M-通货膨胀中的预热引力波: 模型预测与超高频探测

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矩阵通货膨胀(M-flation)是一种新颖的理论框架，它源自弦理论，主要关注于U(N)规范群的伴随表示中的三个标量场矩阵和规范场。这种模型的特点在于，其中的一个标量场表现出类似膨胀的效应，而其他标量场和标尺场则可能作为等曲率场参与预热过程。特别地，研究者在参数空间和初始条件的"山顶区域"找到了一个特殊区域，这里的模型预测与当前Planck卫星数据高度契合。在这个特定的山顶区域，M-flation模型具备一个嵌入式的预热机制。预热阶段是通货膨胀结束后，能量密度转移到标准模型粒子的过程中，其中的耦合关系与膨胀自身的自耦合密切相关。这些耦合可以通过宇宙微波背景(CMB)数据进行观测和约束。本文的重点在于通过辛格积分器进行的数值模拟，作者计算了M-flation预热阶段产生的重力波功率谱。预热场的振动会导致引力波信号，初步的数值结果表明，这种引力波的幅度大约为Ωgwh2 ≈ 10^-16，且在 GHz 波段有显著峰值。这意味着M-flation模型对于未来的超高频重力波探测器具有独特的可检测预测，这一特性使得它在众多通货膨胀模型中脱颖而出，提供了重要的理论依据来区分不同的模型。总结来说，M-flation不仅展示了在理论上的创新，还在实验观测上提出了可验证的预测，尤其是对于高精度的重力波探测。这不仅对理解早期宇宙的演化有着重要意义，也为未来的天体物理学实验提供了一个全新的研究方向。

JCAP03(2014)020

inﬂaton matrices are, by constructi on, in the adjoint representation of the U(N) gauge group;

therefore the y are non-commutative as well as Herm i ti an.

In principle, the dynamics of these matrices is very complicated (increasingly so with

larger N), as one has any number of possible conﬁgurations of the D3-branes within the chosen

backgr ound. However there i s a way to simplify the situation and make it computationally

tractable. As we will elaborate i n the next subsection, the classical dynamics of this model

can be consi st ently truncated to a solution where t he N D3-branes are uniformly distributed

along the surface of a 2-sp he re (within the 6-dimensional orthogonal subspace), and their

positions on this sphere do not change during inﬂation. What instead changes is the sphere’s

radius, which thereby plays the role of an eﬀective scalar inﬂaton.

Aside from the above, many other solutions — that make use of mor e of the available

(classical) degrees of freedom — are of course possible. Th i s possibility was considered in [

27]

and generically appears as a multi ﬁeld inﬂationary model. In this work, however, we focus

on the single ﬁeld model where the other “unused” degrees of freedom in this particular

solution will be identiﬁed with pr eh eat ﬁelds after inﬂation ends .

2.1 Action and equations of motion

We work in the (−, +, +, +) metric signature, and use boldfac e to denote matrices of dimen-

sion N . The eﬀective (3 + 1)-dimensional acti on of M-ﬂation [

26] comprises Einstein gravity,

minimally coupl e d to a Yang-Mills gauge ﬁeld A

and the three inﬂaton matrices Φ

S =

√

−g

(

R −

Tr (F

µν

) −

Tr (D

) − V (Φ

, [Φ

, Φ

])

)

, (2.1)

where, as usual, M

= 1/

√

8πG is the reduced Planck mass, F

µν

= 2∂

[µ

ν]

+ ig

, A

]

is the gauge ﬁeld strength, and D

= ∂

+ ig

, ·] is the gauge covariant derivative.

Moreover, the pote ntial is given by

V (Φ

, [Φ

, Φ

]) = Tr



−

[Φ

, Φ

][Φ

, Φ

] +

iκ

jkl

[Φ

, Φ

]Φ



, (2.2)

where in (

2.1) and ( 2. 2) there is a sum on repeated i, j, k indices and the three coupling

constant s have various stringy meanings: λ = 8πg

= 2g

is related to the string coupling

, κ = ˆκg

√

8πg

is related to the Ramond-Ramond anti sy mm et r i c form strength ˆκ, and m

is a parameter that multiples the three spatial c oordinates along the D3- br ane s in the metric

of the background SUGRA theory [

23]. To ensure a constant dilaton therein, we must also

impose the constraint λm

= 4κ

/9 [

23].

The equations of motion for the scalar and gauge ﬁel ds that follow from the

action (

2.1) are

+ λ[Φ

, [Φ

, Φ

]] − iκǫ

ijk

[Φ

, Φ

] − m

= 0, (2.3)

µν

− ig

[Φ

, D

] = 0. (2.4)

2.2 Truncation to the SU(2) sector

The dynamics determined by the equations of motion (

2.3) and (2.4) can generically be quite

complicated, but this m ay be simpliﬁed considerably as follows. Let J

, i = 1, 2, 3 denote the

– 4 –

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