Spherically Restricted Random Hyperbolic Diffusion

This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short- or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.

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Demonstrating the Tapered Gridded Estimator (TGE) for the Cosmological HI 21-cm Power Spectrum using 150 MHz GMRT observations

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa3831 ◽

2020 ◽

Author(s):

Srijita Pal ◽

Somnath Bharadwaj ◽

Abhik Ghosh ◽

Samir Choudhuri

Keyword(s):

Power Spectrum ◽

Unbiased Estimate ◽

Power Spectra ◽

Neutral Hydrogen ◽

Statistical Error ◽

Temperature Fluctuations ◽

Radio Frequency Interference ◽

Rectangular Region ◽

Angular Power Spectrum ◽

The Mean

Abstract We apply the Tapered Gridded Estimator (TGE) for estimating the cosmological 21-cm power spectrum from 150 MHz GMRT observations which corresponds to the neutral hydrogen (HI) at redshift z = 8.28. Here TGE is used to measure the Multi-frequency Angular Power Spectrum (MAPS) Cℓ(Δν) first, from which we estimate the 21-cm power spectrum P(k⊥, k∥). The data here are much too small for a detection, and the aim is to demonstrate the capabilities of the estimator. We find that the estimated power spectrum is consistent with the expected foreground and noise behaviour. This demonstrates that this estimator correctly estimates the noise bias and subtracts this out to yield an unbiased estimate of the power spectrum. More than $47\%$ of the frequency channels had to be discarded from the data owing to radio-frequency interference, however the estimated power spectrum does not show any artifacts due to missing channels. Finally, we show that it is possible to suppress the foreground contribution by tapering the sky response at large angular separations from the phase center. We combine the k modes within a rectangular region in the ‘EoR window’ to obtain the spherically binned averaged dimensionless power spectra Δ2(k) along with the statistical error σ associated with the measured Δ2(k). The lowest k-bin yields Δ2(k) = (61.47)2 K2 at k = 1.59 Mpc−1, with σ = (27.40)2 K2. We obtain a 2 σ upper limit of (72.66)2 K2 on the mean squared HI 21-cm brightness temperature fluctuations at k = 1.59 Mpc−1.

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Suppressing cosmic variance with paired-and-fixed cosmological simulations: average properties and covariances of dark matter clustering statistics

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa734 ◽

2020 ◽

Vol 496 (3) ◽

pp. 3862-3869 ◽

Cited By ~ 4

Author(s):

Anatoly Klypin ◽

Francisco Prada ◽

Joyce Byun

Keyword(s):

Dark Matter ◽

Power Spectrum ◽

Initial Conditions ◽

Cosmological Parameters ◽

Average Density ◽

Covariance Matrices ◽

Acoustic Oscillations ◽

Galaxy Clustering ◽

The Mean ◽

Statistical Noise

ABSTRACT Making cosmological inferences from the observed galaxy clustering requires accurate predictions for the mean clustering statistics and their covariances. Those are affected by cosmic variance – the statistical noise due to the finite number of harmonics. The cosmic variance can be suppressed by fixing the amplitudes of the harmonics instead of drawing them from a Gaussian distribution predicted by the inflation models. Initial realizations also can be generated in pairs with 180○ flipped phases to further reduce the variance. Here, we compare the consequences of using paired-and-fixed versus Gaussian initial conditions on the average dark matter clustering and covariance matrices predicted from N-body simulations. As in previous studies, we find no measurable differences between paired-and-fixed and Gaussian simulations for the average density distribution function, power spectrum, and bispectrum. Yet, the covariances from paired-and-fixed simulations are suppressed in a complicated scale- and redshift-dependent way. The situation is particularly problematic on the scales of Baryon acoustic oscillations where the covariance matrix of the power spectrum is lower by only $\sim 20{{\ \rm per\ cent}}$ compared to the Gaussian realizations, implying that there is not much of a reduction of the cosmic variance. The non-trivial suppression, combined with the fact that paired-and-fixed covariances are noisier than from Gaussian simulations, suggests that there is no path towards obtaining accurate covariance matrices from paired-and-fixed simulations – result, that is theoretically expected and accepted in the field. Because the covariances are crucial for the observational estimates of galaxy clustering statistics and cosmological parameters, paired-and-fixed simulations, though useful for some applications, cannot be used for the production of mock galaxy catalogues.

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On the Reliability of Initial Conditions for Dissipationless Cosmological Simulations

Publications of the Astronomical Society of Australia ◽

10.1071/as02039 ◽

2003 ◽

Vol 20 (2) ◽

pp. 173-183 ◽

Cited By ~ 3

Author(s):

Alexander Knebe ◽

Alvaro Domínguez

Keyword(s):

Power Spectrum ◽

Fundamental Mode ◽

Initial Conditions ◽

Interparticle Distance ◽

Particle Distributions ◽

The Mean ◽

Internal Properties ◽

Correlation Properties ◽

Random Nature ◽

Simulation Volume

AbstractWe present the study of ten random realisations of a density field characterised by a cosmological power spectrum P(k) at redshift z = 50. The reliability of such initial conditions for N-body simulations is tested with respect to their correlation properties. The power spectrum P(k) and the mass variance σM(r) do not show detectable deviations from the desired behaviour in the intermediate range of scales between the mean interparticle distance and the simulation volume. The estimator for ξ(r) is too noisy to detect any reliable signal at the initial redshift z = 50. The particle distributions are then evolved forward until z = 0. This allows us to explore the cosmic variance stemming from the random nature of the initial conditions. With cosmic variance we mean the fact that a simulation represents a single realisation of the stochastic initial conditions whereas the real Universe contains many realisations of regions of the size of the box; this problem affects most importantly the scales at about the fundamental mode. We study morphological descriptors of the matter distribution such as the genus, as well as the internal properties of the largest object(s) forming in the box. We find that the scatter is at least comparable to the scatter in the fundamental mode.

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A control design procedure for plants with random initial conditions and constraints on the mean square actuator rates

International Journal of Systems Science ◽

10.1080/00207729508929049 ◽

1995 ◽

Vol 26 (3) ◽

pp. 515-524

Author(s):

C. FRANGOS ◽

Y. YAVTN

Keyword(s):

Initial Conditions ◽

Design Procedure ◽

Control Design ◽

Mean Square ◽

Random Initial Conditions ◽

The Mean

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Predictions for measuring the 21-cm multifrequency angular power spectrum using SKA-Low

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa1026 ◽

2020 ◽

Vol 494 (3) ◽

pp. 4043-4056 ◽

Cited By ~ 3

Author(s):

Rajesh Mondal ◽

Abinash Kumar Shaw ◽

Ilian T Iliev ◽

Somnath Bharadwaj ◽

Kanan K Datta ◽

...

Keyword(s):

Power Spectrum ◽

Neutral Hydrogen ◽

Cosmic Time ◽

Observation Time ◽

Angular Power Spectrum ◽

Severe Problem ◽

Statistical Homogeneity ◽

Large Bandwidth ◽

The Mean ◽

Total Observation

ABSTRACT The light-cone effect causes the mean as well as the statistical properties of the redshifted 21-cm signal ${T_{\rm b}}(\hat{\boldsymbol {n}}, \nu)$ to change with frequency ν (or cosmic time). Consequently, the statistical homogeneity (ergodicity) of the signal along the line-of-sight (LoS) direction is broken. This is a severe problem particularly during the Epoch of Reionization (EoR) when the mean neutral hydrogen fraction ($\bar{x}_{\rm {H\,{\small I}}}$) changes rapidly as the Universe evolves. This will also pose complications for large bandwidth observations. These effects imply that the 3D power spectrum P(k) fails to quantify the entire second-order statistics of the signal as it assumes the signal to be ergodic and periodic along the LoS. As a proper alternative to P(k), we use the multifrequency angular power spectrum (MAPS) ${\mathcal {C}}_{\ell }(\nu _1,\nu _2)$, which does not assume the signal to be ergodic and periodic along the LoS. Here, we study the prospects for measuring the EoR 21-cm MAPS using future observations with the upcoming SKA-Low. Ignoring any contribution from the foregrounds, we find that the EoR 21-cm MAPS can be measured at a confidence level ≥5σ at angular scales ℓ ∼ 1300 for total observation time tobs ≥ 128 h across ∼44 MHz observational bandwidth. We also quantitatively address the effects of foregrounds on MAPS detectability forecast by avoiding signal contained within the foreground wedge in $({\boldsymbol {k}}_\perp , k_\parallel)$ plane. These results are very relevant for the upcoming large bandwidth EoR experiments as previous predictions were all restricted to individually analysing the signal over small frequency (or equivalent redshift) intervals.

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An Accurate Reconstruction of CMB E Mode Signal over Large Angular Scales using Prior Information of CMB Covariance Matrix in ILC Algorithm

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa3935 ◽

2020 ◽

Author(s):

Ujjal Purkayastha ◽

Vipin Sudevan ◽

Rajib Saha

Keyword(s):

Power Spectrum ◽

Linear Combination ◽

Covariance Matrix ◽

Large Scale ◽

Prior Information ◽

Future Generation ◽

Temperature Anisotropy ◽

Satellite Mission ◽

Angular Power Spectrum ◽

Accurate Reconstruction

Abstract Recently, the internal-linear-combination (ILC) method was investigated extensively in the context of reconstruction of Cosmic Microwave Background (CMB) temperature anisotropy signal using observations obtained by WMAP and Planck satellite missions. In this article, we, for the first time, apply the ILC method to reconstruct the large scale CMB E mode polarization signal, which could probe the ionization history, using simulated observations of 15 frequency CMB polarization maps of future generation Cosmic Origin Explorer (COrE) satellite mission. We find that the clean power spectra, from the usual ILC, are strongly biased due to non zero CMB-foregrounds chance correlations. In order to address the issues of bias and errors we extend and improve the usual ILC method for CMB E mode reconstruction by incorporating prior information of theoretical E mode angular power spectrum while estimating the weights for linear combination of input maps (Sudevan & Saha 2018b). Using the E mode covariance matrix effectively suppresses the CMB-foreground chance correlation power leading to an accurate reconstruction of cleaned CMB E mode map and its angular power spectrum. We compare the performance of the usual ILC and the new method over large angular scales and show that the later produces significantly statistically improved results than the former. The new E mode CMB angular power spectrum contains neither any significant negative bias at the low multipoles nor any positive foreground bias at relatively higher mutlipoles. The error estimates of the cleaned spectrum agree very well with the cosmic variance induced error.

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The anisotropy of the power spectrum in periodic cosmological simulations

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab874 ◽

2021 ◽

Vol 503 (4) ◽

pp. 5638-5645

Author(s):

Gábor Rácz ◽

István Szapudi ◽

István Csabai ◽

László Dobos

Keyword(s):

Dark Matter ◽

Power Spectrum ◽

Large Scale ◽

Cold Dark Matter ◽

Initial Conditions ◽

Power Spectra ◽

Cosmological Simulations ◽

Spherical Collapse ◽

Lower Amplitude ◽

Hydrodynamical Simulations

ABSTRACT The classical gravitational force on a torus is anisotropic and always lower than Newton’s 1/r2 law. We demonstrate the effects of periodicity in dark matter only N-body simulations of spherical collapse and standard Lambda cold dark matter (ΛCDM) initial conditions. Periodic boundary conditions cause an overall negative and anisotropic bias in cosmological simulations of cosmic structure formation. The lower amplitude of power spectra of small periodic simulations is a consequence of the missing large-scale modes and the equally important smaller periodic forces. The effect is most significant when the largest mildly non-linear scales are comparable to the linear size of the simulation box, as often is the case for high-resolution hydrodynamical simulations. Spherical collapse morphs into a shape similar to an octahedron. The anisotropic growth distorts the large-scale ΛCDM dark matter structures. We introduce the direction-dependent power spectrum invariant under the octahedral group of the simulation volume and show that the results break spherical symmetry.

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Statistical modelling of the cosmological dispersion measure

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab170 ◽

2021 ◽

Vol 502 (2) ◽

pp. 2615-2629

Author(s):

Ryuichi Takahashi ◽

Kunihito Ioka ◽

Asuka Mori ◽

Koki Funahashi

Keyword(s):

Dark Matter ◽

Correlation Function ◽

Probability Distribution ◽

Power Spectrum ◽

Free Electron ◽

Statistical Modelling ◽

Dispersion Measure ◽

Host Galaxies ◽

Angular Power Spectrum ◽

Bias Factor

ABSTRACT We have investigated the basic statistics of the cosmological dispersion measure (DM)—such as its mean, variance, probability distribution, angular power spectrum, and correlation function—using the state-of-the-art hydrodynamic simulations, IllustrisTNG300, for the fast radio burst cosmology. To model the DM statistics, we first measured the free-electron abundance and the power spectrum of its spatial fluctuations. The free-electron power spectrum turns out to be consistent with the dark matter power spectrum at large scales, but it is strongly damped at small scales (≲ Mpc) owing to the stellar and active galactic nucleus feedback. The free-electron power spectrum is well modelled using a scale-dependent bias factor (the ratio of its fluctuation amplitude to that of the dark matter). We provide analytical fitting functions for the free-electron abundance and its bias factor. We next constructed mock sky maps of the DM by performing standard ray-tracing simulations with the TNG300 data. The DM statistics are calculated analytically from the fitting functions of the free-electron distribution, which agree well with the simulation results measured from the mock maps. We have also obtained the probability distribution of source redshift for a given DM, which helps in identifying the host galaxies of FRBs from the measured DMs. The angular two-point correlation function of the DM is described by a simple power law, $\xi (\theta) \approx 2400 (\theta /{\rm deg})^{-1} \, {\rm pc}^2 \, {\rm cm}^{-6}$, which we anticipate will be confirmed by future observations when thousands of FRBs are available.

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