Palatini f(R)引力下的五维厚糙面：非线性修正与局部化效应

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本文探讨了"具有Palatini动力学的f(R)重力下的粗糠"这一主题，主要关注于在五维时空背景下对广义相对论的扩展。Palatini方法在此研究中扮演关键角色，它假设度量和连接是独立的变量，这在一般的理论物理学中是一种重要的非欧几里得引力理论框架。在这个模型中，所谓的"厚厚的Palatini f(R)麸皮"被引入，这是一种描述包含一个无限延伸额外维度的 braneworld模型，这种几何结构受到反德西特空间的扭曲影响。研究者采用了一阶框架来处理标量场和翘曲因子，这是理论分析中的一个重要步骤，因为它允许对这些物理量进行精确的数学描述。他们特别关注了摄动场景，通过Lagrangian f(R) = R + ϵR^n的形式来实现Palatini方法，这里的小参数ϵ作为控制标准粗糠情况与非线性修正间偏差的工具。通过这样的设置，他们观察到非线性效应如何导致额外维度在局部区域得到一定程度的量子化或"局部化"，这是在更高维度理论中一个重要的现象，它可能有助于解释宇宙的某些观测特征或者解决宇宙学问题。文章的详细内容可能包括对理论背景的深入讨论，如f(R)理论如何处理引力的修正，以及这些修正如何影响额外维度的行为。此外，作者可能会分析计算出的具体结果，如翘曲因子的解析表达式，或者对特定参数值下的物理效应的数值模拟。他们也可能讨论这些结果与已知的宇宙学模型（如ΛCDM模型）之间的比较，以及未来实验或观测可能验证这些理论预测的方向。总结来说，这项工作提供了关于f(R)理论在高维度中的一种新颖应用，特别是强调了Palatini动力学对于理解额外维度行为的重要性，并通过数值分析展示了非线性修正可能带来的有趣物理后果。这一研究不仅扩展了我们对引力理论的认识，也为探索多元宇宙和宇宙学难题提供了新的思考视角。

Eur. Phys. J. C (2015) 75:569

DOI 10.1140/epjc/s10052-015-3803-0

Regular Article - Theoretical Physics

Thick brane in f (R) gravity with Palatini dynamics

D. Bazeia

1,a

, L. Losano

1,b

, R. Menezes

2,3,c

, Gonzalo J. Olmo

1,4,d

, D. Rubiera-Garcia

1,5,6,e

Departamento de Física, Universidade Federal da Paraíba, João Pessoa, PB 58051-900, Brazil

Departamento de Ciências Exatas, Universidade Federal da Paraíba, Rio Tinto, PB 58297-000, Brazil

Departamento de Física, Universidade Federal de Campina Grande, Campina Grande, PB 58109-970, Brazil

Departamento de Física Teórica, IFIC, Centro Mixto Universidad de Valencia-CSIC, Universidad de Valencia, 46100 Burjassot, Valencia, Spain

Faculdade de Ciências, Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa, Campo Grande, 1749-016 Lisbon, Portugal

Department of Physics, Center for Field Theory and Particle Physics, Fudan University, 220 Handan Road, Shanghai 200433, China

Received: 4 February 2015 / Accepted: 20 November 2015 / Published online: 30 November 2015

Abstract This work deals with modiﬁed gravity in ﬁve-

dimensional space-time. We study a thick Palatini f (R)

brane, that is, a braneworld scenario described by an anti-

de Sitter warped geometry with a single extra dimension of

inﬁnite extent, sourced by a real scalar ﬁeld under the Palatini

approach, where the metric and the connection are regarded

as independent degrees of freedom. We consider a ﬁrst-order

framework which we use to provide exact solutions for the

scalar ﬁeld and warp factor. We also investigate a perturba-

tive scenario such that the Palatini approach is implemented

through a Lagrangian f (R) = R +  R

, where the small

parameter  controls the deviation from the standard thick

brane case. In both cases it is found that the warp factor

tends to localize the extra dimension due to the nonlinear

corrections.

1 Introduction

Investigations dealing with space-time engendering higher

spatial dimensions started in physics soon after the appear-

ance of general relativity (GR) through the Kaluza–Klein

models (see e.g. [1] for a review), aimed to study uniﬁcation

of the electromagnetic interaction with gravity. Nowadays,

the presence of higher spatial dimensions is very natural in

high energy physics, in string, superstring and other uniﬁca-

tion and fundamental theories [2,3]. However, the addition of

extra spatial dimensions is in conﬂict with the natural world

which, when probed in any experiment, has only revealed the

e-mail: bazeia@ﬁsica.ufpb.br

e-mail: losano@ﬁsica.ufpb.br

e-mail: rmenezes@dce.ufpb.br

e-mail: gonzalo.olmo@csic.es

e-mail: drgarcia@fc.ul.pt

presence of three spatial dimensions, though proposals using

large extra dimensions with potential experimental signatures

in particle accelerators have been discussed [4–7].

To reconcile the constraint of three spatial dimensions of

the natural world with the introduction of extra dimensions,

important scenarios have been proposed. Here we focus our

attention upon the Randall–Sundrum (RS) work [8], where

the relevant portion of the higher-dimensional space-time is

embedded within a ﬁve-dimensional anti-de Sitter (AdS

)

geometry. This scenario assumes that the (3, 1) space-time

that describes the natural world is embedded in an AdS

warped geometry, with a single extra spatial dimension of

inﬁnite extent. This is known as the RS2 braneworld scenario,

and the warp factor identiﬁes a thin brane proﬁle, decaying

along the extra dimension y in the form exp(−2|y|).

Soon after the proposed thin braneworld scenario, it was

modiﬁed with the presence of scalar ﬁelds, giving rise to

a new, very interesting thick braneworld scenario, in which

the warp factor is now described by another function, which

depends on the speciﬁc scalar ﬁeld model one considers [9–

12]. The presence of scalar ﬁelds brought interesting possi-

bilities, as the appearance of a new feature, the splitting of

the brane, which emerges in the presence of distinct effects,

at ﬁnite temperature [13] or with speciﬁc scalar ﬁeld models

[14,15] in the presence of at least one additional parameter,

to be used to control the splitting of the brane.

Like the number of spatial dimensions, there are other

foundational aspects of the idea of gravitation as a geo-

metric phenomenon which are likely to provide interest-

ing new viewpoints on the fundamental open questions of

gravitational physics. In this sense, most extensions of GR

adopt the implicit assumption that space-time is a Rieman-

nian structure completely determined by the metric degrees

of freedom (see e.g. [16–19] for some reviews). However,

though geometry made its appearance in physics through

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