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引力波的数学物理基础，以及D维玻色和费米气体的量子统计。我建议通过以下两个不同的主题来实现这一愿景。*主题1*这个主题的一个关键目标是进行理论研究，以了解脉冲星在黑洞(BH)强引力下的行为。第二代探测器将开始观测，这些探测器提供的状态方程(Eos)信息将主要来自双中子星和BH致密天体(CO)螺旋线内的潮汐相互作用，如中子星(NS)或白矮星(WD)系统，这些系统在NS中引起四极形变。从NS-NS双星测量GW信号是一种非常有希望的方法来了解NS仍然鲜为人知的状态方程，也是LIGO的主要天体物理应用之一。我们将解决在探测连续引力波(GW)源所需的长观测时间内，脉冲星的GW频率自旋下降系数的挑战性问题。在激光干涉仪引力波天文台(LIGO)探测到两个GW辐射后，GW天文学开始了复兴，这两个辐射都是由合并的双星BH系统产生的。我和我的HQP将分析脉冲星围绕BH轨道产生的较弱的GW，以获得对引力物理的更详细的理解。此外，我提议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(LISA)、平方公里阵列(SKA)和即将到来的LIGO-印度相关。SKA独特的灵敏度将定位轨道脉冲星-BH系统，探测单一的GW源，并提供强大的广义相对论测试。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。*主题2*这个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。我们建议研究作为量子统计基础的基本数学物理，以揭示诸如Brown Dwarf(BD)的双多项式模型、热电材料中热电优值系数的增强以及Ising和广义Ising模型中配分函数的零点等问题的特征。我将研究BD和NS等天体物理对象的更精确的状态方程，并研究旋转的NS的超流核心。这些研究将有助于提高透射电子显微镜和太阳能电池的效率。上述两个主题的统一主线是发展数学和理论物理的方法来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.