The Mathematical Physics of Gravitational Waves and the Polylogarithm and Lambert W Functions

我的研究计划的长期愿景是扩展数学物理基础的引力波和量子统计的D维玻色和费米气体。我建议通过以下两个不同的主题来实现这一愿景。主题1 * 本主题的一个关键目标是进行理论研究，以了解黑洞（BH）强引力下的黑洞行为。第二代探测器的观测运行将开始，这些探测器提供的状态方程（EoS）信息将主要来自双星中子星星和BH致密天体（CO）（如中子星星（NS）或白色矮星（WD）系统）螺旋内的潮汐相互作用，这会导致NS的四极形变。测量NS-NS双星的GW信号是了解NS仍然知之甚少的EoS的一种非常有前途的方法，也是LIGO的主要天体物理应用之一。我们将解决具有挑战性的问题的GW频率自旋下降系数的脉冲星，需要长时间的观测时间来检测连续的引力波（GW）源。GW天文学的复兴始于激光干涉引力波天文台（LIGO）探测到两个GW排放，这两个排放都来自合并的BH系统。我的HQP和我将分析由绕BH运行的脉冲星产生的较弱GW，以获得对引力物理学更详细的理解。此外，我提出的多信使天文学研究与LIGO，Virgo（法国-意大利GW天文台），天基激光干涉仪空间天线（丽莎），平方公里阵列（SKA）和即将到来的LIGO-India相关。SKA的独特灵敏度将定位轨道脉冲星BH系统，检测单个GW源，并提供强有力的广义相对论测试。SKA还将对脉冲星进行调查，这将提供脉冲星-BH系统的人口统计数据。主题2 * 本主题的一个关键目标是使用多对数和多分支Lambert W函数的对数结构来理解和统一D维玻色和费米气体的量子自旋统计。我们建议研究量子统计的基础数学物理，以揭示问题的特征，如布朗矮星（BD）的双多方模型，热电材料（TEM）的热电品质因数的增强，以及伊辛和广义伊辛模型中的配分函数的零点。我将调查更精确的EoS天体，如BD和NS，并研究旋转NS的超流体核心。这些研究将有助于提高TEM和太阳能电池的效率。上述两个主题的统一线索是数学和理论物理方法的发展，以解决具有挑战性的物理问题。

The long-term vision of my research program is to extend the mathematical physics that underlies gravitational waves and the quantum statistics of D-dimensional Bose and Fermi gases. I propose to realize this vision through two distinct themes summarized below.******Theme 1 ***A key objective in this theme is a theoretical investigation to understand the behaviour of pulsars in the strong gravity of a black hole (BH). Observation runs will begin for second generation detectors, and the Equation of State (EoS) information provided by these detectors will come mainly from tidal interactions during the in-spiral of binary neutron star and BH-compact object (CO) such as neutron star (NS) or White Dwarf (WD) systems, which induce quadrupolar deformations in the NS. Measuring GW signals from NS-NS binaries is a very promising method to learn about the still poorly understood EoS of NS, and one of the primary astrophysical applications of LIGO. We will address the challenging problem of the GW frequency spin-down coefficients of a pulsar, over long observation times required to detect continuous sources of gravitational wave (GW). A renaissance in GW astronomy began after the Laser Interferometer Gravitational-Wave Observatory (LIGO) detected two GW emissions, which both arose from the merger of coalescing binary BH systems. My HQP and I will analyze the weaker GW produced by a pulsar orbiting a BH to gain a more detailed understanding of gravitational physics. In addition, my proposed multi-messenger astronomy study is relevant for LIGO, Virgo (French-Italian GW Observatory), the space-based Laser Interferometer Space Antenna (LISA), the Square Kilometre Array (SKA) and the upcoming LIGO-India. The unique sensitivity of the SKA will locate orbiting pulsars-BH systems, detect single GW sources and provide strong tests of General Relativity. The SKA will also perform a survey of pulsars and this will provide the population statistics of pulsar-BH systems.******Theme 2 ***A key objective in this theme is to use the logarithmic structure of the Polylogarithm and multi-branched Lambert W functions in the understanding and unification of the quantum spin statistics of D dimensional Bose and Fermi gases. We propose to study the fundamental mathematical physics that underlies quantum statistics to unravel characteristic features in problems such as the double polytrope model for Brown Dwarfs (BD), enhancement of the thermoelectric figure of merit in thermoelectric materials (TEM), as well as the zeros of partition functions in the Ising and generalized Ising Models. I will investigate the more exact EoS for astrophysical objects such as BD and NS, and study the superfluid core of a spinning NS. These studies will help to improve the efficiency of TEM and solar cells.******The unifying thread in both of the themes described above is the development of methods in mathematical and theoretical physics to solve challenging physical problems.