arXiv subjects

Tianye Liu

Publications and source records attributed to Tianye Liu.

4 recordsLinked to original sources

arXiv2025

Creating a Universe from Nothing as an Alternative to the Cosmological Principle

In the cosmological Robertson-Walker geometry required of the cosmological principle both the Weyl tensor $C^{μλνκ}$ and the Bach tensor $W^{μν}=[2\nabla_κ\nabla_λ-R_{λκ}]C^{μλνκ}$ vanish. In general, in perturbations around the cosmological background neither of the fluctuating $δC^{μλνκ}$ or $δW^{μν}$ would vanish. However, it is possible for $δW^{μν}$ to vanish even as $δC^{μλνκ}$ does not. In this paper we construct an explicit model in which this is the case. The model consists of a 3-tensor gravitational wave fluctuating around a background with a constant negative 3-curvature. The model is exactly solvable and consists purely of geometric quantities with no matter fields at all (i.e., $G^{μν}=0$, $δG^{μν}=0$, $W^{μν}=0$, $δW^{μν}=0$, where $G^{μν}$ is the Einstein tensor). The model can thus be created out of nothing, with creating a universe from nothing thus being an alternative principle to the cosmological principle. The fluctuating gravitational wave contributes to the temperature anisotropy in the cosmic microwave background and its $B$ mode polarization in a calculable manner, one for which we provide a simple analytic way of treating spatial modes that is based on the use of a spatial mode addition theorem. In addition, we provide a treatment of the anisotropy that is based on properties of bandwidth limited functions. Classically by ``nothing" we mean that there are no $T^{μν}$ or $δT^{μν}$ matter field terms. Quantum-mechanically by ``nothing" we mean that all fields other than the gravitational field are in a negative energy mode vacuum state, with the only occupied positive energy modes being graviton modes. As well as use the Bach tensor as a diagnostic, we consider dynamics based on it.

gr-qc↗

arXiv2025

Properties of associated Legendre conical functions

We present some new properties of associated Legendre conical functions of the first and second kind, $P^{-1/2-K}_{-1/2+i τ}(χ)$ and $Q^{-1/2-K}_{-1/2+i τ}(χ)$. In particular we show that with the $τ$-independent $\mathcal{R}^{K}_{n}(χ)=(2π)^{-3/2}\tanh^{-K}χ\sinh^{-1/2}χ[Γ(1+K)]^{-1}\int_{0}^{2π} dω\left (1-\cos ω/\coshχ\right)^{K}e^{inω}$ for any general $K$, we can set $P^{-1/2-K}_{-1/2+i τ}(χ)=2\sum_{n}\mathcal{R}^{K}_{n}(χ)\sin[(τ-in)χ)]/(τ-in)$, where $n$ ranges from $-\ell$ to $\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $-\infty$ to $\infty$ in unit steps otherwise. Also we can set $Q^{-1/2-K}_{-1/2+i τ}(χ)=-π\sum_n\mathcal{R}^K_{n-K}(χ)e^{-i χ(τ-i(n-K))}/(τ-i(n-K))$, where $n$ ranges from $0$ to $2\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $0$ to $\infty$ in unit steps otherwise. With these forms isolating the entire $τ$ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over $τ$ of associated Legendre conical functions and their products. The $Q^{-1/2-K}_{-1/2+i τ}(χ)$ have an integral representation containing the integral $\int_χ^{\infty}dωe^{iωτ}(\coshω-\coshχ)^{K}$, an integral that only converges at $ω=\infty$ if ${\rm Re}[K]<{\rm Im}[τ]$. We show how to use the divergence of this integral outside of this range in order to characterize the complex $τ$ plane pole structure of $Q^{-1/2-K}_{-1/2+i τ}(χ)$. We present a new treatment of the Borwein integral and the Nyquist-Shannon sampling theorem.

math-ph↗

arXiv2023

Exceptional points for associated Legendre functions of the second kind

We consider the complex $ν$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_ν(\coshρ)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of poles in the complex $ν$ plane, but for any negative integer $K$ there are no poles at all. For $K=0$ or any positive integer $K$ there is only a finite number of poles, with there only being one single pole (at $ν=0$) when $K=0$. This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts. However, unusually for theories with exceptional points, $Q^{-1/2-K}_ν(\coshρ)$ has an infinite number of them. Other than in the $PT$-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the exceptional points associated with the $Q^{-1/2-K}_ν(\coshρ)$ nonetheless occur at real values of $K$.

math-ph↗

arXiv2021

Exact solution to perturbative conformal cosmology from recombination until the current era

In a previous paper (P. D. Mannheim, Phys. Rev. D 102, 123535 (2020)) we studied cosmological perturbation theory in the cosmology associated with the fourth-order derivative conformal gravity theory, and provided an exact solution to the theory in the recombination era. In this paper we present an exact solution that holds all the way from recombination until the current era.

gr-qc↗