Analytic model for the matter power spectrum, its covariance matrix, and baryonic effects

We develop a model for the matter power spectrum as the sum of Zeldovich approximation and even powers of $k$, i.e., $A_0 - A_2k^2 + A_4k^4 - ...$, compensated at low $k$. With terms up to $k^4$ the model can predict the true power spectrum to a few percent accuracy up to $k\sim 0.7 h \rm{Mpc}^{-1}$, over a wide range of redshifts and models. The $A_n$ coefficients contain information about cosmology, in particular amplitude of fluctuations. We write a simple form of the covariance matrix as a sum of Gaussian part and $A_0$ variance, which reproduces the simulations remarkably well. In contrast, we show that one needs an N-body simulation volume of more than 1000 $({\rm Gpc}/h)^3$ to converge to 1\% accuracy on covariance matrix. We investigate the super-sample variance effect and show it can be modeled as an additional parameter that can be determined from the data. This allows a determination of $\sigma_8$ amplitude to about 0.2\% for a survey volume of 1$({\rm Gpc}/h)^3$, compared to 0.4\% otherwise. We explore the sensitivity of these coefficients to baryonic effects using hydrodynamic simulations of van Daalen (2011). We find that because of baryons redistributing matter inside halos all the coefficients $A_{2n}$ for $n>0$ are strongly affected by baryonic effects, while $A_0$ remains almost unchanged, a consequence of halo mass conservation. Our results suggest that observations such as weak lensing power spectrum can be effectively marginalized over the baryonic effects, while still preserving the bulk of the cosmological information contained in $A_0$ and Zeldovich terms.Comment: 21 pages,11 figures, 1 table; Accepted for publication in MNRA

Flashover performance of lightning protected buildings using scaled models and electric field analysis

In early era, Benjamin Franklin discovered that the application of Lightning Rod (also known as the Franklin Rod) method is found to be effectived as a lightning protective device for buildings. Hence, it was considered among the best solution to overcome the problems facing by publics due to lightning strikes. However, few years later it was found that the corroded Franklin Rod due to the impact of environmental contaminations tends to reduce its ability to effectively capture the lightning strikes. The directly or indirectly impacts of lightning strikes had caused owners to spend huge amount of money just to repair damages on the buildings. Nowadays, there were many professional standards and documents guiding public to properly install the building’s lightning protection system, yet the same damages problems had shown to be frequently occur that related to the strikes often bypasses the of Lightning Air Terminal (LAT) system. The main reason for this could be due to lacking ideas by learned circle of lightning experts as not to fully understand the behavior of Franklin Rods system when it interacts with the lightning leaders. Therefore, this thesis discusses the works that investigated the flashover performances occurred on the buildings with various structural geometry shapes. The case study method is using small scaled models for both laboratory and simulation works, aiming to understand the Franklin Rods performance on capturing lightning leaders. Summarizing the works, about 11 scaled-down building shape models equipped with Franklin Rods system are selected in the case studies such as follows; a conical, gable, triangular, half circle, L-shape, square, cylindrical, butterfly, pyramid, rectangular and inclined like shapes. These models were then injected with 30 lightning flashes each using the 100 kVpeak single stage impulse generator. This number of flashes is considered as total two-years lightning activity frequencies in Malaysia, which the lighting flash density is statistically recorded to be around 15 flashes / year / km2. The maximum applied voltage is about 86.5 kVpeak. The model scaling concept is based on 1:30 cm ratio for every 3 m height of building structure. Interestingly, the overall work data had shown that the pyramid-like shapes is found to be the best structure type to be used in reducing the LAT bypasses and direct strike damages. The structure’s Franklin Rod protection system captured the least number of strikes during competitive tests conducted on all of the scaled down building models. Works of electric field analysis on all building models were conducted using ANSYS Maxwell simulation tool. Utilisation of electric field plot data in this work enables the creation of likelihood factor (ranging from 0.1 to 0.9) method that so useful to capable predict the strikes pattern occurring on dedicated terminal rod. Both laboratory and simulation work also confirm that the edge shapes play crucial roles as intense electric fields is found to accumulate on the edges area when the Franklin Rod intercepts the lightning leaders. These mentioned findings lead to introducing better method of LAT placement on the top of the building, whereby the existing lightning protection system is recommended to have one of installed LAT rods elongated to act as sacrificial point to directly attract lightning strikes. All the work and key findings in this work can contribute to the science and technology field toward having a better LAT lightning protection system and also lead to better decision in selecting / designing the shapes and edges concept as to reduce likelihood of LAT bypasses and damages of the building structure

Perturbative approach to covariance matrix of the matter power spectrum

We evaluate the covariance matrix of the matter power spectrum using perturbation theory up to dominant terms at 1-loop order and compare it to numerical simulations. We decompose the covariance matrix into the disconnected (Gaussian) part, trispectrum from the modes outside the survey (beat coupling or super-sample variance), and trispectrum from the modes inside the survey, and show how the different components contribute to the overall covariance matrix. We find the agreement with the simulations is at a 10\% level up to $k \sim 1 h {\rm Mpc^{-1}}$. We show that all the connected components are dominated by the large-scale modes ($k<0.1 h {\rm Mpc^{-1}}$), regardless of the value of the wavevectors $k,\, k'$ of the covariance matrix, suggesting that one must be careful in applying the jackknife or bootstrap methods to the covariance matrix. We perform an eigenmode decomposition of the connected part of the covariance matrix, showing that at higher $k$ it is dominated by a single eigenmode. The full covariance matrix can be approximated as the disconnected part only, with the connected part being treated as an external nuisance parameter with a known scale dependence, and a known prior on its variance for a given survey volume. Finally, we provide a prescription for how to evaluate the covariance matrix from small box simulations without the need to simulate large volumes.Comment: 22 pages, 17 figures, 1 tabl

Generalisations of Integral Inequalities of Hermite-Hadamard type through Convexity

In this paper, we establish various inequalities for some differentiable mappings that are linked with the illustrious Hermite-Hadamard integral inequality for mappings whose derivatives are $s$-$(\alpha,m)$-convex.The generalised integral inequalities contribute some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.Comment: 11 Page