arXiv subjects

Juliana Kwan

Publications and source records attributed to Juliana Kwan.

At least 19 recordsLinked to original sources

arXiv2025

The FLAMINGO project: the coupling between baryonic feedback and cosmology in light of the $S_8$ tension

Large-scale structure surveys have reported measurements of the density of matter, $Ω_\mathrm{m}$, and the amplitude of clustering, $σ_8$, that are in tension with the values inferred from observations of the cosmic microwave background. While this may be a sign of new physics that slows the growth of structure at late times, strong astrophysical feedback processes could also be responsible. In this work, we argue that astrophysical processes are not independent of cosmology and that their coupling naturally leads to stronger baryonic feedback in cosmological models with suppressed structure formation or when combined with a mechanism that removes dark matter from halos. We illustrate this with two well-motivated extensions of the Standard Model known to suppress structure formation: massive neutrinos and decaying dark matter. Our results, based on the FLAMINGO suite of hydrodynamical simulations, show that the combined effect of baryonic and non-baryonic suppression mechanisms is greater than the sum of its parts, particularly for decaying dark matter. We also show that the dependence of baryonic feedback on cosmology can be modelled as a function of the ratio $f_\mathrm{b}/c^2_\mathrm{v}\sim f_\mathrm{b}/(Ω_\mathrm{m}σ_8)^{1/4}$ of the universal baryon fraction, $f_\mathrm{b}$, to a velocity-based definition of halo concentration, $c^2_\mathrm{v}$, giving an accurate fitting formula for the baryonic suppression of the matter power spectrum. Although the combination of baryonic and non-baryonic suppression mechanisms can resolve the tension, the models with neutrinos and decaying dark matter are challenged by constraints on the expansion history.

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arXiv2024

Bayesian "Deep" Process Convolutions: An Application in Cosmology

The nonlinear matter power spectrum in cosmology describes how matter density fluctuations vary with scale in the universe, providing critical insights into large-scale structure formation. The matter power spectrum includes both smooth regions and highly oscillatory features. Cosmologists rely on noisy, multi-resolution realizations of large N-body simulations to study these phenomena, which require appropriate smoothing techniques to learn about underlying structures. We introduce a Bayesian Deep Process Convolution (DPC) model that flexibly adapts its smoothness parameter across the input space, enabling it to capture both smooth and variable structure within a single framework. The DPC model leverages common patterns across related functions to improve estimation in regions with sparse data. Compared to existing methods, the DPC model offers superior accuracy and uncertainty quantification in simulated data, and qualitatively superior performance with the cosmological data. This methodology will be useful in cosmology and other fields requiring flexible modeling of smooth nonstationary surfaces.

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arXiv2024

Revisiting the effects of baryon physics on small-scale redshift space distortions

Redshift space distortions are an important probe of the growth of large-scale structure and for constraining cosmological parameters in general. As galaxy redshift surveys approach percent level precision in their observations of the two point clustering statistics, it is timely to review what effects baryons and associated processes such as feedback may have on small-scale clustering in redshift space. Contrary to previous studies in the literature, we show using the large-volume BAHAMAS hydrodynamic simulations that the effect of baryons can be as much as 1% in the k ~ 0.1 h/Mpc range for the monopole and 5% for quadrupole, and that this could rise to as much as 10% at k~10 h/Mpc in both measurements. For the halo power spectra, this difference can be as much 3-4% in the monopole on scales of 0.05 < k < 0.3 h/Mpc for 10^{13} M_sun/h haloes. We find that these deviations can be mitigated to the sub-percent level in the both the monopole and quadrupole up to k ~ 0.3 h/Mpc if the baryon corrected halo masses are used to calculate the redshift space power spectra. Finally, we use the cosmo-OWLS simulation suite to explore the changes in the redshift space power spectra with different feedback prescriptions, finding that there is a maximum of 15-20% difference between the redshift space monopole and quadrupole with and without baryons at k ~1-2 h/Mpc within these models.

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arXiv2024

The FLAMINGO Project: Galaxy clusters in comparison to X-ray observations

Galaxy clusters are important probes for both cosmology and galaxy formation physics. We test the cosmological, hydrodynamical FLAMINGO simulations by comparing to observations of the gaseous properties of clusters measured from X-ray observations. FLAMINGO contains unprecedented numbers of massive galaxy groups ($>10^6$) and clusters ($>10^5$) and includes variations in both cosmology and galaxy formation physics. We predict the evolution of cluster scaling relations as well as radial profiles of the temperature, density, pressure, entropy, and metallicity for different masses and redshifts. We show that the differences between volume-, and X-ray-weighting of particles in the simulations, and between cool-core non cool-core samples, are similar in size as the differences between simulations for which the stellar and AGN feedback has been calibrated to produce significantly different gas fractions. Compared to thermally-driven AGN feedback, kinetic jet feedback calibrated to produce the same gas fraction at $R_{\rm 500c}$ yields a hotter core with higher entropies and lower densities, which translates into a smaller fraction of cool-core clusters. Stronger feedback, calibrated to produce lower gas fractions and hence lower gas densities, results in higher temperatures, entropies, and metallicities, but lower pressures. The scaling relations and thermodynamic profiles show almost no evolution with respect to self-similar expectations, except for the metallicity decreasing with redshift. We find that the temperature, density, pressure, and entropy profiles of clusters in the fiducial FLAMINGO simulation are in excellent agreement with observations, while the metallicities in the core are too high.

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arXiv2024

Inferring the dark matter splashback radius from cluster gas and observable profiles in the FLAMINGO simulations

The splashback radius, coinciding with the minimum in the dark matter radial density gradient, is thought to be a universal definition of the edge of a dark matter halo. Observational methods to detect it have traced the dark matter using weak gravitational lensing or galaxy number counts. Recent attempts have also claimed the detection of a similar feature in Sunyaev-Zel'dovich (SZ) observations of the hot intracluster gas. Here, we use the FLAMINGO simulations to investigate whether an extremum gradient in a similar position to the splashback radius is predicted to occur in the cluster gas profiles. We find that the minimum in the gradient of the stacked 3D gas density and pressure profiles, and the maximum in the gradient of the entropy profile, broadly align with the splashback feature though there are significant differences. While the dark matter splashback radius varies with specific mass accretion rate, in agreement with previous work, the radial position of the deepest minimum in the log-slope of the gas density is more sensitive to halo mass. In addition, we show that a similar minimum is also present in projected 2D pseudo-observable profiles: emission measure (X-ray); Compton-$y$ (SZ) and surface mass density (weak lensing). We find that the latter traces the dark matter results reasonably well albeit the minimum occurs at a slightly smaller radius. While results for the gas profiles are largely insensitive to accretion rate and various observable proxies for dynamical state, they do depend on the strength of the feedback processes.

astro-ph.CO↗

arXiv2024

The FLAMINGO project: Baryonic impact on weak gravitational lensing convergence peak counts

Weak gravitational lensing convergence peaks, the local maxima in weak lensing convergence maps, have been shown to contain valuable cosmological information complementary to commonly used two-point statistics. To exploit the full power of weak lensing for cosmology, we must model baryonic feedback processes because these reshape the matter distribution on non-linear and mildly non-linear scales. We study the impact of baryonic physics on the number density of weak lensing peaks using the FLAMINGO cosmological hydrodynamical simulation suite. We generate ray-traced full-sky convergence maps mimicking the characteristics of a Stage IV weak lensing survey. We compare the number densities of peaks in simulations that have been calibrated to reproduce the observed galaxy mass function and cluster gas fraction or to match a shifted version of these, and that use either thermally driven or jet AGN feedback. We show that the differences induced by realistic baryonic feedback prescriptions (typically $5 - 30\%$ for $κ= 0.1 - 0.4$) are smaller than those induced by reasonable variations in cosmological parameters ($20 - 60\%$ for $κ= 0.1 - 0.4$) but must be modeled carefully to obtain unbiased results. The reasons behind these differences can be understood by considering the impact of feedback on halo masses, or by considering the impact of different cosmological parameters on the halo mass function. Our analysis demonstrates that, for the range of models we investigated, the baryonic suppression is insensitive to changes in cosmology up to $κ\approx 0.4$ and that the higher $κ$ regime is dominated by Poisson noise and cosmic variance.

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arXiv2023

Cosmic-Enu: An emulator for the non-linear neutrino power spectrum

Cosmology is poised to measure the neutrino mass sum $M_ν$ and has identified several smaller-scale observables sensitive to neutrinos, necessitating accurate predictions of neutrino clustering over a wide range of length scales. The FlowsForTheMasses non-linear perturbation theory for the massive neutrino power spectrum, $Δ^2_ν(k)$, agrees with its companion N-body simulation at the $10\%-15\%$ level for $k \leq 1~h/$Mpc. Building upon the Mira-Titan IV emulator for the cold matter, we use FlowsForTheMasses to construct an emulator for $Δ^2_ν(k)$ covering a large range of cosmological parameters and neutrino fractions $Ω_{ν,0} h^2 \leq 0.01$, which corresponds to $M_ν\leq 0.93$~eV. Consistent with FlowsForTheMasses at the $3.5\%$ level, it returns a power spectrum in milliseconds. Ranking the neutrinos by initial momenta, we also emulate the power spectra of momentum deciles, providing information about their perturbed distribution function. Comparing a $M_ν=0.15$~eV model to a wide range of N-body simulation methods, we find agreement to $3\%$ for $k \leq 3 k_\mathrm{FS} = 0.17~h/$Mpc and to $19\%$ for $k \leq 0.4~h/$Mpc. We find that the enhancement factor, the ratio of $Δ^2_ν(k)$ to its linear-response equivalent, is most strongly correlated with $Ω_{ν,0} h^2$, and also with the clustering amplitude $σ_8$. Furthermore, non-linearities enhance the free-streaming-limit scaling $\partial \log(Δ^2_ν/ Δ^2_{\rm m}) / \partial \log(M_ν)$ beyond its linear value of 4, increasing the $M_ν$-sensitivity of the small-scale neutrino density.

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arXiv2023

FLAMINGO: Calibrating large cosmological hydrodynamical simulations with machine learning

To fully take advantage of the data provided by large-scale structure surveys, we need to quantify the potential impact of baryonic effects, such as feedback from active galactic nuclei (AGN) and star formation, on cosmological observables. In simulations, feedback processes originate on scales that remain unresolved. Therefore, they need to be sourced via subgrid models that contain free parameters. We use machine learning to calibrate the AGN and stellar feedback models for the FLAMINGO cosmological hydrodynamical simulations. Using Gaussian process emulators trained on Latin hypercubes of 32 smaller-volume simulations, we model how the galaxy stellar mass function and cluster gas fractions change as a function of the subgrid parameters. The emulators are then fit to observational data, allowing for the inclusion of potential observational biases. We apply our method to the three different FLAMINGO resolutions, spanning a factor of 64 in particle mass, recovering the observed relations within the respective resolved mass ranges. We also use the emulators, which link changes in subgrid parameters to changes in observables, to find models that skirt or exceed the observationally allowed range for cluster gas fractions and the stellar mass function. Our method enables us to define model variations in terms of the data that they are calibrated to rather than the values of specific subgrid parameters. This approach is useful, because subgrid parameters are typically not directly linked to particular observables, and predictions for a specific observable are influenced by multiple subgrid parameters.

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arXiv2023

The FLAMINGO project: cosmological hydrodynamical simulations for large-scale structure and galaxy cluster surveys

We introduce the Virgo Consortium's FLAMINGO suite of hydrodynamical simulations for cosmology and galaxy cluster physics. To ensure the simulations are sufficiently realistic for studies of large-scale structure, the subgrid prescriptions for stellar and AGN feedback are calibrated to the observed low-redshift galaxy stellar mass function and cluster gas fractions. The calibration is performed using machine learning, separately for three resolutions. This approach enables specification of the model by the observables to which they are calibrated. The calibration accounts for a number of potential observational biases and for random errors in the observed stellar masses. The two most demanding simulations have box sizes of 1.0 and 2.8 Gpc and baryonic particle masses of $1\times10^8$ and $1\times10^9 \text{M}_\odot$, respectively. For the latter resolution the suite includes 12 model variations in a 1 Gpc box. There are 8 variations at fixed cosmology, including shifts in the stellar mass function and/or the cluster gas fractions to which we calibrate, and two alternative implementations of AGN feedback (thermal or jets). The remaining 4 variations use the unmodified calibration data but different cosmologies, including different neutrino masses. The 2.8 Gpc simulation follows $3\times10^{11}$ particles, making it the largest ever hydrodynamical simulation run to $z=0$. Lightcone output is produced on-the-fly for up to 8 different observers. We investigate numerical convergence, show that the simulations reproduce the calibration data, and compare with a number of galaxy, cluster, and large-scale structure observations, finding very good agreement with the data for converged predictions. Finally, by comparing hydrodynamical and `dark-matter-only' simulations, we confirm that baryonic effects can suppress the halo mass function and the matter power spectrum by up to $\approx20$ per cent.

astro-ph.CO↗

arXiv2023

The FLAMINGO project: revisiting the $S_8$ tension and the role of baryonic physics

A number of recent studies have found evidence for a tension between observations of large-scale structure (LSS) and the predictions of the standard model of cosmology with the cosmological parameters fit to the cosmic microwave background (CMB). The origin of this '$S_8$ tension' remains unclear, but possibilities include new physics beyond the standard model, unaccounted for systematic errors in the observational measurements and/or uncertainties in the role that baryons play. Here we carefully examine the latter possibility using the new FLAMINGO suite of large-volume cosmological hydrodynamical simulations. We project the simulations onto observable harmonic space and compare with observational measurements of the power and cross-power spectra of cosmic shear, CMB lensing, and the thermal Sunyaev-Zel'dovich (tSZ) effect. We explore the dependence of the predictions on box size and resolution, cosmological parameters including the neutrino mass, and the efficiency and nature of baryonic 'feedback'. Despite the wide range of astrophysical behaviours simulated, we find that baryonic effects are not sufficiently large to remove the $S_8$ tension. Consistent with recent studies, we find the CMB lensing power spectrum is in excellent agreement with the standard model, whilst the cosmic shear power spectrum, tSZ effect power spectrum, and the cross-spectra between shear, CMB lensing, and the tSZ effect are all in varying degrees of tension with the CMB-specified standard model. These results suggest that some mechanism is required to slow the growth of fluctuations at late times and/or on non-linear scales, but that it is unlikely that baryon physics is driving this modification.

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arXiv2023

Non-linear CMB lensing with neutrinos and baryons: FLAMINGO simulations vs. fast approximations

Weak lensing of the cosmic microwave background is rapidly emerging as a powerful probe of neutrinos, dark energy, and new physics. We present a fast computation of the non-linear CMB lensing power spectrum which combines non-linear perturbation theory at early times with power spectrum emulation using cosmological simulations at late times. Comparing our calculation with lightcones from the FLAMINGO 5.6 Gpc cube dark-matter-only simulation, we confirm its accuracy to 1% (2%) up to multipoles L = 3000 (L = 5000) for a nuLambdaCDM cosmology consistent with current data. Clustering suppression due to small-scale baryonic phenomena such as feedback from active galactic nuclei can reduce the lensing power by of order 10%. To our perturbation theory and emulator-based calculation we add SP(k), a new fitting function for this suppression, and confirm its accuracy compared to the FLAMINGO hydrodynamic simulations to 4% at L = 5000, with similar accuracy for massive neutrino models. We further demonstrate that scale-dependent suppression due to neutrinos and baryons approximately factorize, implying that a careful treatment of baryonic feedback can limit biasing neutrino mass constraints.

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arXiv2023

Galaxy Clustering in the Mira-Titan Universe I: Emulators for the redshift space galaxy correlation function and galaxy-galaxy lensing

We construct accurate emulators for the projected and redshift space galaxy correlation functions and excess surface density as measured by galaxy-galaxy lensing, based on Halo Occupation Distribution (HOD) modeling. Using the complete Mira-Titan suite of 111 $N$-body simulations, our emulators vary over eight cosmological parameters and include the effects of neutrino mass and dynamical dark energy. We demonstrate that our emulators are sufficiently accurate for the analysis of the BOSS DR12 CMASS galaxy sample over the range 0.5 < r < 50 Mpc/h. Furthermore, we show that our emulators are capable of recovering unbiased cosmological constraints from realistic mock catalogs over the same range. Our mock catalog tests show the efficacy of combining small scale galaxy-galaxy lensing with redshift space clustering and that we can constrain the growth rate and σ_8 to 7% and 4.5% respectively for a CMASS-like sample using only the measurements covered by our emulator. With the inclusion of a CMB prior on H_0, this reduces to a 2% measurement on the growth rate.

astro-ph.CO↗

arXiv2023

SP(k) -- A hydrodynamical simulation-based model for the impact of baryon physics on the non-linear matter power spectrum

Upcoming large-scale structure surveys will measure the matter power spectrum to approximately percent level accuracy with the aim of searching for evidence for new physics beyond the standard model of cosmology. In order to avoid biasing our conclusions, the theoretical predictions need to be at least as accurate as the measurements for a given choice of cosmological parameters. However, recent theoretical work has shown that complex physical processes associated with galaxy formation (particularly energetic feedback processes associated with stars and especially supermassive black holes) can alter the predictions by many times larger than the required accuracy. Here we present $\texttt{SP(k)}$, a model for the effects of baryon physics on the non-linear matter power spectrum based on a new large suite of hydrodynamical simulations. Specifically, the ANTILLES suite consists of 400 simulations spanning a very wide range of the "feedback landscape" and show that the effects of baryons on the matter power spectrum can be understood at approaching the percent level in terms of the mean baryon fraction of haloes, at scales of up to $k \lesssim 10 \, h \, $Mpc$^{-1}$ and redshifts up to $z=3$. For the range of scales and redshifts that will be probed by forthcoming cosmic shear measurements, most of the effects are driven by galaxy group-mass haloes ($M \sim 10^{13-14}$ M$_\odot$). We present a simple Python implementation of our model, available at $\href{https://github.com/jemme07/pyspk}{\mathrm{https{:}//github.com/jemme07/pyspk}}$, which can be used to incorporate baryon effects in standard gravity-only predictions, allowing for marginalisation over baryon physics within cosmological pipelines.

astro-ph.CO↗

arXiv2023

The halo bispectrum as a sensitive probe of massive neutrinos and baryon physics

The power spectrum has been a workhorse for cosmological studies of large-scale structure. However, the present-day matter distribution is highly non-Gaussian and significant cosmological information is also contained in higher-order correlation functions. Meanwhile, baryon physics (particularly AGN feedback) has previously been shown to strongly affect the two-point statistics but there has been limited exploration of its effects on higher-order functions to date. Here we use the BAHAMAS suite of cosmological hydrodynamical simulations to explore the effects of baryon physics and massive neutrinos on the halo bispectrum. In contrast to matter clustering which is suppressed by baryon physics, we find that the halo clustering is typically enhanced. The strength of the effect and the scale over which it extends depends on how haloes are selected. On small scales (k > 1 $h$ Mpc$^{-1}$, dominated by satellites of groups/clusters), we find that the bispectrum is highly sensitive to the efficiency of star formation and feedback, making it an excellent testing ground for galaxy formation models. We show that the effects of feedback and the effects of massive neutrinos are largely separable (independent of each other) and that massive neutrinos strongly suppress the halo bispectrum on virtually all scales up to the free-streaming length (apart from the smallest scales, where baryon physics dominates). The strong sensitivity of the bispectrum to neutrinos on the largest scales and galaxy formation physics on the smallest scales bodes well for upcoming precision measurements from the next generation of wide-field surveys.

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arXiv2022

The Mira-Titan Universe IV. High Precision Power Spectrum Emulation

Modern cosmological surveys are delivering datasets characterized by unprecedented quality and statistical completeness; this trend is expected to continue into the future as new ground- and space-based surveys come online. In order to maximally extract cosmological information from these observations, matching theoretical predictions are needed. At low redshifts, the surveys probe the nonlinear regime of structure formation where cosmological simulations are the primary means of obtaining the required information. The computational cost of sufficiently resolved large-volume simulations makes it prohibitive to run very large ensembles. Nevertheless, precision emulators built on a tractable number of high-quality simulations can be used to build very fast prediction schemes to enable a variety of cosmological inference studies. We have recently introduced the Mira-Titan Universe simulation suite designed to construct emulators for a range of cosmological probes. The suite covers the standard six cosmological parameters $\{ω_m,ω_b, σ_8, h, n_s, w_0\}$ and, in addition, includes massive neutrinos and a dynamical dark energy equation of state, $\{ω_ν, w_a\}$. In this paper we present the final emulator for the matter power spectrum based on 111 cosmological simulations, each covering a (2.1Gpc)$^3$ volume and evolving 3200$^3$ particles. An additional set of 1776 lower-resolution simulations and TimeRG perturbation theory results for the power spectrum are used to cover scales straddling the linear to mildly nonlinear regimes. The emulator provides predictions at the two to three percent level of accuracy over a wide range of cosmological parameters and is publicly released as part of this paper.

astro-ph.CO↗

arXiv2022

The morphology of star-forming gas and its alignment with galaxies and dark matter haloes in the EAGLE simulations

We present measurements of the morphology of star-forming gas in galaxies from the EAGLE simulations, and its alignment relative to stars and dark matter (DM). Imaging of such gas in the radio continuum enables weak lensing experiments that complement traditional optical approaches. Star-forming gas is typically more flattened than its associated stars and DM, particularly for present-day subhaloes of total mass $\sim$$10^{ 12-12.5} \mathrm{M_{ \odot}}$, which preferentially host star-forming galaxies with rotationally-supported stellar discs. Such systems have oblate, spheroidal star-forming gas distributions, but in both less- and more-massive subhaloes the distributions tend to be prolate, and its morphology correlates positively and significantly with that of its host galaxy's stars, both in terms of sphericity and triaxiality. The minor axis of star-forming gas most commonly aligns with the minor axis of its host subhalo's DM, but often aligns more closely with one of the other two principal axes of the DM distribution in prolate subhaloes. Star-forming gas aligns with DM less strongly than is the case for stars, but its morphological minor axis aligns closely with its kinematic axis, affording a route to observational identification of the unsheared morphological axis. The projected ellipticities of star-forming gas in EAGLE are consistent with shapes inferred from high-fidelity radio continuum images, and they exhibit greater shape noise than is the case for images of the stars, owing to the greater characteristic flattening of star-forming gas with respect to stars.

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arXiv2021

The BAHAMAS project: Evaluating the accuracy of the halo model in predicting the non-linear matter power spectrum

The halo model formalism is widely adopted in cosmological studies for predicting the growth of large-scale structure in the Universe. However, to date there have been relatively few direct comparisons of the halo model with more accurate (but much more computationally expensive) cosmological simulations. We test the accuracy of the halo model in reproducing the non-linear matter power spectrum, P(k), when the main inputs of the halo model (specifically the matter density profiles, halo mass function, and linear bias) are taken directly from the BAHAMAS simulations and we assess how well the halo model reproduces P(k) from the same simulations. We show that the halo model generally reproduces P(k) in the deep non-linear regime (1-halo) to typically a few percent accuracy, but struggles to reproduce (approx. 15% error) P(k) at intermediate scales of 0.1 < k [h/ Mpc] < 3 at z=0, marking the transition between the 1-halo and 2-halo terms. We show that the magnitude of this error is a strong function of the halo mass definition (through its effects on radial extent of haloes) and of redshift. Furthermore, we test the accuracy of the halo model in recovering the relative impact of baryons on P(k). We show that the systematic errors in recovering the absolute P(k) largely cancel when considering the relative impact of baryons. This suggests that the halo model can make precise predictions for the baryonic suppression, offering a fast and accurate way to adjust collisionless matter power spectra for the presence of baryons and associated processes.

astro-ph.CO↗

arXiv2021

Testing extensions to LCDM on small scales with forthcoming cosmic shear surveys

We investigate the constraining power of forthcoming Stage-IV weak lensing surveys (Euclid, LSST, and NGRST) for extensions to the LCDM model on small scales, via their impact on the cosmic shear power spectrum. We use high-resolution cosmological simulations to calculate how warm dark matter (WDM), self-interacting dark matter (SIDM) and a running of the spectral index affect the non-linear matter power spectrum, P(k), as a function of scale and redshift. We evaluate the cosmological constraining power using synthetic weak lensing observations derived from these power spectra and that take into account the anticipated source densities, shape noise and cosmic variance errors of upcoming surveys. We show that upcoming Stage-IV surveys will be able to place useful, independent constraints on both WDM models (ruling out models with a particle mass of < 0.5 keV) and SIDM models (ruling out models with a velocity-independent cross-section of > 10 cm^2 g^-1) through their effects on the small-scale cosmic shear power spectrum. Similarly, they will be able to strongly constrain cosmologies with a running spectral index. Finally, we explore the error associated with the cosmic shear cross-spectrum between tomographic bins, finding that it can be significantly affected by Poisson noise (the standard assumption is that the Poisson noise cancels between tomographic bins). We provide a new analytic form for the error on the cross-spectrum which accurately captures this effect.

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