arXiv subjects

Takahiko Matsubara

Publications and source records attributed to Takahiko Matsubara.

At least 19 recordsLinked to original sources

arXiv2026

Expected number density of critical points of smooth Gaussian random fields in arbitrary dimensions

We obtain explicit formulas for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields on $\mathbb{R}^d$. The expected number density formula is expressed in terms of at most one-dimensional integrals, regardless of the dimension $d$. To obtain the formulas, we provide a variant of de Bruijn's theorem, as well as Weierstrass' convolution formula with a Gaussian random variable and its inversion.

math.ST↗

arXiv2024

Integrated perturbation theory for cosmological tensor fields. I. Basic formulation

In order to extract maximal information about cosmology from the large-scale structure of the Universe, one needs to use every bit of signal that can be observed. Beyond the spatial distributions of astronomical objects, the spatial correlations of tensor fields, such as galaxy spins and shapes, are ones of promising sources that can be accessed in the era of large surveys in the near future. The perturbation theory is a powerful tool to analytically describe the behaviors and evolutions of correlation statistics on large scales for a given cosmology. In this paper, we formulate a nonlinear perturbation theory of tensor fields in general, based on the formulation of integrated perturbation theory for the scalar-valued bias, generalizing it to include the tensor-valued bias. To take advantage of rotational symmetry, the formalism is constructed on the basis of the irreducible decomposition of tensors, identifying physical variables which are invariant under the rotation of the coordinates system.

astro-ph.CO↗

arXiv2024

Integrated perturbation theory for cosmological tensor fields. II. Loop corrections

In the previous paper [arXiv:2210.10435], the nonlinear perturbation theory of cosmological density field is generalized to include the tensor-valued bias of astronomical objects, such as spins and shapes of galaxies and any other tensors of arbitrary ranks which are associated with objects that we can observe. We apply this newly developed method to explicitly calculate nonlinear power spectra and correlation functions both in real space and in redshift space. Multi-dimensional integrals that appear in loop corrections are reduced to combinations of one-dimensional Hankel transforms, thanks to the spherical basis of the formalism, and the final expressions are numerically evaluated in a very short time using an algorithm of the fast Fourier transforms such as \textsc{FFTLog}. As an illustrative example, numerical evaluations of loop corrections of the power spectrum and correlation function of the rank-2 tensor field are demonstrated with a simple model of tensor bias.

astro-ph.CO↗

arXiv2024

Integrated perturbation theory for cosmological tensor fields. III. Projection effects

The integrated perturbation theory (iPT) is a set of methods in nonlinear perturbation theory for the structure formation in the Universe. In Papers I and II [arXiv:2210.10435, arXiv:2210.11085], the basic formalism and technical methods of the iPT for cosmological tensor fields are developed, generalizing the corresponding theory for scalar fields. In previous papers, methods to predict statistical quantities, such as power spectra, correlation functions, etc., of three-dimensional tensor fields are developed based on the iPT. However, observations of tensors, such as angular momenta and shapes of galaxies, etc., are only possible after the three-dimensional tensors are projected onto the two-dimensional sky. In this paper, power spectra and correlation functions of projected two-dimensional tensors are related to those of original three-dimensional tensors, so that one can make predictions for the observable statistics of projected tensor fields from the iPT. The relations are consistently represented on the basis of irreducible decomposition of both two- and three-dimensional tensors.

astro-ph.CO↗

arXiv2024

Integrated perturbation theory for cosmological tensor fields. IV. Full-sky formulation

In Papers I-III [arXiv:2210.10435, arXiv:2210.11085, arXiv:2304.13304], we use the flat-sky and distant-observer approximations to develop a formalism with which the correlation statistics of cosmological tensor fields are calculated by the nonlinear perturbation theory, generalizing the integrated perturbation theory for scalar fields. In this work, the formalism is extended to include the full-sky and wide-angle effects in evaluating the power spectra and correlation functions of cosmological tensor fields of any rank. With the newly developed formalism, one can evaluate the nonlinear power spectra and correlation functions to arbitrary higher orders in principle. After describing the general formalism, we explicitly derive and give analytic results of the lowest-order linear theory for an illustrative purpose in this paper. The derived linear formulas with full-sky and wide-angle effects are numerically compared with the previous formulas with flat-sky and distant-observer limits in a simple model of tensor bias.

astro-ph.CO↗

arXiv2024

Kurtosis consistency relation in large-scale structure as a probe of gravity theories

Various gravity theories beyond general relativity have been rigorously investigated in the literature such as Horndeski and degenerate higher-order scalar-tensor (DHOST) theories. In general, numerous model parameters are involved in such theories, which should be constrained to test the theories with experiments and observations. We construct the kurtosis consistency relations, calculated based on matter density fluctuations, in which the information of gravity theories is encoded. We derive two independent consistency relations that should hold in the framework of the DHOST theories and argue that such consistency relations would be useful for testing gravity theories.

astro-ph.CO↗

arXiv2023

Skewness consistency relation in large-scale structure and test of gravity theory

We investigate the skewness of galaxy number density fluctuations as a possible probe to test gravity theories. We find that the specific linear combination of the skewness parameters corresponds to the coefficients of the second-order kernels of the density contrast, which can be regarded as the consistency relation and used as a test of general relativity and modified gravity theories. We also extend the analysis of the skewness parameters from real space to redshift space and derive the redshift-space skewness consistency relation.

astro-ph.CO↗

arXiv2023

Asymptotic expansion of the expected Minkowski functional for isotropic central limit random fields

The Minkowski functionals, including the Euler characteristic statistics, are standard tools for morphological analysis in cosmology. Motivated by cosmic research, we examine the Minkowski functional of the excursion set for an isotropic central limit random field, the $k$-point correlation functions ($k$th order cumulants) of which have the same structure as that assumed in cosmic research. Using 3- and 4-point correlation functions, we derive the asymptotic expansions of the Euler characteristic density, which is the building block of the Minkowski functional. The resulting formula reveals the types of non-Gaussianity that cannot be captured by the Minkowski functionals. As an example, we consider an isotropic chi-square random field and confirm that the asymptotic expansion accurately approximates the true Euler characteristic density.

math.ST↗

arXiv2022

Non-Gaussianity effects on the primordial black hole abundance for sharply-peaked primordial spectrum

We perturbatively study the effect of non-Gaussianities on the mass fraction of primordial black holes (PBHs) at the time of formation by systematically taking its effect into account in the one-point probability distribution function of the primordial curvature perturbation. We focus on the bispectrum and trispectrum and derive formulas that describe their effects on the skewness and kurtosis of the distribution function. Then considering the case of narrowly peaked spectra, we obtain simple formulas that concisely express the effect of the bi- and trispectra. In particular, together with the $g_{\rm NL}$ and $τ_{\rm NL}$ parameters of the trispectrum, we find that non-Gaussianity parameters for various types of the bispectrum are linearly combined to give an effective parameter, $f_{\rm NL}^{\rm eff}$, that determines the PBH mass fraction in the narrow spectral shape limit.

astro-ph.CO↗

arXiv2020

Minkowski functionals and the nonlinear perturbation theory in the large-scale structure: second-order effects

The second-order formula of Minkowski functionals in weakly non-Gaussian fields is compared with the numerical $N$-body simulations. Recently, weakly non-Gaussian formula of Minkowski functionals is extended to include the second-order effects of non-Gaussianity in general dimensions. We apply this formula to the three-dimensional density field in the large-scale structure of the Universe. The parameters of the second-order formula include several kinds of skewness and kurtosis parameters. We apply the tree-level nonlinear perturbation theory to estimate these parameters. First we compare the theoretical values with those of numerical simulations on the basis of parameter values, and next we test the performance of the analytic formula combined with the perturbation theory. The second-order formula outperforms the first-order formula in general. The performance of the perturbation theory depends on the smoothing radius applied in defining the Minkowski functionals. The quantitative comparisons are presented in detail.

astro-ph.CO↗

arXiv2020

Weakly non-Gaussian formula for the Minkowski functionals in general dimensions

The Minkowski functionals are useful statistics to quantify the morphology of various random fields. They have been applied to numerous analyses of geometrical patterns, including various types of cosmic fields, morphological image processing, etc. In some cases, including cosmological applications, small deviations from the Gaussianity of the distribution are of fundamental importance. Analytic formulas for the expectation values of Minkowski functionals with small non-Gaussianity have been derived in limited cases to date. We generalize these previous works to derive an analytic expression for expectation values of Minkowski functionals up to second-order corrections of non-Gaussianity in a space of general dimensions. The derived formula has sufficient generality to be applied to any random fields with weak non-Gaussianity in a statistically homogeneous and isotropic space of any dimensions.

astro-ph.CO↗

arXiv2020

The statistics of peaks of weakly non-Gaussian random fields: Effects of bispectrum in two- and three-dimensions

Analytic expressions for the statistics of peaks of random fields with weak non-Gaussianity are provided. Specifically, the abundance and spatial correlation of peaks are represented by formulas which can be evaluated only by virtually one-dimensional integrals. We assume the non-Gaussianity is weak enough such that it is represented by linear terms of the bispectrum. The formulas are formally given in $N$-dimensional space, and explicitly given in the case of $N=1,2,3$. Some examples of peak statistics in cosmological fields are calculated for the cosmic density field and weak lensing field, assuming the weak non-Gaussianity is induced by gravity. The formulas of this paper would find a fit in many applications to statistical analyses of cosmological fields.

astro-ph.CO↗

arXiv2020

de Sitter duality and logarithmic decay of dark energy

We investigate infrared dynamics of four-dimensional Einstein gravity in de Sitter space. We set up a general framework to investigate dynamical scaling relations in quantum/classical gravitational theories. The conformal mode dependence of Einstein gravity is renormalized to the extent that general covariance is not manifest. We point out that the introduction of an inflaton is necessary as a counterterm. We observe and postulate a duality between quantum effects in Einstein gravity and classical evolutions in an inflation (or quintessence) model. The effective action of Einstein gravity can be constructed as an inflation model with manifest general covariance. We show that $g=G_N H^2/π$: the only dimensionless coupling of the Hubble parameter $H^2$ and the Newton's coupling $G_N$ in Einstein gravity is screened by the infrared fluctuations of the conformal mode. We evaluate the one-loop $β$ function of $g$ with respect to the cosmic time $\log Ht$ as $β(g)=-(1/2)g^2$, i.e., $g$ is asymptotically free toward the future. The exact $β$ function with the backreaction of $g$ reveals the existence of the ultraviolet fixed point. It indicates that the de Sitter expansion stared at the Planck scale with a minimal entropy $S=2$. We have identified the de Sitter entropy $1/g$ with the von Neumann entropy of the conformal zero mode. The former evolves according to the screening of $g$ and the Gibbons-Hawking formula. The latter is found to increase by diffusion in the stochastic process at the horizon in a consistent way. Our Universe is located very close to the fixed point $g=0$ with a large entropy. We discuss possible physical implications of our results such as logarithmic decay of dark energy.

hep-th↗

arXiv2020

Clustering of primordial black holes formed in a matter-dominated epoch

In the presence of the local-type primordial non-Gaussianity, it is known that the clustering of primordial black holes (PBHs) emerges even on super-horizon scales at the formation time. This effect has been investigated in the high-peak limit of the PBH formation in the radiation-dominated epoch in the literature. There is another possibility that the PBH formation takes place in the early matter-dominated epoch. In this scenario, the high-peak limit is not applicable because even initially small perturbations grow and can become a PBH. We first derive a general formula to estimate the clustering of PBHs with primordial non-Gaussianity without assuming the high-peak limit, and then apply this formula to a model of PBH formation in a matter-dominated epoch. Clustering is less significant in the case of the PBH formation in the matter-dominated epoch than that in the radiation-dominated epoch. Nevertheless, it is much larger than the Poisson shot noise in many cases. Relations to the constraints of the isocurvature perturbations by the cosmic microwave background radiation are quantitatively discussed.

astro-ph.CO↗

arXiv2019

Velocity bias and the nonlinear perturbation theory of peaks

The biasing in the large-scale structure of the universe is a crucial problem in cosmological applications. The peaks model of biasing predicts a linear velocity bias of halos, which is not present in a simple model of local bias. We investigate the origin of the velocity bias in the peaks model from the viewpoint of the integrated perturbation theory, which is a nonlinear perturbation theory in the presence of general Lagrangian bias. The presence of the velocity bias in the peaks model is a consequence of the "flat constraint," ${\nabla}δ= 0$; i.e., all the first spatial derivatives should vanish at the locations of peaks. We show that the velocity bias in the peaks model is systematically derived in the framework of the integrated perturbation theory, and then develop a formal theory to perturbatively trace the nonlinear evolution of biased objects with the flat constraint. A formula for the nonlinear velocity dispersion of peaks with the one-loop approximation is also derived.

astro-ph.CO↗

arXiv2019

The large-separation expansion of peak clustering in Gaussian random fields

In the peaks approach, the formation sites of observable structures in the Universe are identified as peaks in the matter density field. The statistical properties of the clustering of peaks are particularly important in this respect. In this paper, we investigate the large-separation expansion of the correlation function of peaks in Gaussian random fields. The analytic formula up to third order is derived, and the resultant expression can be evaluated by a combination of one-dimensional fast Fourier transforms, which are evaluated very fast. The analytic formula obtained perturbatively in the large-separation limit is compared with a method of Monte-Carlo integrations, and a complementarity between the two methods is demonstrated.

astro-ph.CO↗

arXiv2018

Intrinsic galaxy alignment from angular dependent primordial non-Gaussianity

In this paper, we explore a detectable imprint of massive fields with integer spins $s \geq 2$, which may be predicted from string theory. It was shown that such a massive non-zero spin field can generate the squeezed primordial bispectrum which depends on the angle between the two wavenumbers. We show that considering the contribution from the massive spin-2 field, the angular dependent primordial non-Gaussianity (PNG) yields a strong scale dependence in the bias parameter for the galaxy alignment, which becomes prominent at small scales. As another example of an angular dependent PNG, we also consider the primordial bispectrum where the angular dependence was introduced by a vector field, while breaking the global rotational symmetry. As a consequence, we find that the B-mode cosmic shear and non-diagonal components do not vanish. These aspects provide qualitative differences from the PNG sourced by massive non-zero spin fields.

astro-ph.CO↗

arXiv2016

Studying topological structure in the epoch of reionization with 3D-Minkowski functionals of 21cm line fluctuations

The brightness temperature of the redshifted 21cm line brings rich information on the Inter Galactic Medium (IGM) from the Cosmic Dawn and Epoch of Reionization (EoR). While the power spectrum is a useful tool to statistically investigate the 21cm signal, the 21cm brightness temperature field is highly non-Gaussian, and the power spectrum is inadequate to characterize the non-Gaussianity. The Minkowski Functionals (MFs) are promising tools to extract non-gaussian features of the 21cm signal and give topological information such as morphology of ionized bubbles. In this work, we study the 21cm line signal in detail with MFs. To promote understanding of basic features of the 21cm signal, we calculate the MFs of not only the hydrogen neutral fraction but the matter density and spin temperature, which contribute to the brightness temperature fluctuations. We find that the structure of the brightness temperature depends mainly on the ionized fraction and the spin temperature at late and early stages of the EoR, respectively. Further, we investigate the redshift evolution of the MFs at $7 < z < 20$. Then, we find that, after the onset of reionization, the MFs reflect mainly the ionized bubble property. In addition, the MFs are sensitive to model parameters which are related to the topology of ionized bubbles and we consider the possibility of constraining the parameters by the future 21cm signal observations.

astro-ph.CO↗