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宇宙学与宇宙常数  

2010-07-20 16:30:54|  分类: 他科之璞可以攻玉 |  标签: |举报 |字号 订阅

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 Cosmology with a Cosmological Constant

2.1 2.1 Cosmological parameters宇宙学参数

From the Friedmann equation ( 5从弗里德曼方程( 5 查看方程 ) (where henceforth we take the effects of a cosmological constant into account by including the vacuum energy density )(其中,今后我们采取能量密度的影响,包括真空的宇宙常数考虑 ρΛ into the total density进入总密度 ρ ), for any value of the Hubble parameter ),对于任何参数值哈勃 ? there is a critical value of the energy density such that the spatial geometry is flat (有一个临界值的能量密度等,空间几何平面( 当k ): ):

3H2ρcrit≡----. (21)8πG
It is often most convenient to measure the total energy density in terms of the critical density, by introducing the density parameter它往往是最方便的措施的总能量密度的密度计算的关键,通过引入密度参数
()-ρ - 8πG -
One useful feature of this parameterization is a direct connection between the value of其中的一个参数化特征是有用的价值之间的直接联系 Ω and the spatial geometry:和空间几何:
K二存在sgn(Ω - 1)。 (23)
[Keep in mind that some references still use “ [请记住,有些引用仍然使用“ Ω ” to refer strictly to the density parameter in matter, even in the presence of a cosmological constant; with this definition ( 23 “严格是指在物质密度参数不变,即使在宇宙存在的;这个定义( 23 查看方程 ) no longer holds.] )不再举行。]

In general, the energy density在一般情况下,能量密度 ρ will include contributions from various distinct components.将包括来自各个不同组成部分的贡献。 From the point of view of cosmology, the relevant feature of each component is how its energy density evolves as the universe expands.从宇宙学的角度来看,每个组件相关的特征是如何扩展其能量密度随着宇宙的演变。 Fortunately, it is often (although not always) the case that individual components幸运的是,往往(但并非总是)的情况下,个别组件 我 have very simple equations of state of the form有国家的形式很简单方程

圆周率=wiρi,(24)
with 无线 a constant.一个常数。 Plugging this equation of state into the energy-momentum conservation equation插上的能量动量守恒方程这种状态方程 μν?μT的=0 , we find that the energy density has a power-law dependence on the scale factor,我们发现,有一个能量密度的规模因子幂律的依赖,
ρα1妮,(25)我
where the exponent is related to the equation of state parameter by其中指数是关系到国家的参数方程
妮=3(1 +无线)。 (26)
The density parameter in each component is defined in the obvious way,在每个元件密度参数是定义在明显的方式,
which has the useful property that它具有有用的属性,
Ωi-α1 -(镍新泽西州)。 (28)Ωj

The simplest example of a component of this form is a set of massive particles with negligible relative velocities, known in cosmology as “dust” or simply “matter”.这种形式最简单的例子,一个组件是一个“集”的事,大量粒子或只是相对的微不足道的速度,在已知的宇宙学的“灰尘”。 The energy density of such particles is given by their number density times their rest mass; as the universe expands, the number density is inversely proportional to the volume while the rest masses are constant, yielding在这样的粒子的能量密度是给予他们数密度倍静止质量;随着宇宙膨胀,数密度是成反比的数量,而其余的群众不断,高产 ρMα1 - 3 . For relativistic particles, known in cosmology as “radiation” (although any relativistic species counts, not only photons or even strictly massless particles), the energy density is the number density times the particle energy, and the latter is proportional to相对论性粒子,称为“(在宇宙学的”辐射虽然任何物种数相对论,不仅严格光子甚至无质量粒子),能量密度是多少倍的粒子的能量密度,而后者是成正比  - 1 1 (redshifting as the universe expands); the radiation energy density therefore scales as (扩大redshifting作为宇宙);的辐射能量密度,因此要跟 ρRα1 - 4 . Vacuum energy does not change as the universe expands, so真空能量是不会改变的宇宙膨胀,使 ρΛα的A0 ; from ( 26 ,从( 26 查看方程 ) this implies a negative pressure, or positive tension, when the vacuum energy is positive. ),这意味着一个负压力,或正紧张,当真空能量是积极的。 Finally, for some purposes it is useful to pretend that the最后,对于某些目的是有益的假装 -2 -2 - R 0的家 term in ( 5任期( 5 查看方程 ) represents an effective “energy density in curvature”, and define )代表了密度曲率“有效”的能源,并确定 2 -2ρk≡ - (3k/8πGR0)1 . We can define a corresponding density parameter我们可以定义一个相应的密度参数

Ωk=1 - Ω;(29)
this relation is simply ( 5这种关系是根本( 5 查看方程 ) divided by )除以 二期H . Note that the contribution from请注意,捐款 Ωk is (for obvious reasons) not included in the definition of是(出于明显的原因)不包括在定义 Ω . The usefulness of是有益的 Ωk is that it contributes to the expansion rate analogously to the honest density parameters是,它有助于扩张速度比照诚实的密度参数 Ωi ; we can write ,我们可以写出
where the notation其中的符号 Σ我(十一) reflects the fact that the sum includes反映这一事实的总和,包括 Ωk in addition to the various components of除了各个组成部分的 Ω=ΣiΩi . The most popular equations of state for cosmological energy sources can thus be summarized as follows:流行方程来源的大多数州的宇宙能量因而可归纳如下:
| | --------------的Wi - 镍事项| 0 3 | |辐射| 1 / 3 4(31)“弯曲” - 1 / 3 2真空| - 1 0 |

The ranges of values of the对值的范围 Ωi 's which are allowed in principle (as opposed to constrained by observation) will depend on a complete theory of the matter fields, but lacking that we may still invoke energy conditions to get a handle on what constitutes sensible values.之获准在原则(而不是观察到的限制),将取决于一个问题领域的完整的理论的,但我们可能仍然缺乏援引能源条件得到处理什么是合理的价值观。 The most appropriate condition is the dominant energy condition (DEC), which states that最合适的条件是主导的能源状况(12月),其中指出, μνTμνl升≥0 , and ,和 μμ?νl is non-spacelike, for any null vector属于非类空,任何空载体 lμ ; this implies that energy does not flow faster than the speed of light [ 117 ,这意味着能源不流[比光的速度更快的117 ] . ]。 For a perfect-fluid energy-momentum tensor of the form ( 4对于一个完美的流体能量动量(张量的形式4 查看方程 ), these two requirements imply that ),这两项要求意味着 ρ+ P值≥0 and |ρ|≥| p | , respectively.分别。 Thus, either the density is positive and greater in magnitude than the pressure, or the density is negative and equal in magnitude to a compensating positive pressure; in terms of the equation-of-state parameter因此,无论是正面的密度和密度越大,增加的幅度大于压力,或者是消极的,平等的幅度为正压补偿;在参数计算公式的其他州的 瓦特 , we have either positive我们要么积极 ρ and |瓦特|≤1 or negative或负 ρ and 瓦特=- 1 . That is, a negative energy density is allowed only if it is in the form of vacuum energy.也就是说,一个负能量密度只有当它被允许在真空中的能量形式。 (We have actually modified the conventional DEC somewhat, by using only null vectors (事实上,我们已经修改了传统的12月有点,只使用空载体 lμ rather than null or timelike vectors; the traditional condition would rule out a negative cosmological constant, which there is no physical reason to do.)而不是空或类时向量;传统的条件会排除负宇宙常数,不得有任何身体原因做。)

There are good reasons to believe that the energy density in radiation today is much less than that in matter.有充分理由相信,在辐射能量密度比今天少得多,在此事。 Photons, which are readily detectable, contribute光子,很容易被检测到,贡献 Ωγ?5 × 10-5 , mostly in the ,主要集中在 2.73 K表 cosmic microwave background [ 211宇宙微波背景[ 211 , 87 , 87 , 225 , 225 ] . ]。 If neutrinos are sufficiently low mass as to be relativistic today, conventional scenarios predict that they contribute approximately the same amount [ 149如果中微子有质量的相对论今天以足够低,传统的情景预测,它们有助于大约相同数量[ 149 跳转到下引点 ] . ]。 In the absence of sources which are even more exotic, it is therefore useful to parameterize the universe today by the values of在异国缺乏来源,甚至更多,因此它是有用的参数化的价值观在今天的宇宙 ΩM and ΩΛ , with Ωk=1 - ΩM - ΩΛ , keeping the possibility of surprises always in mind. ,保持惊喜的可能性始终铭记。

One way to characterize a specific Friedmann–Robertson–Walker model is by the values of the Hubble parameter and the various energy densities一种方法来描述一个特定的弗里德曼,罗伯逊,沃克模型是由哈勃参数值和各种能量密度 ρi . (Of course, reconstructing the history of such a universe also requires an understanding of the microphysical processes which can exchange energy between the different states.) It may be difficult, however, to directly measure the different contributions to (当然,重建宇宙历史的这样一个国家也需要一个了解的微物理过程,可以对不同汇率之间的能量。)可能会很难,但是,直接测量到不同的贡献 ρ , and it is therefore useful to consider extracting these quantities from the behavior of the scale factor as a function of time. ,因此它是有用的考虑提取时间从这些量函数的行为的规模的一个因素作为。 A traditional measure of the evolution of the expansion rate is the deceleration parameter作者:进化的扩张速度是传统的测量参数的减速

where in the last line we have assumed that the universe is dominated by matter and the cosmological constant.如果在最后一行我们假设宇宙是由物质和不断为主的宇宙。 Under the assumption that根据假设 ΩΛ=0 , measuring ,测量 Q0的 provides a direct measurement of the current density parameter密度参数提供了直接测量的电流 ΩM0 ; however, once ,然而,一旦 ΩΛ is admitted as a possibility there is no single parameter which characterizes various universes, and for most purposes it is more convenient to simply quote experimental results directly in terms of被接纳为一的可能性不存在单一的宇宙的各种参数,特点,对大多数的目的是更方便,简单引述计算,实验结果直接 ΩM and ΩΛ . [Even this parameterization, of course, bears a certain theoretical bias which may not be justified; ultimately, the only unbiased method is to directly quote limits on [即使这当然参数,负有一定的理论偏差,可能没有道理的,最终,唯一公正的方法是直接报价期限 1(吨) .] 。]

Notice that positive-energy-density sources with请注意,积极的能量密度与来源 2" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article141x.gif" width=43< cause the universe to decelerate while导致宇宙减速,而 ? <2 leads to acceleration; the more rapidly energy density redshifts away, the greater the tendency towards universal deceleration.导致加速;更迅速的能量密度红移距离,更大的减速趋势,实现普遍。 An empty universe (一个空的宇宙( Ω=0 , Ωk=1 ) expands linearly with time; sometimes called the “Milne universe”, such a spacetime is really flat Minkowski space in an unusual time-slicing. )随时间线性扩展,有时被称为“米尔恩宇宙”,这样的单位实在是闵可夫斯基时空切片的空间在一个不寻常的时间。

2.2 2.2 Model universes and their fates宇宙模型及其命运

In the remainder of this section we will explore the behavior of universes dominated by matter and vacuum energy,在本节的其余部分,我们将探讨能源行为真空和宇宙物质为主, Ω=ΩM+ΩΛ=1 - Ωk . According to ( 33根据( 33 查看方程 ), a positive cosmological constant accelerates the universal expansion, while a negative cosmological constant and/or ordinary matter tend to decelerate it. The relative contributions of these components change with time; according to ( 28 ),正不断加快宇宙的普遍扩张,而负宇宙常数和/或普通物质会有所减慢了。改变的时候,相对的贡献,这些成分与根据( 28 查看方程 ) we have )我们

ΩΛαa2Ωkαa3ΩM。 (33)
For ΩΛ<0 , the universe will always recollapse to a Big Crunch, either because there is a sufficiently high matter density or due to the eventual domination of the negative cosmological constant. ,宇宙将永远坍缩到一个大紧缩,或者是因为有一个足够高的物质密度,或由于宇宙常数的负最终统治。 For 0" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article148x.gif" width=56< the universe will expand forever unless there is sufficient matter to cause recollapse before宇宙将永远膨胀下去,除非有足够的物质引起坍缩之前 ΩΛ becomes dynamically important.成为重要的动态。 For Ω=0Λ we have the familiar situation in which我们已经熟悉的情况,就是 ΩM≤1 universes expand forever and宇宙永远膨胀和 1" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article152x.gif" width=58< universes recollapse; notice, however, that in the presence of a cosmological constant there is no necessary relationship between spatial curvature and the fate of the universe.宇宙坍缩;通知,但是,在一个宇宙常数的存在是没有必要的宇宙的空间曲率之间的关系和命运。 (Furthermore, we cannot reliably determine that the universe will expand forever by any set of measurements of (此外,我们不能可靠地确定宇宙将永远扩大的测量任何一套 ΩΛ and ΩM ; even if we seem to live in a parameter space that predicts eternal expansion, there is always the possibility of a future phase transition which could change the equation of state of one or more of the components.) ,即使我们似乎生活在一个永恒的参数空间的扩展,可以预测,但始终是一个未来的可能性阶段过渡,可以改变状态方程的一个或多个组件的。)

Given特定 ΩM , the value of ,其价值 ΩΛ for which the universe will expand forever is given by为此,宇宙将永远膨胀,给出了

1。 (34)3ΩM3" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article157x.gif" width=650<
Conversely, if the cosmological constant is sufficiently large compared to the matter density, the universe has always been accelerating, and rather than a Big Bang its early history consisted of a period of gradually slowing contraction to a minimum radius before beginning its current expansion.相反,如果宇宙常数是足够大的物质密度比,宇宙一直加快,而不是大爆炸的早期历史的扩展包括目前一期开始之前,其最小半径逐渐放缓收缩到。 The criterion for there to have been no singularity in the past is那里的标准为已没有过去奇异的是
3 [1 -  - 1(1 ---ΩM - )]ΩΛ≥4ΩMcoss3cossΩ,(35)小
where “coss” represents其中,“科斯”代表 护身用手杖 when何时 ΩM<1 / 2 , and ,和 余弦 when何时 1 / 2" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article162x.gif" width=77< .

The dynamics of universes with对宇宙的动态与 Ω=ΩM+ΩΛ are summarized in Figure 1总结在图1 查看图片 , in which the arrows indicate the evolution of these parameters in an expanding universe. ,其中箭头表示参数在不断扩大的宇宙演化这些。 (In a contracting universe they would be reversed.) This is not a true phase-space plot, despite the superficial similarities. (在一个收缩的宇宙,他们将得到扭转。)这不是一个真正的相空间图相似,尽管表面上。 One important difference is that a universe passing through one point can pass through the same point again but moving backwards along its trajectory, by first going to infinity and then turning around (recollapse).一个重要的区别是,宇宙途经一点可以通过相同点,但再次向后移动轨迹沿坍缩,首先要到无穷远,然后转身()。

查看图片

Figure 1: Dynamics for 图1: 动力学 Ω=ΩM+ΩΛ . The arrows indicate the direction of evolution of the parameters in an expanding universe. 箭头表明宇宙的膨胀在参数的方向进化。

Figure 11 查看图片 includes three fixed points, at包括三个固定点,在 (ΩM,ΩΛ) equal to等于 (0,0) , (0,1) , and ,和 (1,0) . The attractor among these at在这些之间的吸引 (0,1) is known as de Sitter space – a universe with no matter density, dominated by a cosmological constant, and with scale factor growing exponentially with time. The fact that this point is an attractor on the diagram is another way of understanding the cosmological constant problem.被称为de Sitter时空-宇宙的密度与宇宙学常数不管,占主导地位的一个,规模和时间呈指数增长的因素。事实上,这一点是吸引上图是另外一个问题的方式理解宇宙常数。 A universe with initial conditions located at a generic point on the diagram will, after several expansion times, flow to de Sitter space if it began above the recollapse line, and flow to infinity and back to recollapse if it began below that line.阿图宇宙的初始点位于一个通用的条件后,便会将几个扩张的时候,流向de Sitter空间,如果它开始坍缩以上的路线,流向无穷,回到坍缩,如果低于该行开始。 Since our universe has expanded by many orders of magnitude since early times, it must have begun at a non-generic point in order not to have evolved either to de Sitter space or to a Big Crunch. The only other two fixed points on the diagram are the saddle point at由于我们的宇宙有规模扩大的订单年初以来,很多时候,它必须有一个开始或大坍缩。在非泛型点,以免已经发展要么de Sitter空间只有其他两个固定点的图是鞍点 (ΩM,ΩΛ)=(0,0) , corresponding to an empty universe, and the repulsive fixed point at ,相应的空的宇宙,而令人厌恶的固定点 (ΩM,ΩΛ)=(1,0) , known as the Einstein–de Sitter solution. ,溶液被称为爱因斯坦德保姆。 Since our universe is not empty, the favored solution from this combination of theoretical and empirical arguments is the Einstein–de Sitter universe.由于我们的宇宙是不是空的,经验参数的最佳解决方案,并从这种组合理论是爱因斯坦德保姆宇宙。 The inflationary scenario [ 113通货膨胀的情况[ 113 跳转到下引点 , 159 , 159 跳转到下引点 , 6 , 6 跳转到下引点 ] provides a mechanism whereby the universe can be driven to the line ]提供一个机制,使宇宙线可驱车前往 Ω+Ω=1海里Λ (spatial flatness), so Einstein–de Sitter is a natural expectation if we imagine that some unknown mechanism sets (空间平整度),所以爱因斯坦德保姆是一个自然的期望,如果我们想象,一些未知的机制套 Λ=0 . As discussed below, the observationally favored universe is located on this line but away from the fixed points, near正如下面讨论中,对观察宇宙青睐位于这条线,但离固定点,近 (ΩM,ΩΛ)=(0.3,0.7) . It is fair to conclude that naturalness arguments have a somewhat spotty track record at predicting cosmological parameters.这是公平的论点,认为自然宇宙学参数记录在预测有点参差不齐的轨道。

2.3 2.3 Surveying the universe测量宇宙

The lookback time from the present day to an object at redshift从今天回望时间的对象在红移 ? * is given by由下式给出

with 的H(1) given by ( 30由( 30 查看方程 ). )。 The age of the universe is obtained by taking the宇宙的年龄以获得 ? *→∞ ( *→0吨 ) limit. For )限制。对于 Ω=ΩM=1 , this yields the familiar answer这熟悉的答案产量 T0的=(2 / 3)H的-01 ; the age decreases as ;的年龄会随着 ΩM is increased, and increases as增加,并增加为 ΩΛ is increased.增加。 Figure 22 查看图片 shows the expansion history of the universe for different values of these parameters and显示了这些参数和扩张的历史价值为不同的宇宙 您 fixed; it is clear how the acceleration caused by固定不变的,它是明确如何加速造成 ΩΛ leads to an older universe.导致旧的宇宙。 There are analytic approximation formulas which estimate ( 36有分析近似公式的估计数( 36 查看方程 ) in various regimes [ 264 )在制度[264个 跳转到下引点 , 149 149 跳转到下引点 , 48 48 跳转到下引点 ] , but generally the integral is straightforward to perform numerically. ],但一般的积分是直接进行数值。
查看图片

Figure 2: Expansion histories for different values of 图2: 价值膨胀的不同的历史 ΩM and ΩΛ . From top to bottom, the curves describe 从上到下,曲线描述 (ΩM,ΩΛ)=(0.3,0.7) , (0.3,0.0) , (1.0,0.0) , and ,和 (4.0,0.0) .

In a generic curved spacetime, there is no preferred notion of the distance between two objects. Robertson–Walker spacetimes have preferred foliations, so it is possible to define sensible notions of the distance between comoving objects – those whose worldlines are normal to the preferred slices.在一个通用的弯曲时空,没有沃克时空中首选的概念,在两个物体之间的距离。罗伯逊倾向于面理,所以它可以定义对象-明智的概念comoving之间的距离是正常的,其worldlines片的首选。 Placing ourselves at我们在配售 "?=0" in the coordinates defined by ( 2在坐标定义为( 2 查看方程 ), the coordinate distance ),坐标距离 ? to another comoving object is independent of time.另一comoving对象是时候独立。 It can be converted to a physical distance at any specified time它可以被转换为物理距离在任何特定时间 吨* by multiplying by the scale factor由因子乘以规模 R0A值(吨*) , yielding a number which will of course change as the universe expands. ,产生了多少宇宙的膨胀过程中改变。 However, intervals along spacelike slices are not accessible to observation, so it is typically more convenient to use distance measures which can be extracted from observable quantities.然而,随着时间的类空片是不容易观察,所以它通常更方便地使用观察距离的措施,从数量可以提取。 These include the luminosity distance,这些措施包括光度距离,

where哪里 L is the intrinsic luminosity and是内在的亮度和 F the measured flux; the proper-motion distance,实测流量;适当的动作距离,
马克≡ü,(38)˙θ
where哪里 ü is the transverse proper velocity and是适当的速度和横向 θ˙ the observed angular velocity; and the angular-diameter distance,观测到的角速度和角直径距离,
where哪里 ? is the proper size of the object and是的目标和合适的大小 θ its apparent angular size.其明显的角大小。 All of these definitions reduce to the usual notion of distance in a Euclidean space.所有这些定义的距离缩短到通常的概念在欧氏空间。 In a Robertson–Walker universe, the proper-motion distance turns out to equal the physical distance along a spacelike slice at在罗伯逊,沃克宇宙,正确的动作距离原来是平等的类空片的物理距离在沿 "吨=T0代" :
"马克=R0r。
The three measures are related by这三项措施是相关的
"2分升=(1
so any one can be converted to any other for sources of known redshift.因此,任何一个可以转换为任何其他已知红移来源。

The proper-motion distance between sources at redshift来源之间适当的动作距离在红移 z1 and z2的 can be computed by using可以计算出 "二局副局长=0" along a light ray, where沿着光线,在那里 二局副局长 is given by ( 2公式为( 2 查看方程 ). )。 We have我们已

where we have used ( 5如果我们用( 5 查看方程 ) to solve for )来解决 ?----- , 的H(1) is again given by ( 30又由( 30 查看方程 ), and “ )和“ 辛恩(十) ” denotes “表示 双曲正弦(十) when何时 0" src="http://relativity.livingreviews.org/Articles/lrr-2001-1/article219x.gif" width=60< , 罪(十) when何时 Ωk0<零 , and ,和 x when何时 Ωk0=0 . An analytic approximation formula can be found in [ 193一种解析近似公式中可以找到[ 193 ] . ]。 Note that, for large redshifts, the dependence of the various distance measures on请注意,大红移,这些措施的依赖各种距离 ? is not necessarily monotonic.不一定是单调的。

The comoving volume element in a Robertson–Walker universe is given by罗伯逊在1 -沃克宇宙comoving体积元是给予

which can be integrated analytically to obtain the volume out to a distance它可以集成解析获取量出来的距离 马克 :
where “sinn” is defined as before ( 42其中,“新芬”是指前( 42 查看方程 ). )。

2.4 2.4 Structure formation结构的形成

The introduction of a cosmological constant changes the relationship between the matter density and expansion rate from what it would be in a matter-dominated universe, which in turn influences the growth of large-scale structure.从引进一个膨胀的宇宙密度不断变化率之间的关系问题,它会在一个问题为主的宇宙,从而影响经济增长的结构大的规模。 The effect is similar to that of a nonzero spatial curvature, and complicated by hydrodynamic and nonlinear effects on small scales, but is potentially detectable through sufficiently careful observations.效果是类似的曲率是一个非零空间,复杂的非线性效应和规模小的水动力和,但仔细观察可能是通过探测充分。

The analysis of the evolution of structure is greatly abetted by the fact that perturbations start out very small (temperature anisotropies in the microwave background imply that the density perturbations were of order 10 –5 at recombination), and linearized theory is effective.结构分析的演变有很大的背景助长了这一事实扰动开始温度在微波各向异性出非常小的(意味着密度扰动的顺序为:在10个重组-5),和线性理论是有效的。 In this regime, the fate of the fluctuations is in the hands of two competing effects: the tendency of self-gravity to make overdense regions collapse, and the tendency of test particles in the background expansion to move apart.在这种制度下,波动的命运是在竞争的影响手中的两个:重力倾向自我崩溃,使稠密的地区,以及扩大趋势测试粒子在后台移动分开。 Essentially, the effect of vacuum energy is to contribute to expansion but not to the self-gravity of overdensities, thereby acting to suppress the growth of perturbations [ 149从本质上讲,节能效果真空是有助于扩大而不是自我overdensities严重性,从而采取行动,制止[扰动增长149 , 189 , 189 ] . ]。

For sub-Hubble-radius perturbations in a cold dark matter component, a Newtonian analysis suffices. (We may of course be interested in super-Hubble-radius modes, or the evolution of interacting or relativistic particles, but the simple Newtonian case serves to illustrate the relevant physical effect.) If the energy density in dynamical matter is dominated by CDM, the linearized Newtonian evolution equation is对于元器件,1牛顿分析足以分哈勃半径扰动在寒冷的暗物质。(我们当然可以颗粒感兴趣超哈勃半径模式,或相对论的演变或相互作用,但简单的牛顿的实例,已说明有关的物理效应。方程)如果在动态的能量密度问题的演变主要由清洁发展机制,是牛顿的线性

¨δ+
The second term represents an effective frictional force due to the expansion of the universe, characterized by a timescale第二项代表一个有效的摩擦力,由于宇宙膨胀的,其特点是一个时间表 -1 -1(˙1 / , while the right hand side is a forcing term with characteristic timescale ,而右边是一个强迫项与特征时间表 (4πGρM) - 1 / 2≈Ω-1/2H -1 M . Thus, when因此,当 ΩM≈1 , these effects are in balance and CDM perturbations gradually grow; when这些影响都是在平衡与清洁发展机制扰动逐渐长大,当 ΩM dips appreciably below unity (as when curvature or vacuum energy begin to dominate), the friction term becomes more important and perturbation growth effectively ends.团结逢低明显低于(或真空曲率作为能源时开始主宰),长期的摩擦变得更加重要和有效的扰动增长的目的。 In fact ( 45 (其实45 查看方程 ) can be directly solved [ 119 )可以直接解决[ 119 ] to yield ]屈服
∫1δ(1)=5 - H2的Ω˙1 - [a'H(1')] -3大',(46)小M0余额1 0 2 0
where哪里 的H(1) is given by ( 30公式为( 30 查看方程 ). )。 There exist analytic approximations to this formula [ 48存在这个公式解析近似[ 48 跳转到下引点 ] , as well as analytic expressions for flat universes [ 81 ],以及宇宙[解析表达式为单位81 ] . ]。 Note that this analysis is consistent only in the linear regime; once perturbations on a given scale become of order unity, they break away from the Hubble flow and begin to evolve as isolated systems.请注意,这种分析是一致的唯一的线性制度;秩序的统一性成为一次规模上给予扰动,他们摆脱哈勃流,并开始演变成为孤立的系统。
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