSD). Its implementation into RSD modeling is often faciliated by expansion into a series of pairwise velocity moments . Motivated by the logrithmic transformation of the cosmic density field, we investigate an alternative expansion into series of pairwise velocity cumulants . We numerically evaluate the convergence rate of the two expansions, with three 30723 particle simulations of the CosmicGrowth N-body simulation series. (1) We find that the cumulant expansion performs significantly better, for all the halo samples and redshifts investigated. (2) For modeling RSD at k ∥ < 0.1h Mpc−1, including only the n = 1,2 cumulants is sufficient. (3) But for modeling RSD at k ∥ = 0.2h Mpc−1, we need and only need the n = 1,2,3,4 cumulants. These results provide specific requirements on RSD modeling in terms of m-th order statistics of the large-scale structure.

AI-assisted superresolution cosmological simulations – II. Halo substructures, velocities, and higher order statistics

ABSTRACT In this work, we expand and test the capabilities of our recently developed superresolution (SR) model to generate high-resolution (HR) realizations of the full phase-space matter distribution, including both displacement and velocity, from computationally cheap low-resolution (LR) cosmological N-body simulations. The SR model enhances the simulation resolution by generating 512 times more tracer particles, extending into the deeply nonlinear regime where complex structure formation processes take place. We validate the SR model by deploying the model in 10 test simulations of box size 100 h−1 Mpc, and examine the matter power spectra, bispectra, and two-dimensional power spectra in redshift space. We find the generated SR field matches the true HR result at per cent level down to scales of k ∼ 10 h Mpc−1. We also identify and inspect dark matter haloes and their substructures. Our SR model generates visually authentic small-scale structures that cannot be resolved by the LR input, and are in good statistical agreement with the real HR results. The SR model performs satisfactorily on the halo occupation distribution, halo correlations in both real and redshift space, and the pairwise velocity distribution, matching the HR results with comparable scatter, thus demonstrating its potential in making mock halo catalogues. The SR technique can be a powerful and promising tool for modelling small-scale galaxy formation physics in large cosmological volumes.

Velocity-dependent J-factors for annihilation radiation from cosmological simulations

We determine the dark matter pair-wise relative velocity distribution in a set of Milky Way-like halos in the Auriga and APOSTLE simulations. Focusing on the smooth halo component, the relative velocity distribution is well-described by a Maxwell-Boltzmann distribution over nearly all radii in the halo. We explore the implications for velocity-dependent dark matter annihilation, focusing on four models which scale as different powers of the relative velocity: Sommerfeld, s-wave, p-wave, and d-wave models. We show that the J-factors scale as the moments of the relative velocity distribution, and that the halo-to-halo scatter is largest for d-wave, and smallest for Sommerfeld models. The J-factor is strongly correlated with the dark matter density in the halo, and is very weakly correlated with the velocity dispersion. This implies that if the dark matter density in the Milky Way can be robustly determined, one can accurately predict the dark matter annihilation signal, without the need to identify the dark matter velocity distribution in the Galaxy.

The moment generating function of pairwise velocity in the context of redshift space distortion

We study the connections between the pairwise velocity moment generating function G(k ∥, r) and redshift space distortion (RSD) modeling. Here k ∥ is the Fourier wavevector parallel to the line of sight, and r is the pair separation vector. We demonstrate its usage by two examples. (1) Besides the known relations between G and the RSD power spectrum (and the correlation function), we propose a new RSD statistics Ps (k ∥,r ⊥) whose connection to G is convenient to evaluate numerically. (2) We then develop a fast method to numerically evaluate G, and apply it to a high resolution N-body simulation. We find that G (ln G) shows complicated dependence on k ∥ beyond the linear and quadratic dependencies. This not only shows inaccuracy in some existing models and identifies sources of inaccuracy but also provides possible ways of improving the RSD modeling. Consequently, more comprehensive investigations on G are needed to fully explore the usage of G in RSD modeling.

A Semianalytic Model of the Pairwise Velocity Distribution between Dark Matter Halos

Abstract We study the probability distribution function (PDF) of relative velocity between two different dark matter halos (i.e., pairwise velocity) with high-resolution cosmological N-body simulations. We revisit a non-Gaussian framework to predict pairwise velocity statistics developed in Tinker. We investigate the pairwise velocity PDFs over a wide range of halo masses of 1012.5 ≲ M [h −1 M ⊙] ≲ 1015 and redshifts of 0 &lt; z &lt; 1. At a given set of masses, redshift, and separation length between two halos, our model requires three parameters to set the pairwise velocity PDF, whereas previous non-Gaussian models in the literature assumed four or more free parameters. At length scales of 5 &lt; r [h −1 Mpc] &lt; 40, our model predicts the mean and dispersion of the pairwise velocity for dark matter halos with masses of 1012.5 ≲ M [h −1 M ⊙] ≲ 1013.5 at 0.3 &lt; z &lt; 1 with a 5%-level precision. We demonstrate that our model of the pairwise velocity PDF provides an accurate mapping of the two-point clustering of massive-galaxy-sized halos at scales of O(10)h −1 Mpc between redshift and real space for a given real-space correlation function. For a mass-limited halo sample with masses greater than 1013.5 h −1 M ⊙ at z = 0.55, our model can explain the monopole and quadrupole moments of the redshift-space two-point correlations with a precision better than 5% at the scales of 5−40 and 10–30 h −1 Mpc, respectively. Our model of the pairwise velocity PDF will give a detailed explanation of the statistics of massive galaxies at intermediate scales in redshift surveys.

Imprint of baryons and massive neutrinos on velocity statistics

We explored the impact of baryonic effects (namely stellar and active galactic nuclei feedback) on the moments of pairwise velocity using the Illustris-TNG, EAGLE, cosmo-OWLS, and BAHAMAS suites of cosmological hydrodynamical simulations. The assumption that the mean pairwise velocity of the gas component follows that of the dark matter is studied here at small separations, and we find that even at pair separations of 10–20h−1Mpc, there is a 4–5% velocity bias. At smaller separations, it gets larger with varying strength depending on the sub-grid prescription. By isolating different physical processes, our findings suggest that the large-scale velocity bias is mainly driven by stellar rather than active galactic nuclei feedback. If unaccounted for, this velocity offset could possibly bias cosmological constraints from the kinetic Sunyaev-Zel’dovich effect in future cosmic microwave background (CMB) surveys. Furthermore, we examined how the first and the second moment of the pairwise velocity are affected by both the baryonic and the neutrino free-streaming effects for both the matter and gas components. For both moments, we were able to disentangle the effects of baryonic processes from those of massive neutrinos; and for pair separations below 20h−1Mpc, we find that these moments of the pairwise velocity decrease with increasing neutrino mass. Our work thus sets out a way in which the pairwise velocity statistics can be utilised to constrain the summed mass of neutrinos from future CMB surveys and peculiar velocity surveys.