f(T)重力与宇宙学背景下的不均匀状态方程重建

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"这篇论文详细探讨了在标量张量理论和f(T)重力框架下，如何重建不均匀状态方程的过程。作者Jackson Levi Said分析了如何利用辅助标量场来研究宇宙学背景下的f(T)重力，并对不同形式的哈勃参数进行了考察，以构建f(T)拉格朗日。研究发现，辅助标量场和弗里德曼-罗伯逊-沃alker (FRW) 场方程中的奇异项提供了相同的解，暗示哈勃参数的变化可能需要通过f(T)重力来重新构造引力机制。该文章是一篇开放访问的物理期刊论文，发表在Eur.Phys.J.C(2017)77:883，链接为https://doi.org/10.1140/epjc/s10052-017-5460-y。" 详细知识点: 1. 广义相对论（GR）: GR是爱因斯坦提出的一种描述引力的理论，它将引力视为时空曲率的结果。在GR中，物质和能量分布导致时空弯曲，而物体则沿着这种弯曲的路径运动。 2. 伸缩平行（Teleparallelism）: 这是一种替代GR的理论，它描述重力通过四维空间的扭率。伸缩平行理论与GR在等式层面上可以等价，但其物理基础不同，因为它依赖于平行传输而不是曲率。 3. f(R)引力: f(R)引力是GR的一种扩展，其中R是Ricci曲率标量，f(R)是R的任意函数。这种理论允许对引力进行量子修正或其他修改，以适应观测或理论需求。 4. f(T)引力: 类似于f(R)，f(T)引力是另一种对GR的修改，这里的T是torsion scalar（扭率标量），在伸缩平行理论中扮演着R的角色。f(T)引力的场方程是二阶的，不同于f(R)引力的四阶方程。 5. 宇宙学背景: 在研究引力理论时，宇宙学提供了一个理想的框架，因为宇宙的大尺度结构和演化能显著地体现出重力效应。在这项工作中，f(T)重力被应用于宇宙尺度以重构引力模型。 6. 辅助标量场: 在研究中，辅助标量场被用来帮助重构f(T)引力的解。这个标量场可以模拟宇宙学过程中的某些特性，例如宇宙膨胀。 7. 哈勃参数: 描述宇宙膨胀速率的关键参数，H = (d/dt)(a/a)，其中a是宇宙尺度因子，t是时间。通过改变哈勃参数的形式，可以研究不同的宇宙模型和引力理论。 8. 不均匀状态方程（EoS）: EoS通常用p/ρ表示，其中p是压力，ρ是密度。在不均匀的宇宙模型中，EoS可以变化，反映物质分布和引力相互作用的复杂性。 9. 拉格朗日方程: 在物理学中，拉格朗日方程描述了系统动力学的数学表达，由系统的总能量（拉格朗日量）推导得出。 10. 异常项匹配: 论文指出，辅助标量场和FRW场方程的奇异项给出相同的结果，这表明在考虑哈勃参数变化时，f(T)重力可能是理解宇宙动态的一种有效工具。通过以上知识点，我们可以看到这项工作不仅深入研究了f(T)引力理论的性质，还揭示了在宇宙学背景下如何利用这些理论来解释观测到的现象。这种方法为理解和可能改进当前的引力理论提供了新的途径。

Eur. Phys. J. C (2017) 77:883

https://doi.org/10.1140/epjc/s10052-017-5460-y

Regular Article - Theoretical Physics

Reconstruction from scalar–tensor theory and the inhomogeneous

equation of state in f (T) gravity

Jackson Levi Said

1,2,a

Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta

Department of Physics, University of Malta, Msida MSD 2080, Malta

Received: 12 July 2017 / Accepted: 8 December 2017 / Published online: 18 December 2017

Abstract General relativity (GR) characterizes gravity as

a geometric properly exhibited as curvature on spacetime.

Teleparallelism describes gravity through torsional proper-

ties, and can reproduce GR at the level of equations. Similar

to f (R) gravity, on taking a generalization, f (T ) gravity can

produce various modiﬁcations its gravitational mechanism.

The resulting ﬁeld equations are inherently distinct to f (R)

gravity in that they are second order. In the present work,

f (T ) gravity is examined in the cosmological context with a

number of solutions reconstructed by means of an auxiliary

scalar ﬁeld. To do this, various forms of the Hubble param-

eter are considered with an f (T ) Lagrangian emerging for

each instance. In addition, the inhomogeneous equation of

state (EoS) is investigated with a particular Hubble parame-

ter model used to show how this can be used to reconstruct

the f (T ) Lagrangian. Observationally, the auxiliary scalar

ﬁeld and the exotic terms in the FRW ﬁeld equations give the

same results, meaning that the variation in the Hubble param-

eter may be interpreted as the need to reformulate gravity in

some way, as in f (T ) gravity.

1 Introduction

Modiﬁed gravity is one of the two direct approaches for

reproducing the late-time acceleration observed in the uni-

verse [1–3]. Additionally, there are other consistency prob-

lems that must eventually be tackled in the general relativity

(GR) approach to gravity [4,5]. The question then becomes

what reformulation of gravity should be adopted, or whether

we should take an extension of GR as our starting position.

This is indeed a popular approach to the problem; many ref-

erences consider an extension to the GR Einstein–Hilbert

action in which the predominant attempt can be represented

by the f (R) gravity formulations [6,7].

e-mail: jackson.said@um.edu.mt

One proposition that has gained interest in recent years

is that of teleparallel gravity [8–11]. The concept was ini-

tiated by Einstein himself shortly after his introduction of

GR [12]. At ﬁrst this was simply an equivalent reformulation

of GR, called the teleparallel equivalent of general relativ-

ity (TEGR), which replaces the Einstein–Hilbert Lagrangian

with an analog teleparallel torsion quantity, T (known as the

torsion scalar).

As with the Einstein–Hilbert action and the f (R)

approach, the simplest generalization is to an arbitrary func-

tion of the torsion scalar, f (T ) [8]. Similarly, the TEGR

formulation is equivalent to GR at the level of equations,

however, its generalization is distinctly different from f (R)

in a number of important ways. The ﬁrst way is that the result-

ing ﬁeld equations continue to be of second order while those

of f (R) are fourth order. This has led to interesting insights

both cosmologically and otherwise [8,9,11,13,14].

Any alternative to GR should answer some of the open

questions in GR. From the cosmological perspective, it would

be interesting to put together a consistent theory that predicts

the expansion behavior of both the early- and the late-time

universe. In fact this is one of the primary motivations behind

the CDM modiﬁcation to Einstein’s GR formulation. While

the ﬁrst realization of the need for modiﬁcations to Einstein’s

GR comes from the observation in 1998 of the accelerating

expansion of the universe [1,2], there is now overwhelming

observational evidence for the need to modify the Einstein

formulation [15].

The goal of the present work is to produce known and new

Lagrangian models within the f (T ) gravity context using

several Hubble parameter models [16], using an auxiliary

scalar ﬁeld as a conduit to perform this reconstruction. This

scalar ﬁeld, essentially, takes on the role of dark energy, and

would be the mediator through which these cosmological

effects would take over.

Unless stated otherwise, geometric units are used where

G = 1 = c. Also Latin indices are used to refer to local iner-

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