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LANCS-TH/9501astro-ph/9501044Large scale perturbations in the open universe David H.Lyth ?and Andrzej Woszczyna ??School of Physics and Materials,University of Lancaster,Lancaster LA14YB.U.K.and Isaac Newton Institute,20Clarkson Road,Cambridge CB30EH.U.K.?Astronomical Observatory,Jagiellonian University,ul.Orla 171,Krakow 30244.Poland.Abstract When considering perturbations in an open (?0<1)universe,cosmologists retain only sub-curvature modes (de?ned as eigenfunctions of the Laplacian whose eigenvalue is less than ?1in units of the curvature scale,in contrast with the super-curvature modes whose eigenvalue is between ?1and 0).Mathematicians have known for almost half a century that all modes must be included to generate the most general homogeneous Gaussian random ?eld ,despite the fact that any square integrable function can be generated using only the sub-curvature modes.The former mathematical object,not the latter,is the relevant one for physical applications.The mathematics is here explained in a language accessible to physicists.Then it is pointed out that if the perturbations originate as a vacuum ?uctuation of a scalar ?eld there will be no super-curvature modes in nature.Finally the e?ect on the cmb of any super-curvature contribution is considered,which generalizes to ?0<1the analysis given by Grishchuk and Zeldovich in 1978.A formula is given,which is used to estimate the e?ect.In contrast with the case ?0=1,the e?ect contributes to all multipoles,not just to the quadrupole.It is important to ?nd out whether it has the same l dependence
as the data,by evaluating the formula numerically.
1Introduction
On grounds of simplicity,the present energy density ?0of the universe is generally assumed to be equal to unity (working as usual in units of the critical density).1It is not however well determined by observation [1].The density of baryonic matter can only be of order 0.1or there will be a con?ict with the nucleosynthesis calculation,and although non-baryonic matter seems to be required by observation [2]there is no guarantee that it will bring the total up to ?0=1.Nor should one assume that a cosmological constant or other exotic contribution to the energy density will play this role.
From a theoretical viewpoint the value ?0=1is the most natural,because any other value of ?is time dependent.The preference for ?0=1is sharpened if,as is widely believed,the hot big bang is preceded by an era of in?ation.In that case ?has its present value at the epoch when the present Hubble scale leaves the horizon,and for a generic choice of the in?aton potential this indeed implies that ?0is very close to 1more or less independently of
the initial value of?.It is also easier for in?ation to explain the homogeneity and isotropy of the observable universe if?0=1.On the other hand it is certainly not the case that ?0=1is an unambiguous prediction of in?ation[3,4].
The literature on the?0<1cosmology is small compared with the enormous output on the case?0=1,because the latter is simpler and observations that can distinguish the two are only now becoming available.This is especially true in regard to the subject of the present paper,which is the e?ect of spatial curvature on cosmological perturbations.The only data relevant to this subject are the lowest few multipoles of the cosmic microwave background(cmb)anisotropy,that were measured recently by the COBE satellite[5,6,7].
This article is concerned both with the basic formalism that one should use in describing cosmological perturbations,and with the cmb multipoles.To describe its contents,let us be-gin by recalling the presently accepted framework within which cosmological perturbations are discussed.
Cosmological perturbations are expanded in a series of eigenfunctions of the Laplacian for two separate reasons.One is that each mode(each term in the series)evolves indepen-dently with time,which makes it easier to evolve a given initial perturbation forward in time.The other is that by assigning a Gaussian probability distribution to the amplitude of each mode,one can generate a homogeneous Gaussian random?eld.Such a?eld consists of an ensemble of possible perturbations,and it is supposed that the perturbation seen in the observable universe is a typical member of the ensemble.The stochastic properties of a Gaussian random?eld are determined by its two point correlation function f(1)f(2) , where f is the perturbation and the brackets denote the ensemble average,and the adjective ‘homogeneous’indicates that the correlation function depends only on the distance between the two points.
The question arises which eigenfunctions to use,and in particular what range of eigen-values to include.If?0=1space is?at and it is known that the Fourier expansion,which includes all negative eigenvalues,is the correct choice.It is complete in two distinct re-spects.First,it gives the most general square integrable function,so that initial conditions in a?nite region of the universe can be evolved forward in time.Secondly,it gives the most general homogeneous Gaussian random?eld.Instead of the Fourier expansion one can use the entirely equivalent expansion in spherical polar coordinates.
If?0<1,the curvature of space de?nes a length scale.The spherical coordinate expansion can still be used,and it is known[8,9]that the modes which have real negative eigenvalue less than?1in units of the curvature scale provide a complete orthonormal basis for square integrable functions.Presumably for this reason,only these modes have been retained by cosmologists.We will call them sub-curvature modes,because they vary signi?cantly on a scale which is less than the curvature scale.The other modes,with eigenvalues between?1and0in units of the curvature scale,we will call super-curvature modes.
It is certainly enough to retain only sub-curvature modes if all one wishes to do is to track the evolution of a given initial perturbation,since the region of interest is always going to be?nite and any function de?ned in a?nite region can be expanded in terms of the sub-curvature modes.(In fact,to describe the observations that we can make it is enough to specify initial conditions within our past light cone.)But this is not what one does in cosmology.2Rather,one uses the mode expansion to generated a Gaussian perturbation, by assigning a Gaussian probability distribution to the amplitude of each mode.In this context the inclusion of only sub-curvature modes looks restrictive.For example,it leads to a correlation function which necessarily becomes small at distances much bigger than the
curvature scale(to be precise,it is less than r/sinh r times its value at r=0,where r is the distance in curvature units).
Faced with this situation,we queried the assumption that only sub-curvature modes should be included,and the results of our investigation are reported here.
First we describe the mathematical situation,showing that indeed a more general Gaus-sian random?eld is generated by including also the super-curvature modes.As expected the correlation function can now be constant out to arbitrarily large distances.
Then we go on to ask whether nature has chosen to use the super-curvature modes, focussing on the low multipoles of the cmb anisotropy which are the only relevant observa-tional data,and on the curvature perturbation which is thought to be responsible for these multipoles.If,as is usually supposed,this perturbation originates as a vacuum?uctuation of the in?aton?eld,there will be no super-curvature modes.On the other hand,like any other statement about the universe one expects this assumption to be at best approximately valid.Supposing that it fails badly on some very large scale,but that the curvature per-turbation still corresponds to a typical realization of a homogeneous Gaussian random?eld, one is lead to ask if a failure of the assumption could be detected by observing the cmb anisotropy.We note that for?0=1this question has already been discussed by Grishchuk and Zeldovich[10],and we extend their discussion to the case?0<1.
After our investigation was complete,and the draft of this paper was almost complete, M.Sasaki suggested to one of us(DHL)that a mathematics paper written by Yaglom in 1961[11]might be relevant.From this paper we learned that the need to include both sub-and super-curvature modes in the expansion of a homogeneous Gaussian random?eld in negatively curved space has been known to mathematicians since at least1949[12].It would appear therefore that the assumption by cosmologists that only the sub-curvature modes are needed is a result of a complete failure of communication between the worlds of mathematics and science,which has persisted for many decades.We have retained the mathematics part of our paper because it gives the relevant results in the sort of language that is familiar to physicists,though it is strictly speaking redundant.
Let us end this introduction by saying a bit more about the cosmology literature.Start-ing with the paper of Lifshitz in1946[13],there are many papers on the treatment of cosmological perturbations for the case?<1.However,most of them deal with the def-inition and evolution of the perturbations,which is not our main concern.We have not attempted a full survey of this part of the literature,but have just cited useful papers that we happen to be aware of.By contrast,the cosmology literature on stochastic properties is very small for the case?0<1,and as we have mentioned it is out of touch with the relevant pure mathematics literature where the theory of random?elds is discussed.The?rst serious treatment of stochastic properties is by Wilson in1983[14].He developed the theory from scratch,and not surprisingly included only the sub-curvature modes which he knew were su?cient for the description of the non-stochastic properties.His notation is defective and much is left unsaid,but subsequent papers have not made basic advances in the formulation of the subject,though they have gone much further in calculating the cmb multipoles and comparing them with observation.We believe our referencing to be reasonable complete, as far as the cosmology literature on the stochastic properties is concerned.
The layout of this paper is as follows.In Section2some basic formulas are given for the Robertson-Walker universe with?<1.In Section3the standard procedure is described, and in the next section it is extended to the super-curvature modes.In?ation is discussed in Section5,and the cmb anisotropy is treated in Section6.In an Appendix we give various mathematical results in the sort of language that is familiar to us as physicists.
2Distance scales
Ignoring perturbations,the universe is homogeneous and isotropic.There is a universal scale factor a(t),with t the universal time measured by the synchronized clocks of comoving observers,and the distance between any two such observers is proportional to a.
According to the Einstein?eld equation,the time dependence of a is governed by the Friedmann equation which may be written
1??=?
K
2r ph=??10?1.Even the smallest conceivable value?0?0.1gives r ph=3.6,
so e?ect of curvature is negligible except on scales comparable with the size of the observable universe.
From Eq.(1),the physical distance of the particle horizon is
a0r ph=(1??0)?1/2H?10r ph(3) For?0=1it is2H?10,and even for?0=0.1it is only3.8H?10.Thus it is not very much bigger than the Hubble distance H?10.
3Sub-curvature modes
We are concerned with the?rst order treatment of cosmological perturbations.To this order, the perturbations‘live’in unperturbed spacetime,because the distortion of the spacetime geometry is itself a perturbation.
The perturbations satisfy linear partial di?erential equations,in which derivatives with respect to comoving coordinates occur only through the Laplacian.When the perturbations are expanded in eigenfunctions of the Laplacian with eigenvalues?(k/a)2,each mode(term in the expansion)decouples.
Denoting the eigenvalue by?(k/a)2,it is known[8,9]that the modes with real k2>1 provide a complete orthonormal basis for L2functions,and the usual procedure is to keep
only them.Since they all vary appreciably on scales less than the curvature scale a we will call them sub-curvature modes.It will be useful to de?ne the quantity
q2=k2?1(4) 3.1The spherical expansion
Spherical coordinates are de?ned by the line element
d l2=a2[d r2+sinh2r(dθ2+sin2θdφ2)](5) In th
e region r?1curvature is negligible and this becomes the?at-space line element written in spherical polar coordinates.The volume element between adjacent spheres is 4πsinh2r d r,so for r?1the volume V and area A o
f a sphere are related by V=A/2. In contrast with the?at-space case this relation is independent of r,because most of the volume of a very large sphere is near its surface.
Since the spherical harmonics Y lm are a complete set on the sphere,any eigenfunction can be expanded in terms of them.The radial functions depend only on r,and they satisfy a second order di?erential equation.As in the?at-space case,only one of the two solutions is well behaved at the origin,so the radial functions are completely determined up to normalisation.The mode expansion of a generic perturbation f is therefore of the form
f(r,θ,φ,t)= ∞0dq lm f klm(t)Z klm(r,θ,φ)(6)
where
Z klm=Πkl(r)Y lm(θ,φ)(7)
A compact expression for the radial functions is[16,13,17,9,18]
Πkl=Γ(l+1+iq)1
2
iq?1
sinh r
d
2
3These expressions correct some misprints in[19,4].
The un-normalised radial functions?Πkl satisfy a recurrence relation[20]
?Π
k,l+2=? (l+1)2+q2 ?Πkl+(2l+3)coth r?Πk,l+1(15) and the?rst three functions are
?Π
k0=1
q (16)
?Π
k1=1
q (17)
?Π
k2=1
q (18)
The case?=1corresponds to q→∞with qr?xed,and in that limitΠkl(r)reduces to the familiar radial function,
Πkl(r)→ πqj l(qr).(19)
Near the originΠkl(r)has the same behaviour as j l(qr),namelyΠkl∝r l,which ensures that the Laplacian is well de?ned there.The other linearly independent solution of the radial equation,which corresponds to the substitution cos(qr)→sin(qr)in Eq.(13),has the same behaviour as the other Bessel function h l(qr)and is therefore excluded.
3.2Stochastic properties
We are interested in the stochastic properties of the perturbations,at?xed time.To de?ne them we will take the approach of considering an ensemble of universes of which ours is supposed to be one.
The stochastic properties of a generic perturbation f(r,θ,φ)are de?ned by the set of probability distribution functions,relating to the outcome of a simultaneous measurement of a perturbation at a given set of points.From the probability distributions one can calculate ensemble expectation values,such as the correlation function for a pair of points r1,θ1,φ1 and r2,θ2,φ2,
ξf≡ f(r1,θ1,φ1),f(r2,θ2,φ2) (20) and the mean square f2(r,θ,φ) .
If the probability distributions depend only on the geodesic distances between the points, the perturbation is said to be homogeneous with respect to the group of transformations that preserve this distance.(For?at space this is the group of translations and rotations, and for homogeneous negatively curved space it is isomorphic to the Lorentz group[21].) Then the correlation function depends only on the distance between the points,and the mean square is just a number.
Cosmological perturbations are assumed to be homogeneous,and except for the curva-ture perturbation that we discuss in Section6their correlation functions are supposed to be very small beyond some maximum distance,called the correlation length.
An ergodic universe?
If there is a?nite correlation length,one ought to be able to dispense with the concept of an ensemble of universes,in favour of the concept of sampling our own universe at di?erent locations.In this approach one de?nes the probability distribution for simultaneous measurements at N points with by considering random locations of these points,subject to the condition that the distances between them are?xed.The correlation function is
de?ned by averaging over all pairs of points a given distance apart,and the mean square is the spatial average of the square.For a Gaussian perturbation in?at space this‘ergodic’property can be proved under weak conditions[22]and there is no reason to think that spatial curvature causes any problem though we are not aware of any literature on the subject.
For the ergodic viewpoint to be useful,the observable in question has to be measured in a region that is big compared with the correlation length.This is the case for the distributions and peculiar velocities of galaxies and clusters,where surveys have been done out to several hundred Mpc to be compared with a correlation length of order10Mpc,and accordingly the ergodic viewpoint is always adopted there[23].However,even a distance of a few hundred Mpc is only ten percent or so of the Hubble distance H?10,and therefore at most a few percent of the curvature scale(1??0)?1/2H?10.Thus galaxy and cluster surveys do not probe spatial curvature.The only observables that do,which are the low multipoles of the cmb anisotropy,are measured only at our position so there is no practical advantage in going beyond the concept of the ensemble even if the mathematics turns out to be straightforward.
In addition to the interpretation that the ensemble corresponds to di?erent locations within the smooth patch of the universe that we inhabit,there are two other possibilities. One is that the ensemble corresponds to di?erent smooth patches,which are indeed supposed to exist both in‘chaotic’[24]and bubble nucleation[25,26,27,28]scenarios of in?ation. The other,adopting the usual language of quantum mechanics,is to regard the ensemble as the set of all possible outcomes of a‘measurement’performed on a given state vector.
A concrete realization of this‘quantum cosmology’viewpoint is provided by the hypothesis that the perturbations originate as a vacuum?uctuation of the in?aton?eld,which we consider later.
3.3Gaussian perturbations
It is generally assumed that cosmological perturbations are Gaussian,in the regime where they are evolving linearly.A Gaussian perturbation is normally de?ned as one whose probability distribution functions are multivariate Gaussians[29,22,30],and its stochas-tic properties are completely determined by its correlation function.The perturbation is homogeneous if the correlation function depends only on the distance between the points.
The simplest Gaussian perturbation is just a coe?cient times a given function,the co-e?cient having a Gaussian probability distribution.A more general Gaussian perturbation is a linear superposition of functions[29],
f(r,θ,φ)= n f n X n(r,θ,φ)(21)
with each coe?cient having an independent Gaussian distribution.Its stochastic properties are completely determined by the mean squares f2n of the coe?cients.(For the moment we are taking the expansion functions X n to be real,and to be labelled by a discrete index.) The correlation function corresponding to the above expansion is
f(r1,θ1,φ1)f(r2,θ2,φ2) = n f2n X n(r1,θ1,φ1)X n(r2,θ2,φ2)(22)
For it to depend only on the distance between the points requires very special choices of the expansion functions,and of the mean squares f2n .
It is very important to realise that the functions in such an expansion need not be linearly independent.Suppose for example that X3=X1+X2,and that f23 is much bigger than f21 and f22 .Then most members of the ensemble are of the form f=const X3,which would clearly not have been the case if the function X3had been dropped because of its linear dependence.
So far all our considerations have been at a?xed time.The time dependence is trivial if we expand in eigenfunctions of the Laplacian,because each coe?cient f n then evolves independently of the others.Let us therefore replace the discrete,real expansion above by the complex,partially continuous expansion Eq.(6).The coe?cients now satisfy the reality condition f?klm=f kl?m,and a Gaussian perturbation is constructed by assigning independent Gaussian probability distributions to the real and imaginary parts of the co-e?cients with m≥0.We demonstrate in the Appendix that the correlation function being dependent only on the distance between the points is equivalent to the mean squares of their real and imaginary parts being equal,and independent of l and m.One can therefore de?ne the spectrum of a generic perturbation f by[4]
f?klm f k′l′m′ =2π2
k3P f(k)δ(k?k′)δll′δmm′(24) The correlation function is given by
ξf= ∞0d q2π2
k P f(k)
sin(qr)
k =
q d q
kr
d k
k P f(k)(30)
The?at-space limit is
ξf(0)≡ f2 = ∞0d k
ξf(0)<
r
In order for f2 to be well de?ned,the spectrum must have appropriate behaviour at q=∞and0.As q→∞one needs P→0.As q→0one needs P→0in the?at case,but only q2P f(k)→0in the curved case.
Note that in the curved case the limit q→0does not correspond to in?nite large scales, but rather to scales of order the curvature scale.This means that one cannot tolerate a divergent behaviour there(unless of course the curvature scale happens to be larger than any relevant scale,in which case we are back to?at space).
For future reference,we note that most other authors have used a di?erent de?nition of the spectrum.This is usually denoted by P f,and it is related to our P f by
P f(k)=q(q2+1)
2π2 ∞0d qq2P f(k)sin(qr)
sinh r
d
2
2
sinh r sinh(|q|r)
4One of us(DHL)is indebted to R.Gott and P.J.E.Peebles for pointing out this
fact.
?Π
k1=1
|q|
(40)
?Π
k2=1
|q|
(41)
At large r the super-curvature modes go like exp[?(1?|q|)r].Because the volume element is d V=sinh2r sinθdrdθd?the integral over all space of a product of any two of them diverges.As a result they are not orthogonal in the sense of Eq.(9),let alone orthonormal. In any?nite region of space(and of course we are only going to do physics in such a region) they are not even linearly independent of the sub-curvature eigenfunctions,since the latter are complete(for the set of L2functions de?ned over all space).None of this matters for the purpose of generating a Gaussian perturbation.
The super-curvature modes add an additional term to the expansion Eq.(6),
f SC(r,θ,φ)= 10d(iq) lm f klm Z klm(r,θ,φ)(42) Let us de?ne the correspondin
g spectrum by analogy wit
h Eq.(23),
f klm f?k′l′m′ =2π2
k P f(k)
sinh(|q|r)
k P f(k)(45) Uni?ed expressions including all modes
The use of q in the mode expansion Eq.(6)is natural for the sub-curvature modes,and we are using in this paper to facilitate comparison with existing literature.Uni?ed expressions including all modes on an equal footing would use k in the mode expansion,so de?ning new coe?cients?f klm.One would then have the following expressions,which include both sub-and super-curvature modes.
f(r,θ,φ,t)= ∞0d k lm?f klm(t)Z klm(r,θ,φ)(46)
?f?klm?f k′l′m′ =2π2
k P f(k)
sin(qr)
for r ?1.Thus the correlation length,in units of the curvature scale a ,is of order k ?2.This is in contrast with the ?at-space case,where the contribution from a mode with k ?1gives a correlation length of order 1/k .The di?erence can be understood in terms of the di?erent behaviour of the volume element,in the following way.In both cases,the r dependence is that of the l =0mode,and as long as r is small enough that the mode is approximately constant the divergence theorem gives
r
d r ??k 2r V (r )
T =w .e +∞ l =2+l m =?l
a lm Y lm (e ).(53)
The dipole term w .e is well measured,and is the Doppler shift caused by our velocity w relative to the rest frame of the cmb.Unless otherwise stated,?T will denote only the intrinsic,non-dipole contribution from now
on.
If the perturbations in the universe are Gaussian,the real and imaginary part of each
multipole will have an independent Gaussian probability distribution(subject to the con-dition a?lm=a l,?m).The expectation values of the squares of the real and imaginary parts are equal so one need only consider their sum,
C l≡ |a lm|2 .(54) Rotational invariance is equivalent to the independence of this expression on m.
Even if it can be identi?ed with an average over observer positions,the expectation value C l cannot be measured.Given a theoretical prediction for C l,the best guess for |a lm|2measured at our position is that it is equal to C l,but one can also calculate the variance of this guess,which is called the cosmic variance.Since the real and imaginary
part of each multipole has an independent Gaussian distribution the cosmic variance of m|a lm|2is only2/(2l+1)times its expected value,and by taking the average over several l’s one can reduce the cosmic variance even further.Nevertheless,for the low multipoles that are sensitive to curvature it represents a serious limitation on our ability to distinguish between di?erent hypotheses about the C l.Any hypothesis can be made consistent with observation by supposing that the region around us is su?ciently atypical.
The surface of last scattering of the cmb is practically at the particle horizon,whose coordinate distance isη0with sinh2η0/2=??10?1.An angleθsubtends at this surface a coordinate distance d given by[23]
θ=1
2
(a0H0?0d)(55)
Spatial curvature is negligible when d?1,corresponding to
θ?30(1??0)?1/2?0degrees(56) A structure with angular sizeθradians is dominated by multipoles with
l～1/θ(57) one expects that spatial curvature will be negligible for the multipoles
l?2
√
?0
(58)
This is the regime l?20if?0=0.1,and the regime l?6if?0=0.3.
This restriction need not apply to super-curvature modes with k2?1because the spatial gradient involved is then small in units of the curvature scale.The contribution of these modes is called the Grishchuk-Zeldovich e?ect,and we discuss it later.
The linear scale probed by the multipoles decreases as l increases,and for l～1000it becomes of order100Mpc.On these scales one can observe the distribution and motion of galaxies and clusters in the region around us.On the supposition that they all have a common origin,the cmb anisotropy and the motion and distribution of galaxies and clusters are collectively termed‘large scale structure’.
A promising model of large scale structure is that it originates as an adiabatic density perturbation,or equivalently[42,43,44,45]as a perturbation in the curvature of the hypersurfaces orthogonal to the comoving worldlines.This model has has been widely investigated for the case?0=1[46],and recently it has been advocated also for the case ?0<1[35,39,37].In this paper we consider the model only in relation to the cmb anisotropy since the galaxy and cluster data are insensitive to spatial curvature.We note though that the full data set may impose a signi?cant lower bound on?0[47].
5.1The curvature perturbation
The curvature perturbation is conveniently characterised by a quantity R
,which is de?ned in terms of the perturbation in the curvature scalar by 5
4(k 2+3)R klm /a 2=δR (3)klm
(59)In the limit ?→1,
4k 2R klm /a 2=δR (3)klm (60)
On cosmologically interesting scales,R klm is expected to be practically constant in the early universe.To be precise,it is practically constant on scales far outside the horizon in the regime where ?(t )is close to 1(assuming that the density perturbation is adiabatic)
[43,44,45,52].During matter domination the former condition can be dropped,so that R klm is constant on all scales until ?breaks away from 1.After that it has the time dependence R klm =F ?R
klm where ?R klm is the early time constant value and F =5sinh 2η?3ηsinh η+4cosh η?4
k 2+3
δρklm
5
R klm (63)For ?0=1this reduces to a 2H 2
ρ=
2k ?.007
5
The quantity R was called φm by Bardeen who ?rst considered it [42],R m by Kodama and Sasaki [48].It is equal to 3/2times the quantity δK/k 2of Lyth [43,44],which is in turn equal to the ζof Mukhanov,Feldman and Brandenberger [49].After matter domination it is equal to ?(3/5)Φ,where Φis the peculiar gravitational potential (and one of the ‘gauge invariant’variables introduced in [42]).On scales far outside the horizon,in the case ?=1,it is the ζof [50],and three times the ζof [51].
Assuming an initial adiabatic perturbation,these multipoles are dominated by the e?ect of the distortion of the spacetime metric between us and the surface of last scattering,which is called Sachs-Wolfe e?ect.If ?0=1the Sachs-Wolfe approximation accounts for about
90%of
C
l at l =10,and about 50%at l =30[7].
The Sachs-Wolfe e?ect is determined by the curvature perturbation.In the case ?0=1it is given by [23,46,54]?T (e )/T =?
13Φ(η0e )where Φis the peculiar gravitational potential.)Using Eq.(19)the multipoles are therefore given by
a lm =?
1225 ∞0
d k
5Πkl (η0)+6
k P R (k )I 2kl (73)
When k →1,I kl tends to a ?nite and nonzero limit for each l .This means that the C l are ?nite provided that q 2P R →0.Two things should be noted about the regime q →0in the curved space case.First,all multipoles receive contributions from this regime;in contrast with the ?at case the quadrupole does not dominate as is claimed in [41]).Second,the limit q →0corresponds to scales of order the curvature,not to in?nitely large scales as is claimed in [35,41].Because of this last fact,one cannot tolerate a divergence of the C l as q →0(unless the curvature scale is much bigger than any scale of interest in which case one is back to the ?at-space case).
5.3In?ation and horizon exit
It is widely supposed that the hot big bang is preceded by an era of in?ation,during which gravity is by de?nition repulsive.A very attractive hypothesis is that the curvature per-turbation originates as a vacuum ?uctuation during in?ation,so that the ensemble average appearing in the de?nition of the spectrum (Eq.(23))is just the vacuum expectation value.Made originally for the case ?0=1[55,51,43,56,57],this hypothesis was later extended to the case ?0<1by Lyth and Stewart [4].Before discussing it,let us see how in?ation works with special reference to the case ?0<1.
The Hubble distance H ?1is usually termed the horizon (to be distinguished from the particle horizon),and the comoving length scale a/k associated with a given mode is said
to be outside the horizon if aH/k>1,and inside the horizon if aH/k<1.The evolution of perturbations outside the horizon is very simple,because it is not a?ected by causal processes.Instead,the perturbation evolves independently in each comoving region[46].
Super-curvature scales,a/k>1,are always outside the horizon(from Eq.(1)),but sub-curvature scales can be either outside or inside it.In the usual cosmology where gravity is attractive,aH≡˙a decreases with time and at each epoch some scale is entering the horizon.The Hubble scale H?10is entering the horizon now,and smaller scales entered the horizon earlier.Also,from Eq.(1),?is driven away from1as time passes,so that|1??| must have been extraordinarily small at early times even if it is not small now.
In?ation may be de?ned as an early era of repulsive gravity,when aH≡˙a increases with time,and it is widely supposed that such an era preceded the hot big bang.At each epoch during in?ation some scale is leaving the horizon,and as time goes by?is driven towards1.The standard assumption is that in?ation occurs because the scalar?eld potential dominates the energy density,which falls slowly with time owing to the evolution of one of the scalar?elds,termed the in?aton?eld.Constant energy density corresponds to?∝H?2,and combining this dependence with Eq.(1)gives,for the case?<1,
a=?H sinh(?Ht)(74) and
H=?H coth(?Ht)(75) After?has been driven to1,H achieves the almost constant value?H,and
a∝exp(Ht)(76) It is related to the scalar?eld potential V(in turn practically equal to the energy density) by
?H2=8π
3.The homogeneity(horizon)problem Without in?ation,the observable universe is far
outside the horizon(Hubble distance)at early times.This means that causal processes cannot determine the initial conditions,which is usually held to be a problem,termed the‘horizon problem’.If?0is close to1,the observable universe is typically far inside the horizon at the beginning of in?ation,which solves the horizon problem and is usually said to‘explain’the homogeneity of the observable universe.In?ation with ?0<1cannot solve the horizon problem because the observable universe(or to be precise the comoving length presently equal to the Hubble distance)never occupies less than a fraction1??0of the Hubble distance.
However,no causal mechanism has ever been proposed for actually establishing homo-geneity at the beginning of in?ation,even after the horizon problem has been solved.
It seems to us therefore that the‘horizon problem’is a red herring,and that one should therefore look elsewhere for an explanation of the homogeneity of the universe.
For the case?0?1a fruitful avenue seems to be the following[46].As smaller and smaller scales are considered one expects to?nd homogeneity below some minimum scale,but this is not the Hubble distance even though that is the only scale available at the classical level.Rather it is the scale,available only at the quantum level,
ρ?1/4= 3m P l 1/2H?1(78) (we are settingˉh as well as c equal to1).Indeed,within a volume with this radius, even the vacuum?uctuation of a massless scalar?eld generates energy density and pressure of orderρ1/4,which would spoil in?ation.As in the case of?at spacetime this vacuum contribution to the energy density is to be discounted(ie.,one has to solve the cosmological constant problem by?at at our present level of understanding).But one cannot allow a signi?cant occupation number for the particle states de?ned on this vacuum.In other words,if?0=1the universe has to be absolutely homogeneous at the classical level,on scales smaller thanρ?1/4.This guarantees the homogeneity of the observable universe at the classical level,provided that in?ation starts at least [ln(m P l/H1)]Hubble times before the observable universe leaves the horizon,where H1is the value of H at this latter epoch.In order to respect the isotropy of the cmb one requires(H1/m P l)1/2～<10?3[59],and the bound is saturated in typical models of in?ation.Thus,homogeneity of the observable universe is typically guaranteed if it leaves the horizon more than7or so Hubble times after the beginning of exponential in?ation[46].
If?0<1it is unclear how to de?ne the vacuum as we discuss below,but with the mathematically simple conformal vacuum the vacuum?uctuation again generates an energy density and pressure of order d?4on the scale d.The criterion that this should not spoil the in?ationary behaviour Eq.(74)is that d?4be much less than the critical density,which requires as before d～>(H/m P l)1/2H?1.But now the observable universe is never far inside the Hubble distance H?1,so its homogeneity is not guaranteed by this type of argument.A di?erent avenue would be to invoke quantum cosmology,along the lines of[60]which however deals only with the case ?0>1.
The bubble nucleation model of in?ation
All of the above discussion assumes a classical evolution for the in?aton?eld,leading to a smooth evolution of?(t).It might happen,however,that the scalar?eld potential allows quantum tunneling in scalar?eld space at some point during in?ation.In that case a bubble of scalar?eld can form,whose interior is an??1universe[25,26,27,28].Provided that the scalar?eld potential is still?at enough,?will again be driven to1.
If?0turns out to be less than1the bubble nucleation model will be very attractive. Homogeneity is automatic.Also,?0is determined by the form of the scalar?eld potential and can easily be less than1[26].Assuming the usual‘chaotic’scenario for the beginning of in?ation[24,61,62],the in?aton?eld rolls slowly down a valley in scalar?eld space,and then the bubble nucleation model might correspond to sideways tunneling out of this valley [63].The only problem would be to?nd a potential of the required form that looks sensible in the context of modern particle theory;as has recently been pointed out[58,64],this constraint makes it di?cult even to?nd a potential that leads to ordinary,non-tunneling in?ation.
5.4Sub-curvature contributions and the vacuum?uctuation
During in?ation,the curvature perturbation is related to the perturbationδφof the in?aton ?eld by[57,4,46]
R=?(H/˙φ)δφ(79) where the dot denotes di?erentiation with respect to time t.This expression holds at all epochs,not just when R is constant.In it,δφis de?ned[57]on hypersurfaces which have zero perturbation in their curvature scalar(it is often called the‘gauge invariant’in?aton ?eld perturbation).
A very attractive hypothesis is thatδφoriginates as a vacuum?uctuation,so that the ensemble average appearing in the de?nition of the spectrum(Eq.(23))is just the vacuum expectation value[55,51,43,56,57,65].
The vacuum?uctuation during in?ation also generates a spectrum of gravitational waves,which is well understood for the case?0=1[66],and is under investigation for the case?0<1[67].We will not consider it here.
To calculate the vacuum?uctuation one uses quantum?eld theory in negatively curved space[19,68],and the?rst step in setting up this theory is to expandδφin terms of the sub-curvature mode functions.In this context there is no question of including additional modes,because many results of quantum?eld theory(such as the vanishing of?eld com-mutators outside the light cone)depend essentially on the fact that one is using a complete orthonormal set.As a result the spectrum predicted by the vacuum?uctuation will include only sub-curvature modes.
The same restriction holds for the?uctuation in any quantum state that is homogeneous (with respect to the group of coordinate transformations leaving the distance between each pair of points invariant).But one can give the in?aton?eld perturbation any desired stochastic properties by choosing a suitable quantum state(pure or mixed),and in particular one can generate an arbitrary homogeneous Gaussian perturbation.The absence of super-curvature modes in the vacuum?uctuation prediction is not a feature of quantum?eld theory per se.
The coe?cients in the mode expansion of the quantum?eldφklm satisfy the classical ?eld equation(in the Heisenberg representation),which?xes them up to a one-parameter ambiguity once a convention is made for their normalisation.Breaking this ambiguity is equivalent to de?ning the vacuum.In the case?0=1each mode starts out well inside the horizon,where the spacetime curvature is negligible.In that case the vacuum is de?ned to be the usual?at spacetime vacuum,and assuming the usual slow roll conditions one?nds [43]
P R(k)1/2=82πV′(80) In this expression V(φ)is the in?aton potential,and the right hand side is to be evaluated at the epoch of horizon exit k=aH.It gives an almost scale-independent result for typical models of in?ation.
In the case?0<1,without bubble nucleation,it is not clear how to de?ne the vacuum because a given scale is never far inside the horizon.The mathematically simplest choice is the‘conformal vacuum’,and using it one?nds(after suitably generalising the slow roll conditions)that P R is still given by the above expression[4,52].(For the special case of a linear potential this result has been reproduced recently,using a di?erent calculational technique[39].6)
In the bubble nucleation model the quantum state of the in?aton?eld perturbation inside the bubble can be calculated[27,26,67],and it is found not to be in the conformal vacuum.As a result[26,28],P R(k)is multiplied by a factor coth(πq)compared with Eq.(80).
Comparison of the vacuum?uctuation with observation
If?0=1,Eq.(70)with a scale independent P R gives
2π
l(l+1)C l=
20μK?8.0×10?10(82) which corresponds to
P R?3×10?9(83) For?0<1Eq.(73)has to be evaluated numerically.With the?at spectrum coming from the conformal vacuum assumption,l(l+1)C l has in general a negative slope for 0.1
As already noted,the spectrum is not?at in the bubble nucleation model,but rather is proportional to coth(πq).It turns out however[28]that if?0is substantially below1 the integral in Eq.(72)dominates,with I2kl peaking at q2～>1even for the quadrupole(and at higher values for higher multipoles).As a result the bubble nucleation prediction is not signi?cantly di?erent from the?at spectrum prediction,when cosmic variance is taken into account.
Power law parameterization of the density perturbation spectrum
The spectrum P R of the curvature perturbation is directly related to the Sachs-Wolfe e?ect, and from this viewpoint the assumption that it is e?ectively?at seems natural.This assumption was not,however,the assumption made in the literature before the(very recent) advent of the vacuum?uctuation prediction.Rather,it was assumed that the spectrum Pδ(de?ned by Eq.(33))of the density perturbation is proportional to q.This choice is equivalent to the?atness of P R for?0=1,but otherwise it is equivalent to
P R∝q2(1+q2)
6The authors of[39]do not establish the identity of their result with the earlier one,but it follows by evaluating Eq.(2)of[37](multiplied by16π/m2P l to bring the conventions of[39]into line with the usual ones)during matter domination before?breaks away from unity.To do this one has to replace1??0by 1???1in the quantity W1/c1,leading to[23]W1/c1→(2/5)(aH)?2.Remembering that the energy density scales like a?3during matter domination and like a?4during radiation domination,one then indeed reproduces the spectrum of the energy density given by Eqs.(63)and(80).
The right hand side tends to1in the limit q2→∞of negligible curvature,but is much less than1for q2～<4.With this parameterization,l(l+1)C l acquires a positive slope[36]for 0.1
The main cause of this di?erence is the factor(4+q2)2coming from the relation between the density and curvature perturbations,and a similar result would probably be obtained if Pδwere used instead of Pδ,or k instead of q.In other words,the predicted C l can be regarded as coming simply from a linearly rising density perturbation spectrum,as opposed to a?at curvature perturbation spectrum.These two parameterizations are equivalent if ?0is equal to1,but if?0is signi?cantly smaller than1the?rst one gives less power on small scales,leading to a signi?cantly di?erent prediction for the C l’s.The central values of the present data points lie between the two predictions,and the present error bars are big enough that they are indistinguishable.But in the future the data should be able to distinguish between the two parameterizations,ruling out one or both of them for small values of?0.
5.5Super-curvature scales and the Grishchuk-Zeldovich e?ect
Like any statement in physics,the statement that the in?aton?eld is in the vacuum will be at best approximate,and its validity will presumably depend on the scale under consideration. When considering departures from it there is no reason to exclude super-curvature scales.
The contribution of super-curvature scales to the mean square multipole C l is just the extension of Eq.(73)to super-curvature scales,
C SC l=2π2 10d k
small-scale cuto?to be the Hubble distance at the end of in?ation(which is equivalent to the usual procedure of dropping the contribution of the vacuum?uctuation to the energy density in?at spacetime),this scale as a multiple of the Hubble distance is
a0H0
2516
a0H0 4 R2 (91)
The quadrupole measured by COBE is not signi?cantly in excess of the typical values l(l+1)C l?8×10?10of the other multipoles(in fact it is somewhat smaller),so we conclude that C VL
2
is absent at this level.This means that
d VL>70 R2 1/4H?10(92) As Grishchuk and Zeldovich pointed out,this bound on R2 becomes weaker as th
e scale increases.It can be o
f order1provided that(cf.[71])
a0H0
The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because
The way的用法及其含义（二） 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语： 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势，忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中
定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...
“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法
不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.
表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法
The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.
way 的用法 【语境展示】 1. Now I’ll show you how to do the experiment in a different way. 下面我来演示如何用一种不同的方法做这个实验。 2. The teacher had a strange way to make his classes lively and interesting. 这位老师有种奇怪的办法让他的课生动有趣。 3. Can you tell me the best way of working out this problem? 你能告诉我算出这道题的最好方法吗？ 4. I don’t know the way (that / in which) he helped her out. 我不知道他用什么方法帮助她摆脱困境的。 5. The way (that / which) he talked about to solve the problem was difficult to understand. 他所谈到的解决这个问题的方法难以理解。 6. I don’t like the way that / which is being widely used for saving water. 我不喜欢这种正在被广泛使用的节水方法。 7. They did not do it the way we do now. 他们以前的做法和我们现在不一样。 【归纳总结】 ●way作“方法，方式”讲时，如表示“以……方式”，前面常加介词in。如例1； ●way作“方法，方式”讲时，其后可接不定式to do sth.，也可接of doing sth. 作定语，表示做某事的方法。如例2，例3；
t h e-w a y-的用法
The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.
way的用法总结大全 way的用法你知道多少，今天给大家带来way的用法，希望能够帮助到大家，下面就和大家分享，来欣赏一下吧。 way的用法总结大全 way的意思 n. 道路,方法,方向,某方面 adv. 远远地，大大地 way用法 way可以用作名词 way的基本意思是“路,道,街,径”,一般用来指具体的“路,道路”,也可指通向某地的“方向”“路线”或做某事所采用的手段,即“方式,方法”。way还可指“习俗,作风”“距离”“附近,周围”“某方面”等。 way作“方法,方式,手段”解时,前面常加介词in。如果way前有this, that等限定词,介词可省略,但如果放在句首,介词则不可省略。
way作“方式,方法”解时,其后可接of v -ing或to- v 作定语,也可接定语从句,引导从句的关系代词或关系副词常可省略。 way用作名词的用法例句 I am on my way to the grocery store.我正在去杂货店的路上。 We lost the way in the dark.我们在黑夜中迷路了。 He asked me the way to London.他问我去伦敦的路。 way可以用作副词 way用作副词时意思是“远远地,大大地”,通常指在程度或距离上有一定的差距。 way back表示“很久以前”。 way用作副词的用法例句 It seems like Im always way too busy with work.我工作总是太忙了。 His ideas were way ahead of his time.他的思想远远超越了他那个时代。 She finished the race way ahead of the other runners.她第一个跑到终点,远远领先于其他选手。 way用法例句
t h e_w a y的用法大全
The way 在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了 另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的
The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.
the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms. 从那天起,其他同学是夹着书本来上课,而他们却带着"失败"的思想负担来上课.
The way的用法及其含义（三） 三、the way的语义 1. the way=as（像） Please do it the way I’ve told you.请按照我告诉你的那样做。 I'm talking to you just the way I'd talk to a boy of my own.我和你说话就像和自己孩子说话一样。 Plant need water the way they need sun light. 植物需要水就像它们需要阳光一样。 2. the way=how（怎样，多么） No one can imagine the way he misses her.没人能够想象出他是多么想念她！ I want to find out the way a volcano has formed.我想弄清楚火山是怎样形成的。 He was filled with anger at the way he had been treated.他因遭受如此待遇而怒火满腔。That’s the way she speaks.她就是那样讲话的。 3. the way=according as （根据） The way you answer the questions, you must be an excellent student.从你回答问题来看，你一定是名优秀的学生。 The way most people look at you, you'd think a trash man was a monster.从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物。 The way I look at it, it’s not what you do that matters so much.依我看，重要的并不是你做什么。 I might have been his son the way he talked.根据他说话的样子，好像我是他的儿子一样。One would think these men owned the earth the way they behave.他们这样行动，人家竟会以为他们是地球的主人。
一．Way：“方式”、“方法” 1.表示用某种方法或按某种方式 Do it (in) your own way. Please do not talk (in) that way. 2.表示做某事的方式或方法 It’s the best way of studying [to study] English.。 There are different ways to do [of doing] it. 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句 正：I don’t like the way he spoke. I don’t like the way that he spoke. I don’t like the way in which he spoke.误：I don’t like the way how he spoke. 4. the way 的从句 That’s the way (=how) he spoke. I know where you are from by the way you pronounce my name. That was the way minority nationalities were treated in old China. Nobody else loves you the way(=as) I do. He did not do it the way his friend did. 二．固定搭配 1. In a/one way：In a way he was right. 2. In the way /get in one’s way I'm afraid your car is in the way， If you are not going to help，at least don't get in the way. You'll have to move－you're in my way. 3. in no way Theory can in no way be separated from practice. 4. On the way (to……) Let’s wait a few moments. He is on the way Spring is on the way. Radio forecasts said a sixth-grade wind was on the way. She has two children with another one on the way. 5. By the way By the way，do you know where Mary lives? 6. By way of Learn English by way of watching US TV series. 8. under way 1. Elbow one’s way He elbowed his way to the front of the queue. 2. shoulder one’s way 3. feel one‘s way 摸索着向前走；We couldn’t see anything in the cave, so we had to feel our way out 4. fight/force one’s way 突破。。。而前进The surrounded soldiers fought their way out. 5.. push/thrust one‘s way（在人群中）挤出一条路He pushed his way through the crowd. 6. wind one’s way 蜿蜒前进 7. lead the way 带路，领路；示范 8. lose one‘s way 迷失方向 9. clear the way 排除障碍，开路迷路 10. make one’s way 前进，行进The team slowly made their way through the jungle.
在the way+从句中, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 如果怕弄混淆，下面的可以不看了 另外，在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮. the way=according to the way/judging from the way 4)The way you answer the qquestions, you must be an excellent student. 从你回答就知道,你是一个优秀的学生. 5)The way most people look at you, you'd think a trashman was a monster. 从大多数人看你的目光中,你就知道垃圾工在他们眼里是怪物. the way=how/how much 6)I know where you are from by the way you pronounce my name. 从你叫我名字的音调中,我知道你哪里人. 7)No one can imaine the way he misses her. 人们很想想象他是多么想念她. the way=because 8) No wonder that girls looks down upon me, the way you encourage her. 难怪那姑娘看不起我, 原来是你怂恿的 the way =while/when(表示对比) 9)From that day on, they walked into the classroom carrying defeat on their shoulders the way other students carried textbooks under their arms.
“the way+从句”结构的意义及用法 首先让我们来看下面这个句子： Read the following passage and talk about it with your classmates. Try to tell what you think of Tom and of the way the children treated him. 在这个句子中，the way是先行词，后面是省略了关系副词that 或in which的定语从句。 下面我们将叙述“the way+从句”结构的用法。 1．the way之后，引导定语从句的关系词是that而不是how,因此，<<现代英语惯用法词典>>中所给出的下面两个句子是错误的：This is the way how it happened. This is the way how he always treats me. 2. 在正式语体中，that可被in which所代替；在非正式语体中，that则往往省略。由此我们得到the way后接定语从句时的三种模式：1) the way +that-从句2) the way +in which-从句3) the way +从句 例如：The way(in which ,that) these comrades look at problems is wrong.这些同志看问题的方法不对。
The way(that ,in which)you’re doing it is completely crazy.你这么个干法，简直发疯。 We admired him for the way in which he faces difficulties. Wallace and Darwin greed on the way in which different forms of life had begun.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 This is the way (that) he did it. I liked the way (that) she organized the meeting. 3.the way(that)有时可以与how(作“如何”解)通用。例如： That’s the way (that) she spoke. = That’s how she spoke. I should like to know the way/how you learned to master the fundamental technique within so short a time. 4．the way的其它用法：以上我们讲的都是用作先行词的the way，下面我们将叙述它的一些用法。
定冠词the的12种用法 定冠词the 的12 种用法，全知道?快来一起学习吧。下面就和大家分享，来欣赏一下吧。 定冠词the 的12 种用法，全知道? 定冠词the用在各种名词前面，目的是对这个名词做个记号，表示它的特指属性。所以在词汇表中，定冠词the 的词义是“这个，那个，这些，那些”，可见，the 即可以放在可数名词前，也可以修饰不可数名词，the 后面的名词可以是单数，也可以是复数。 定冠词的基本用法： (1) 表示对某人、某物进行特指，所谓的特指就是“不是别的，就是那个!”如： The girl with a red cap is Susan. 戴了个红帽子的女孩是苏珊。 (2) 一旦用到the，表示谈话的俩人都知道说的谁、说的啥。如：
The dog is sick. 狗狗病了。(双方都知道是哪一只狗) (3) 前面提到过的，后文又提到。如： There is a cat in the tree.Thecat is black. 树上有一只猫，猫是黑色的。 (4) 表示世界上唯一的事物。如： The Great Wall is a wonder.万里长城是个奇迹。(5) 方位名词前。如： thenorth of the Yangtze River 长江以北地区 (6) 在序数词和形容词最高级的前面。如： Who is the first?谁第一个? Sam is the tallest.山姆最高。 但是不能认为，最高级前必须加the，如： My best friend. 我最好的朋友。 (7) 在乐器前。如： play the flute 吹笛子
Way用法 A:I think you should phone Jenny and say sorry to her. B:_______. It was her fault. A. No way B. Not possible C. No chance D. Not at all 说明：正确答案是A. No way，意思是“别想！没门！决不！” 我认为你应该打电话给珍妮并向她道歉。 没门！这是她的错。 再看两个关于no way的例句： （1）Give up our tea break? NO way! 让我们放弃喝茶的休息时间？没门儿！ （2）No way will I go on working for that boss. 我决不再给那个老板干了。 way一词含义丰富，由它构成的短语用法也很灵活。为了便于同学们掌握和用好它，现结合实例将其用法归纳如下： 一、way的含义 1. 路线
He asked me the way to London. 他问我去伦敦的路。 We had to pick our way along the muddy track. 我们不得不在泥泞的小道上择路而行。 2. （沿某）方向 Look this way, please. 请往这边看。 Kindly step this way, ladies and gentlemen. 女士们、先生们，请这边走。 Look both ways before crossing the road. 过马路前向两边看一看。 Make sure that the sign is right way up. 一定要把符号的上下弄对。 3. 道、路、街，常用以构成复合词 a highway（公路），a waterway（水路），a railway（铁路），wayside（路边）
way/time的特殊用法 1、当先行词是way意思为”方式.方法”的时候,引导定语从句的关系词有下列3种形式： Way在从句中做宾语 The way that / which he explained to us is quite simple. Way在从句中做状语 The way t hat /in which he explained the sentence to us is quite simple. 2、当先行词是time时,若time表示次数时,应用关系代词that引导定语从句,that可以省略; 若time表示”一段时间”讲时,应用关系副词when或介词at/during + which引导定语从句 1.Is this factory _______ we visited last year? 2.Is this the factory-------we visited last year? A. where B in which C the one D which 3. This is the last time _________ I shall give you a lesson. A. when B that C which D in which 4.I don’t like the way ________ you laugh at her. A . that B on which C which D as 5.He didn’t understand the wa y ________ I worked out the problem. A which B in which C where D what 6.I could hardly remember how many times----I’ve failed. A that B which C in which D when 7.This is the second time--------the president has visited the country. A which B where C that D in which 8.This was at a time------there were no televisions, no computers or radios. A what B when C which D that