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Preprint typeset in JHEP style -HYPER VERSION Hiranya V.Peiris ?Kavli Institute for Cosmological Physics and Enrico Fermi Institute,University of Chicago,Chicago IL 60637,USA.hiranya@https://www.doczj.com/doc/fd16829942.html, Richard Easther Department of Physics,Yale University,New Haven CT 06520,USA richard.easther@https://www.doczj.com/doc/fd16829942.html, Abstract:We study the constraints on the in?ationary parameter space derived from the 3year WMAP dataset using “slow roll reconstruction”,using the SDSS galaxy power spectrum to gain further leverage where appropriate.This approach inserts the in?ationary slow roll parameters directly into a Monte Carlo Markov chain estimate of the cosmological parameters,and uses the in?ationary ?ow hierarchy to compute the parameters’scale-dependence.We work with the ?rst three parameters (?,ηand ξ)and pay close attention to the possibility that the 3year WMAP dataset contains evidence for a “running”spectral index,which is dominated by the ξterm.Mirroring the WMAP team’s analysis we ?nd that the permitted distribution of ξis broad,and centered away from zero.However,when we require that in?ationary parameters yield at least 30additional e-folds of in?ation after the largest observable scales leave the horizon,the bounds on ξtighten dramatically.We
make use of the absence of an explicit pivot scale in the slow roll reconstruction formalism to determine the dependence of the computed parameter distributions on the pivot.We show that the choice of pivot has a signi?cant e?ect on the inferred constraints on the in?ationary variables,and the spectral index and running derived from them.Finally,we argue that the next round of cosmological data can be expected to place very stringent constraints on the region of parameter space open to single ?eld models of slow roll in?ation.
1.Introduction
The primordial perturbation spectrum poses two distinct problems for cosmology.Obser-vationally,we wish to extract the spectrum’s amplitude and scale dependence from data, while theoretically we seek to understand the origin of the perturbations.After the release of the3-year WMAP dataset[1,2,3]there is considerable optimism that we will soon mea-sure the departure from scale invariance in the spectrum at signi?cance levels greater than 3σ.1On the theoretical front,in?ation[4,5,6]remains the leading theoretical paradigm for understanding the very early universe and the origin of the perturbations.In general terms,the latest round of cosmological data has con?rmed the broad predictions of in?a-tion–the universe is very close to being?at,it has a roughly scale-invariant perturbation spectrum and T E ,the cross correlation between the E mode of the polarization and the temperature,has the form predicted by in?ation.
Beyond this rosy picture,however,lies a number of quali?cations.Firstly,the observed form of T E is not a unique prediction of in?ation,but is also a generic feature of models where the perturbations are laid down during a collapsing phase prior to a bounce,such as the ekpyrotic scenario[7].2Secondly,in?ation can be implemented in a vast number of di?erent ways,and each mechanism produces a potentially distinctive perturbation spec-trum.Consequently,the in?ationary paradigm does not make an unambiguous prediction for the detailed form of the perturbation spectrum.Some authors have posited that in-?ation has a“default”perturbation spectrum[8].The underlying arguments leading to this position rely on simplicity and naturalness,rather than a watertight“no go”theorem.
Consequently,in?ation itself will not be ruled out if the perturbation spectrum turns out
to di?er from this simple expectation,but it is certainly true that many well-known models
would be excluded.
Looking at data,it is always tempting to focus on possible anomalies–parameters
that di?er noticeably from their“expected”values,but for which the discrepancy falls well
short of a3σdetection.Since the release of the original WMAPI dataset[9,10,11],three
aspects of the power spectrum have received considerable attention:the“low”values of
C?at small?,the glitches around the?rst Doppler peak and at?～40,and the apparent
evidence for a running,or scale-dependent,spectral index that arose when the WMAPI
data was combined with other probes of the power spectrum at shorter scales.The status
of all of these anomalies has evolved since the WMAPI release.The quadrupole is still
～2σlower than the LCDM best-?t value,but the octupole has moved closer to the LCDM value,thanks to an optimal extraction of the spectrum.The glitch near the?rst Doppler
peak is not present in the WMAPII dataset with its smaller errorbars,while the?～40
discrepancy is still seen[12,13].Finally,the WMAPII dataset contains a preference for a
running spectral index when viewed alone,in combination with other CMB data,and/or
large scale structure information such as2dFGRS[14]and the Sloan Digital Sky Survey
[15].However,a reanalysis by Seljak et al.,which includes SN1a data and information
from the Lyman-αforest,puts a tight constraint on any running[16]—this result seems
to hinge on the inclusion of the Lyman-αdata.Further analysis of a possible running index
is provided by[17].
In this paper,we focus on the running of the spectral index,and the constraints it
places on the in?ationary parameter space.Conventionally,one speci?es the spectrum at
a pivot point in terms of the spectral index n s and the running dn s
d ln k in terms of th
e slow roll parameters,which are functions o
f the potential and its
derivatives.In[18]we showed that if the running is as large as the central value extracted from the WMAPII dataset,no simple model of in?ation can simultaneously provide both this running and a su?cient amount of in?ation.3Currently,there are simple models of in?ation within the errorbars,but if the constraints from the data continue to tighten around the median value of the running,they will be ruled out.We arrived at this result by considering the Hubble Slow Roll hierarchy,otherwise known as the in?ationary?ow equations[19,20,21,22].These codify the scale dependence of the slow roll parameters: given the values of the slow roll parameters at a single moment during the in?ationary era, we can compute their values at all other times.A further virtue of this approach is that when the in?nite slow roll hierarchy is truncated at?nite order,the truncation is preserved by the evolution https://www.doczj.com/doc/fd16829942.html,ing the slow roll hierarchy means that we e?ectively dispense with the pivot point,other than when comparing our results to treatments using the standard n s and dn s
3Which we take to be a minimum of30e-foldings,and by“simple”we mean a single in?aton,minimally coupled to gravity,whose potential can be well speci?ed in terms of its?rst three derivatives.
ing in?ationary potential,leading us to dub this approach“slow roll reconstruction.”4Of course,this presupposes that the perturbations were produced by in?ation,so slow roll reconstruction does not replace?ts to n s and dn s
d ln k is arguably th
e simplest measure o
f the
scale dependence,it is by no means unique.In particular,by adopting it we are implicitly assuming that while n s varies with k,dn s
d ln k ,whereas in
order to implement slow roll reconstruction we have to numerically solve the di?erential equation that maps the?eld evolution into the comoving wavenumber k.8However,slow roll reconstruction can be more easily extended to include higher order terms in slow roll, and copes naturally with heterogeneous datasets that span a wide range of wavelengths–this is not impossible in the formalism of[26,27,28],but would require more algebraic
labor.Secondly,slow roll reconstruction can naturally include constraints based on the total duration of in?ation.Finally,and perhaps most importantly,because[26,27,28] use n s and dn s
d ln k yields a tighter constraint than that of Seljak
et al.[16]when we add the N>30prior.When we drop the constraint on N the central value ofξis unexpectedly large,andξ=0is apparently excluded at the2σlevel by the WMAPII+SDSS dataset.When we compute N for these chains we?nd a distribution peaked at10-15e-folds,consistent with[18],but in con?ict with successful cosmological in?ation.We then turn to the pivot dependence of the bounds on the in?ationary and spectral parameters,and show that in the absence of a prior on N,choosing a di?erent pivot scale(comparing constraints at0.002Mpc?1to those at0.02Mpc?1,in this case)
somewhat tightens the bounds on ξand dn s
4π H ′
(φ)
4π ?(H ′)??1
dφ(?+1);?≥1.(2.2)
The usual slow roll parameters are η=1λH and ξ=2λH .If all terms with ?>M are zero at some ?ducial point,these di?erential equations ensure they vanish at all other times.Liddle showed that the hierarchy can be solved exactly when truncated at order M [21],so
H (φ)
m Pl +···+B M +1 φ
4π?0B ?+1=(4π)?
8πH 2(φ) 1?1
while N is given by dN
m 2Pl H
d ln k =?m Pl π√1??.(2.7)Finally,w
e express the scalar and tensor primordial power spectra in terms o
f these param-eters.A ?rst order expansion around an exact solution for the case of power law in?ation [37]gives P R =[1?(2C +1)?+Cη]2m Pl 2 k =aH ,(2.8)P h =[1?(C +1)?]216m Pl 2 k =aH
,(2.9)where η=1λH ,C =?2+ln 2+γ≈?0.729637and γis the Euler-Mascheroni constant.One minor inconsistency in our approach is that (2.8)and (2.9)are calculated using the slow roll approximation,and truncated at second order.In practice,we do not believe this is a signi?cant drawback,as the datasets we are working with provide meaningful constraints on (at most)the lowest three slow-roll parameters.In the context of future high-precision data,this problem could be surmounted by either using a higher order expansion for the spectra,or even integrating the mode equations directly.
We can write the usual observables in terms of the slow roll parameters,
n s =1+2η?4??2(1+C )?2?12
(3?C )ξ(2.10)r =16?[1+2C (??η)](2.11)dn s 1?? 2ξ+8?2?10?η+7C ?92
ξη (2.12)n t =?2??(3+C )?2+(1+C )?η(2.13)
where C =4(ln 2+γ)?5,and we have introduced the customary notation α=dn s /d ln k ;r is the tensor/scalar ratio 10and n t is the tensor spectral index.We retain all terms in dn s k 0 n s (k 0)?1+1k 0
n t (k 0)+110Beware of the distinction between C and C –these quantities are sometimes confused in the literature.
Hence,while it is common to assume that dn t/d ln k=0and n t=?r(k0)/8is constant, the scale-dependence of the tensor/scalar ratio r(k)=P h(k)/P R(k)is given by
r(k)=r(k0) k2(dn s/d ln k)ln(k/k0)+r(k0)/8(2.16) and it is apparent that the consistency condition does not hold away from the pivot.This is an unsatisfactory state of a?airs if one wishes to impose a single-?eld slow-roll prior,and slow roll reconstruction does not su?er from this shortcoming.
An alternative to slow roll reconstruction is given by the work of Leach et al.[26,27, 28,29],who specify the spectrum in terms of the slow roll parameters,but then compute its scale dependence via equations2.10to2.12.Conversely,we dispense with an explicit pivot point and compute the spectra directly from equations2.8and2.9,while using the ?ow equations to obtain the appropriate scale dependent values of the slow roll parameters. When only?andηare included in the analysis,these two approaches(i.e.ours vs.Leach et al.’s)overlap to with a few percent over the full observable range.However,whenξis included the discrepancy can be much larger.For instance with a pivot at k=0.002Mpc?1, the spectra computed using the formalism of[26,27,28]di?er from that obtained using the?ow equations by as much as10%at very short scales,as illustrated in Figure1.This discrepancy could be corrected by extending the approach of[26,27,28]to higher order in slow roll.
We perform standard MCMC?ts to cosmological data,using modi?ed versions of CAMB11[39]to provide the CMB spectra and CosmoMC12[38]to handle the Markov
Dataset Combination
1.
–
WMAPII
3.N >30Parameter
WMAPII (no N restriction)
(N >30)<0.013(95%CL)
<0.015(95%CL)0.029±0.021
0.0±0.0140.029±0.013
0.0±0.0023.17±0.063.17±0.06
π?0 H 0
13https://www.doczj.com/doc/fd16829942.html,/product/map/dr2/
WMAPII+SDSS WMAPII+SDSS
(N>30)
n s0.99±0.02
dn s/d ln k0.004±0.006
r<0.56(95%CL)
n t>?0.067(95%CL)
Table3:Constraints on derived primordial parameters de?ned at k=0.002Mpc?1.Constraints are at the68%con?dence level unless otherwise noted.
the WMAPII dataset on its own,without a prior on N.When SDSS data was included, convergence was comparatively prompt,re?ecting the greater leverage provided by the combination of CMB and LSS data.
The prior(or lack thereof)on N is implemented as follows.Each draw of{?,η,ξ}in the chains maps uniquely onto a potential.For this potential,we can evolve it forward in time from when the?ducial scale k0leaves the horizon and compute when?=1,corresponding to the end of in?ation.In the case with no prior on N,we only check that in?ation lasts for at least the few e-folds encompassing the CMB and LSS data that we actually use–if in?ation ends within this observable window,that model is discarded.For the case where N>30,we require that the potential so speci?ed provides at least an additional30e-folds of in?ation after the?ducial scale leaves the horizon.Otherwise it is rejected from the chain.
The parameter bounds derived from each set of chains are given in Tables2and3,while Figures2through9show the results derived from the chains we ran.In particular,Figure2 shows the limits on the slow roll parameters derived from Set1,where we are?tting to the combined power spectrum derived from WMAPII and SDSS,with no constraint on the number of e-folds.In Figure3we show the same results,after they have been mapped into the“standard”slow roll variables.These plots are analogous to those shown in[22], except for the presence of theξhttps://www.doczj.com/doc/fd16829942.html,paring these plots with those from our previous paper,we see that includingξgreatly increases the allowed range of values of the running,compared to the{?,η}chains.Indeed,a naive interpretation of the data suggests that there is the evidence forξ>0beyond the2σlevel–but there are a number of issues surrounding this result,and we stress that we make no claim to have found a non-trivial value ofξ.In Figure4we compare these results to chains run by Spergel et al.[3]for the same dataset.The two sets of chains are not exactly equivalent as the Spergel results have an additional constraint on the bias for the SDSS,which we do not include in the chains described here.We again see that a naive inspection of the allowed portion of the parameter space shows that working with an in?ationary prior imposed via the use of the slow roll parameters in the chains leads to a stronger preference for a negative running than was found by Spergel et al.A further di?erence is that upper bound on the tensor contribution found using a slow roll prior is considerably lower than that obtained from the {n s,r,dn s
between?andηin terms of their contribution to n s is broken by the way these quantities contribute to the running index and r.
A further,more general,message to be drawn from the two sets of chains shown in Figure4is that the strength of the evidence for a running spectrum is dependent on the way we parameterize the running.One might assume that imposing a strong in?ationary prior by directly including the slow roll parameters in the chains would decrease the available region of parameter space,relative to that found from using the conventional{n s,r,dn s
This would require re-running our model with the SDSS bias prior;we leave this issue for possible future study.
As noted above,using the slow roll prior puts tighter constraints on the amplitude of the tensor spectrum than the{n s,r,dn s
number of e-folds into the parameter estimation algorithm.In practice,we only work with lower limits on N.While it is not possible for an arbitrary amount of in?ation to occur after cosmological scales leave the horizon,there is a lower bound on the number of e-folds if the universe is to successfully reheat.In many models,the end of in?ation occurs when an instability develops in a direction in?eld-space that is orthogonal to the in?ationary trajectory.In that case there is no“advance warning”that in?ation is about to end,and the slow roll parameters may remain small right up until the transition to non-in?ationary expansion.Here a naive computation of N using the values the slow roll parameters as the CMB scales leave the horizon yields a misleadingly large number.Conversely,a low value of N(which is practice means anything less than30),implies that in?ation will end too quickly unless some further e?ect comes into play.This could be a contribution from higher order terms in slow roll,or a second period of in?ation–both of which are excluded by our prior.Consequently,we can usefully include a lower bound on N in our?ts,but we do not impose an upper bound.
In Sets2and3we impose the prior N>30on both WMAPII,and the combination
5.These chains put much tighter limits
WMAPII+SDSS,and plot the results in Figure
onξthan Set1,which has no constraint on N.Further,the addition of the SDSS data to WMAPII signi?cantly tightens the parameter https://www.doczj.com/doc/fd16829942.html,paring our results here with Figure5of[22],where we perform a?t to the?rst two slow roll parameters,we see that the permitted region of the{?,η}plane does not change substantially when theξparameter is added along with the N>30constraint.Without the N>30constraint addingξsigni?cantly expands the allowed range of{?,η}.
In Figure6the two sets of N>30chains are converted into constraints on n s,r and
dn s
d ln k ar
e an order o
f magnitude tighter than those we obtained in
the absence of the N>30prior.The tightest published constraints on dn s
d ln k =(?1.5±1.2)×10?2,where
the error range spans the68%con?dence interval.We present our constraints on the slow roll parameters in Table2,while Table3contains the corresponding limits on the spectral parameters.With the N>30constraint,our bound on dn s
due to the fact that they report their constraints at a pivot point (0.05Mpc ?1)which is at signi?cantly smaller scales than ours.We do not include the dark energy equation of state (w )in our analysis,whereas this is a free parameter in
[16].The
di?erences between our results and those of [16]re?ects di?erent underlying assumptions,rather than any intrinsic “con?ict”between the two analyses.The authors of [16]set out to work with as large a range of data sources as possible,whereas our approach is to work with only one or two probes of the fundamental power spectrum,and to impose a strong theoretical prior –namely slow roll in?ation –on the mechanism responsible for the generation of perturbations.
4.Interpretation and Analysis
While slow roll reconstruction hinges on a very simple idea –inserting the slow roll param-eters directly into the cosmological parameter estimation process –interpreting the results calls for a good deal of sophistication.Without a constraint on the number of e-folds,we see an apparent preference for a large,positive value of ξ,which translates into substantial and negative values of dn s
d ln k ～?0.05(near
Figure 6:Primordial parameter constraints for WMAPII+SDSS,(black)and WMAPII (red),imposing a prior N >30,in conventional variables.The top plot in each column shows the probability distribution function for each of the parameters,while the other plots show their joint 68%and 95%con?dence levels.The power law variables are derived parameters which are not used in the MCMC chains themselves.
the center of the distribution derived from the combination WMAPII+SDSS),the running term dominates this expression as we move away from k0.As is well known,conventional cosmological observations probe the primordial spectrum over a range of around three or-ders of magnitude in scale.This may stretch to?ve orders in the future as we better probe the Lyman-αforest or obtain a power spectrum from observations of high redshift 21cm emission.With|dn
s
d ln k≈?0.05,th
e power spectrum at k=e10k0is an order o
f magnitude smaller than at k0.However,if we have a spectrum that is strongly de?cient in short scale power,the other variables in our parameter set will move to o?set this de?cit.As we can see from Figure7,this is exactly what happens here,as?m has increased relative to its concordance value,where?Λand H0have moved downwards.The former is simply a result of our assumption of a?at universe,so that?Total=1,and the increased?m compensates for the reduced power in the perturbation spectrum at short scales.
Imposing additional constraints on?m or?Λwould put pressure on the models that produce large?m and small?Λ.In Figure8we show the distribution of{n s,dn s/d ln k} for Set1(WMAPII+SDSS),with the points color-coded by?m.This plot illustrates the correlation between an extreme running and an anomalously large?m.Consequently,a strong upper bound on?m from observations independent of the CMB and LSS data we employed in our MCMC analysis would exclude the models with the most extreme running. Consequently,the long tail of the allowed region in the{n s,dn s/d ln k}plane for Set1can be understood as a marginalization artefact.
In models with largeξ,ηand?are typically strongly scale-dependent,thanks to feedback from theξterm in the?ow equations.Once?is equal to unity,in?ation will come to an end.As we have explained,imposing N>30leads to a dramatic change in the permitted region of parameter space,consistent with our qualitative analysis in [18].In Figure9,we show the remaining number of e-folds as a functionξ,obtained by post-processing our chains.Note that only models in which in?ation explicitly ends are retained from the chains in order to make these plots.Models which are drawn to a“late-time attractor”and in?ate forever(thus needing an orthogonal mechanism to end in?ation)cannot be assigned an unambiguous number of remaining e-folds after the?ducial scale leaves the horizon.For this reason they are excluded from the?gure(and we have checked that the Markov chains cut in this way still retain their convergence properties). We immediately see that Set1–chains which ran without any constraint on N–has a distribution in the{ξ,N}plane that peaks near N=10,and excludes N=30at more than2σ.On the other hand,the corresponding distribution for chains with N>30is peaked around N=30,but N=60is no longer excluded.
While we have dubbed the methodology employed here“slow roll reconstruction”,we are not in a position to present an unambiguous form for the in?ationary potential.With-out a non-zero lower bound on?the height of the potential is essentially undetermined. Secondly,as we noted above,our chains prefer a positive value ofξ,but we emphasize
that this is still well short of a convincing detection of a running,nor have we performed a Bayesian analysis(e.g.,[41,42])to test the signi?cance of the inclusion of theξparam-eter.On the other hand,the underlying power spectrum is uniquely determined by any given choice of{A s,?,η,ξ},and in Figure10we plot the permitted range of P(k)for the three sets of chains.In particular,without the N>30prior we see the preference for a scale dependent spectral index in the WMAP+SDSS dataset.Conversely,it is easy to understand why including the Lyman-αdataset has such a dramatic impact on the allow-able running,since this samples the power spectrum at very short scales,and thus has substantial“leverage”.
Figures9and10show that large values ofξhave a strong tendency to make in?ation end after an insu?cient number of e-folds to satisfy cosmological requirements.These ?gures imply that in order to satisfy the requirement that N>30,the allowed range ofξfrom the data becomes dramatically restricted,leading to a small allowed range on dn s
d ln k .Her
e we have chosen a pivot near the“waist”
of the P(k)distribution in Figure10,or k0=0.02Mpc?1.For the case where no N prior was imposed,we now?nd that there is a considerable shift to the red in n s but the bounds onξand dn s
d ln k degeneracy has rotated somewhat,encapsulating both th
e tendency seen in
the top panel of Figure10for the power spectrum to run from blue on large scales to red on small scales,and the fact that specifying the constraints at the“waist”decorrelate these two parameters as much as possible.When we impose the N prior,the constraints do not change signi?cantly when we translate the pivot–in this case the running was already tightly constrained and the pivot dependence of n s is correspondingly small.We can compare these plots directly to speci?c in?ationary models,and we have superimposed the trajectories for several canonical models on the plots in Figure11.A representative natural in?ation[43]model14and m2φ2in?ation lie inside the95%con?dence level with and without the N prior for the combination of WMAPII+SDSS.Conversely,λφ4lies outside this con?dence level for both cases.We do not show results for power-law or hybrid in?ation.In the former case,n s and the slow roll parameters do not evolve with time,and so any given model would be represented by a single point on these plots,rather than a trajectory.Likewise,in hybrid models the end of in?ation is not speci?ed by the
value of the slow roll parameters,so we cannot unambiguously plot their trajectories as a function of N.
Looking at Figure11we are again reminded that interpreting MCMC estimates of cosmological parameters is a complicated task.At present,the publicly available Lyman-αlikelihood codes are not compatible with the Hubble slow roll formalism used here. Since these data probe the fundamental spectrum at small scales,incorporating it into our analysis would greatly enhance our ability to constrain the spectrum–particularly for models which are strongly scale dependent.Likewise,we expect that the release of high quality CMB spectra for?>1000will greatly enhance the ability of slow roll reconstruction to constrain the in?ationary parameter space.
A further lesson to be drawn from Figure11is that the data is now approaching the point where we must directly address the N dependence of the parameters in simple in?ationary models.In[45],Kinney and Riotto show that our ignorance of reheating physics means that we cannot make a unique mapping between the remaining number of e-folds N and a given comoving scale k.Looking at the trajectories we plot,we see that the N dependence of the slow roll and spectral parameters over a ten e-fold range is only a few times smaller than the widths of the parameter distributions we recover from our MCMC analyses.The quality of cosmological data is thus reaching the point where this degeneracy will need to be included explicitly in any experimental tests of speci?c in?ationary models –especially those where the?eld point evolves comparatively rapidly.
5.Discussion
In this paper we have used slow roll reconstruction to extract bounds on the in?ationary potential from observational data.These bounds make strong use of an in?ationary prior, and thus apply to cosmological models where the density perturbation spectrum is laid down by single,slow rolling,minimally coupled in?aton?eld.Our results align with the broad picture supplied by previous analyses of the WMAPII dataset:there is noticeable but by no means compelling evidence for a signi?cant scale dependence(running)in the power spectrum when one considers WMAPII and SDSS on their own.However,adding a prior constraint that rules out models with an unacceptably small number of e-folds signi?cantly tightens the bounds on any possible running.
The combination of WMAPII and SDSS on their own are compatible with a broad class of perturbation spectra,but this range shrinks substantially when further constraints are added to the estimation process.In particular,we consider in?ationary models where at least30e-folds of in?ation occur after CMB scales have left their horizon.Similar results are seen in conventional analyses of the spectrum when Lyman-αforest data is included [16].We further analyze the impact on the choice of pivot on the quoted ranges for the spectral parameters.Slow roll reconstruction provides an estimate for the in?ationary potential over its entire range,thanks to the closed nature of the truncated slow roll hierarchy.Consequently,we actually have access to the underlying power spectrum and in?ationary potential,although these quantities are fully speci?ed by the truncated slow roll hierarchy–which here consists of three independent parameters.However,we can use the
?ow equations to post-process our chains to give the constraints on the standard spectral parameters at an arbitrary pivot.We make use of this feature,and study the change in constraints on the in?ationary parameter space when changing k0from0.002Mpc?1to 0.02Mpc?1.Without a prior on the number of e-folds N,the constraints on n s and dn s
d ln k space.In our view,it is a better aid to interpretation to present constraints over
the entire k range implied by one’s parameterization,as in Figure10–in principle,this is a simple thing to do in the MCMC methodology for any parameterization of P(k)–and then compare with speci?c in?ationary models over this entire range of scales.What does matter is whether the constraints at a wide range of k either encompass or exclude the pre-diction from that speci?c potential.For absolute completeness,the“band”of constraints from data on P(k)should be compared with a“band”for the model under consideration, encompassing the theoretical uncertainty in both the model parameters and the reheating process[45].
Where they are directly comparable,our results broadly agree with those of Finelli et al.[29].However,the authors of[29]use a di?erent formulation of the slow roll expansion from the one adopted here,which allows their?3to be of order unity,and they?nd they must pay careful attention to their prior for this parameter.15Conversely,our“third”parameter,ξneeds no such special handling,and we simply take all our slow roll parameters to lie within[?1,1],which is far larger than the range allowed by the data.Secondly,our running(α)is e?ectively scale-dependent,sinceξis a function of scale,but[29]specify their running at a?xed pivot,and it is not a function of scale.Finally,by using slow roll reconstruction we are able to include constraints on the remaining number of e-folds N,a possibility which is not considered in the analysis of[29].The tension between the running and the number of e-folds con?rms our theoretical analysis in[18].
Algebraically,the running index is dominated byξ,since?andηare both required by the data to be relatively small,and these terms only contribute quadratically to dn s
d ln k
shrinks considerably.
At large values,ξhas a long degeneracy because of the lack of constraining power in the current dataset.This is possibly a marginalization artifact,as we found that a large running is correlated with values of?m beyond the range seen in the standard concordance cosmology.If“simple”models of in?ation are a good description of the universe,the rising quality of astrophysical data will eventually break this degeneracy,and the running will fall somewhere within the range found when the N>30prior is applied here.Conversely if the data contracts around the large negative median value of the running,we would learn that in?ation is non-minimal in some way,as discussed in[18].In either case,slow roll reconstruction will be able to put tighter constraints on N as the constraints onξtighten.
In a subsequent paper we plan to assess the ability slow roll reconstruction to put bounds on the in?ationary parameter space with the data from di?erent proposed and pro-jected experiments.In addition,we will need to includeλ3,the fourth slow roll parameter, in these calculations,to be sure than any conclusions we reach are not a function of a premature truncation of the slow roll hierarchy.On the other hand,if this latter analysis shows thatλ3is necessary to describe the data four independent parameters would be needed to describe the in?ationary potential,which may be an indication that in?ation is non-minimal.
Finally we stress that while we?nd evidence that a signi?cant breaking of scale-invariance in the primordial power spectrum is consistent with the3-year WMAP dataset in the presence of an in?ationary prior,there is no compelling evidence for a running spectral index.However,the analysis here gives us cause for optimism that this question can be settled in the near future,and that slow roll reconstruction provides an elegant and powerful framework for analyzing the cosmological constraints on slow roll in?ation. Acknowledgments
This work has made use of the LAMBDA archive at https://www.doczj.com/doc/fd16829942.html,.RE is supported in part by the United States Department of Energy,grant DE-FG02-92ER-40704.HVP is supported by NASA through Hubble Fellowship grant#HF-01177.01-A awarded by the Space Telescope Science Institute,which is operated by the Association of Universities for Research in Astronomy,Inc.,for NASA,under contract NAS5-26555. HVP wishes to thank the organizers and participants at the“In?ation+25”conference in Paris and the Benasque Cosmology Workshop in Summer2006for stimulating discussions which have helped to improve this paper,and Andrew Liddle for useful conversations. She acknowledges the hospitality of the IoA in Cambridge where part of this work was carried out.We thank Antony Lewis for prompt responses to questions about the GetDist parameter estimation package.We thank David Spergel and Lyman Page for valuable comments on an earlier draft.
References
[1]G.Hinshaw et al.,arXiv:astro-ph/0603451.
[2]L.Page et al.,arXiv:astro-ph/0603450.
比较级和最高级 1.用“as+原级+as”表示 Tom is as tall as Mike. 2.用“not as(so) +原级+as”或“less than”表示 I didn’t do my homework so(as) carefully as you. The picture is less attractive than that one. 3.用“比较级+than”表示 Our city is more beautiful than any other city in our country. 注意：1) 为了避免重复，在从句中常用one, that, those等词来代替前面提过的名词。 The weather here is warmer than that of Shanghai. The radios made in our factory are better than those in your factory. 2)比较等级应注意避免和包括自己的对象比。 比较级+than+ any other + 单数名词 all the other + 复数名词 anyone else any of the other + 复数名词 3）如果形容词作定语修饰一个单数可数名词，一般将不定冠词a/an放在形容词之后。 Our neighbour has _____ ours. A. as a big house as
B. as big a house as C. the same big house as D. house the same big as 4）比较级前一般不用冠词，但若表示“两者中较……时”。比较级前要加定冠词。若比较级后有名词，常在比较级前加不定冠词，表示泛指。 E.g. 他是两者中较高的一个 He is the taller of the two. 她唱得真动听！我可从未听过比这更好的嗓音了。 How beautifully she sings! I have never heard a better voice. 4. 三者或三者以上相比，表示最高级时，用“the +最高级”的结构表示，这种句式一般常有表示比较范围的介词短语。 Zhang Hua is the tallest of the three. He works (the) hardest in his class. That was the least exciting football game I’ve ever watched. This hotel is the most comfortable I’ve ever stayed. 注意：当最高级的前面无限定词the或有不定冠词a/an时，仅表示“很……，非常……” Monday is my busiest day. 星期一是我很忙的一天。 Qingdao is a most (very) beautiful coastal city. 青岛是一个非常美丽的海滨城市。 一、请写出下列形容词的比较级和最高级。 big ______ ______ small ______ ________ new ____?__ ________ tall ______ ______ short______ ________ old____?__ ________ weak ______ ______ strong ______ ______ fat____?__ ________ hot ______ ______ cold ______ ________ thin ____?__ ________ nice ______ _____ good ______ ________ high____?__ ________ low____?__ ________cheap______ ______ easy ______ ________
图解驾考科目二连续过障碍物（轧饼子）技巧 根据c1科目二过连续障碍考试要求，通过连续障碍时，车轮不得碰压圆饼，那么最好的路线就是每通过一个圆饼时都打正车轮，使车辆呈“Z”字形直线通过。搞清楚了车辆通过的路线，其实就两个关键，就是何时打轮何时回轮。 下面给大家介绍一种通过连续障碍的方法：通过连续障碍前，提前摆正车身，打正车轮，使左右前轮与1饼保持横向距离相等直行（圆饼直径为70cm，左右前轮横向间距为130cm，所以两前轮与圆饼两 切 边 横 向 距 离 为 30cm），如图： 作为初学的学员，有些不知道如何将车摆正，将第一个饼摆到两轮中间，在这里告诉大家一个方法：首先从车头看过去，看到150m之外的地方，判断车身是否正了,以桑塔纳车型为例，从视野上看过去，左边凸起的“筋”延 长至圆饼左切边，并与其保持30cm的横向距离，这样就将1饼摆在两轮中间了（视野上看过去的距离与实际距离是不一样的）。如图： 在骑越1饼的时候，当感觉1饼到了前排座位底下，也就是手刹底下，马上向右转半圈方向盘（打方向的时机），目标是2饼，从视野上看过去，当车头左前角尖与4饼右切边差不多对齐时，马上向左回半圈方向盘（回方向的时机），目的就是回正两前轮（注意，方向盘
打多少回多少，回正端平，这样才能把轮回正），这样就把2饼摆在两前轮中间了，继续前行；如图： 骑越2饼时，当感觉2饼到前排座位底下时，迅速向左打一圈方向盘，发现车头右前角尖差不多对正5饼左切边30cm处迅速向右回一圈，端正方向盘，继续前行；如图： 骑越3饼时，当感觉3饼到前排座位底下时，迅速向右转一圈再1/4方向盘（多打1/4方向的目的就是为了在短距离之内迅速将车移至目的位置），目的是4饼，发现车头左前角尖差不多对齐6饼右切边时，迅速往左回正方向盘（打多少回多少），继续前行；如图：
形容词、副词的比较级和最高级的用法： 当两种物体之间相互比较时，我们要用形容词或副词的比较级； 当相互比较的物体是三个或三个以上时，我们就要用形容词或副词的最高级。※形容词、畐I」词的比较级和最高级的变化规律： 1. 单音节形容词或副词后面直接加-er或-est tall —taller —tallest fast —faster —fastest 2. 以-e结尾的单音节形容词或副词直接加-r或-st large —larger —largest n ice —ni cer —ni cest 3. 以-y结尾的形容词或副词，改-y为-i再加-er或-est busy—busier —busiest early —earlier —earliest 4. 形容词或副词是重读闭音节时，双写最后的辅音字母，再加-er或-est hot ——hotter — hottest big ——bigger — biggest 5. 多音节形容词或副词前面直接加more或most delicious —more delicious —most delicious beautiful ——more beautiful ——most beautiful 6. 不规则变化 good (well) —better —best bad (badly) —worse—worst man y(much)-more-most little-less-least old-older(elder)-oldest(eldest) far-farther(further)-farthest(furthest) 以下笔记请手动记录！！！
方法一:摆斜倒车法 第一步：首先判断车位是否“合格”，如何可以的话，则稍微靠近车位。当前轮超过车位时，开始打方向，让车让车朝背靠车位的方向驶去。 第二步：利用可用的通道宽度，尽量将车的位置“摆斜”。这样就可以让车子与车位的夹角减少，从而减少倒车时打方向的幅度。同时，车辆“摆斜”还可以让驾驶员通过后视镜观察车辆后方的两个危险点。
如上图所示，当车辆“摆斜”到驾驶员通过左后视镜可以看到车位左侧车辆的边角时，就可以开始倒车了。倒车时要观察左后视镜，留意左后轮与旁边车辆的距离，这是倒车过程中出现的第一个“危险点”。 第三步：当自己车的左后轮越过了左侧车辆车头后，“危险点”就转移到另一个地方——车子右后角。这时目光应从左侧后视镜转移到右侧后视镜，判断尾部与右侧车辆的距离是否安全。 当右侧后视镜里看到自己与两侧车辆之间出现“缝隙”时，说明你已经成功通过了所有危险点。这时就可以继续打方向，调整车子的后退轨迹，尽量摆正入库。
方法二:定点倒车法 非字型停车位也是生活中比较常见的车位，通常在地下车库、露天停车场出现。这种车位停车入库技巧第一步，是先将你的车和车位间的横向距离保持在1.5米左右。（仅针对三厢轿车） 第二步：调整车辆，将车停在隔一个车位中间(车位旁边的旁边车位)，此时驾驶员刚好与这个车位的车辆在同一平行线。
注：此步骤中，车辆停靠的位置会因车辆大小有所区别，车主在实际倒车过程中需要根据自己车辆做适当调节。 第三步：将车辆往左打满方向，然后倒车入位!需要注意的是，倒车的速度一定要慢，因为倒车时难免有盲区，如果速度不快的话，有突发状况也能从容应对。 第四步：当车辆与车位平行时回正方向，然后继续往后倒，之后顺利停车入位。
英语比较级和最高级的用法 一、形容词、副词的比较级和最高级的构成规则 1.一般单音节词和少数以-er，-ow结尾的双音节词，比较级在后面加-er，最高级在后面加-est; (1)单音节词 如：small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest (2)双音节词 如：clever→cleverer→cleverest narrow→narrower→narro west 2.以不发音e结尾的单音节词，比较在原级后加-r，最高级在原级后加-st; 如：large→larger→largest nice→nicer→nicest able→abler→ablest 3.在重读闭音节(即：辅音+元音+辅音)中，先双写末尾的辅音字母，比较级加-er，最高级加-est; 如：big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4.以“辅音字母+y”结尾的双音节词，把y改为i，比较级加-er，最高级加-est; 如：easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5.其他双音节词和多音节词，比较级在前面加more，最高级在前面加most; 如：beautiful→more beautiful→most beautiful different→more different→most different easily→more easily→most e asily 注意：(1)形容词最高级前通常必须用定冠词 the，副词最高级前可不用。 例句： The Sahara is the biggest desert in the world. (2) 形容词most前面没有the，不表示最高级的含义，只表示"非常"。 It is a most important problem. =It is a very important problem. 6.有少数形容词、副词的比较级和最高级是不规则的，必须熟记。 如：good→better→best well→better→best bad→worse→worst ill→worse→worst
比较级和最高级的讲解 变化规则 1．一般单音节词和少数以-er，-ow结尾的双音节词，比较级在后面加-er，最高级在后面加-est； （1）单音节词 如：small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest （2）双音节词 如：clever→cleverer→cleverest narrow→narrower→narrowest 2．以不发音e结尾的单音节词，比较在原级后加-r，最高级在原级后加-st； 如：large→larger→largest nice→nicer→nicest able→abler→ablest 3．在重读闭音节（即：辅音＋元音＋辅音）中，先双写末尾的辅音字母，比较级加-er，最高级加-est； 如：big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4．以“辅音字母＋y”结尾的双音节词，把y改为i，比较级加-er，最高级加-est； 如：easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5．其他双音节词和多音节词，比较级在前面加more，最高级在前面加most； 如：beautiful→more beautiful→most beautiful different→more different→most different easily→more easily→most easily 注意： （1）形容词最高级前通常必须用定冠词the，副词最高级前可不用。 例句：The Sahara is the biggest desert in the world. （2）形容词most前面没有the，不表示最高级的含义，只表示"非常"。
一、比较级和最高级的讲解 1．一般单音节词和少数以-er，-ow结尾的双音节词，比较级在后面加-er，最高级在后面加-est； （1）单音节词 如：small→smaller→smallest short→shorter→shortest tall→taller→tallest great→greater→greatest （2）双音节词 如：clever→cleverer→cleverest narrow→narrower→narrowes t 2．以不发音e结尾的单音节词，比较在原级后加-r，最高级在原级后加-st；如：large→larger→largest nice→nicer→nicest able→abler→ablest 3．在重读闭音节（即：辅音＋元音＋辅音）中，先双写末尾的辅音字母，比较级加-er，最高级加-est； 如：big→bigger→biggest hot→hotter→hottest fat→fatter→fattest 4．以“辅音字母＋y”结尾的双音节词，把y改为i，比较级加-er，最高级加-est；如：easy→easier→easiest heavy→heavier→heaviest busy→busier→busiest happy→happier→happiest 5．其他双音节词和多音节词，比较级在前面加more，最高级在前面加most；如：beautiful→more beautiful→most beautiful different→more different→most different easily→more easily→most e asily 注意：（1）形容词最高级前通常必须用定冠词the，副词最高级前可不用。例句：The Sahara is the biggest desert in the world. （2）形容词most前面没有the，不表示最高级的含义，只表示"非常"。 It is a most important problem. =It is a very important problem. 6．有少数形容词、副词的比较级和最高级是不规则的，必须熟记。 如：good→better→best well→better→best bad→worse→worst ill→worse→worst old→older/elder→oldest/eldest many/much→more→most little→less→least far →further/farther→ furthest/farthest 二、形容词、副词的比较级和最高级的用法 1．“A + be +形容词比较级+ than + B” 意思为“A比B更……”。 如：This tree is taller than that one. 这棵树比那棵树高。 注意： ①在含有连词than的比较级中，前后的比较对象必须是同一范畴，即同类事物之间的比较。 ②在比较级前面使用much，表示程度程度“强得多”。 如：A watermelon is much bigger than an apple. ③very, quite一般只能修饰原级，不能修饰比较级。 2．“比较级+ and + 比较级”或“more and more +原级”表示“越来越……”
科目二 桩考内容： 倒桩+定点停车+侧方位停车+七选一：连续障碍，单边桥，曲线行驶，限速限宽门，起伏路，直角转弯，百米加减档 本文介绍了科目二考试的一些技巧，希望对大家有所帮助：PART 1.到桩技巧 PART 2.通过连续障碍、过单边桥，侧方位停车的方法 PART 3.侧方位停车 PART 4.离合器半联动的使用 PART 1.到桩技巧
倒桩图 倒桩是一个长期实践得到的驾驶技巧 下文完全是应付考试而作,其中有二点要特别注意。 1、稳住离合,完全掌控车速 2、选定合适的参照物 参照物稍有不同,练车时一定要选当地考试同车型练习 1 2 3(杆) 4 5 6(杆) 1、右后到：右转回头看杆，当3,4杆标齐的时候，开始向右大方向，打到底。这时头向前倾靠向方向盘,右转回头,保持右车窗三角玻璃右下角始终出现2号杆,看不到时,回一点方向,然后继续向右把方向打到底.然后注视左后车镜,当你可以从左后视镜正中位置看到4号杆时,刹车.这时看看左右后视镜,哪边车距大方向就向哪边打,打完后立刻调正方向,将车到入乙库,这时稳住车速,右转回头看5号杆快消失就停车(回头要随意点,距离自己要多感觉一下)。 2、第一进：先换1档,把方向往右打到底，再慢慢抬离合往前走，当左大灯与2号杆一齐时候，快速回方向到底，车身快直的时候，再快速的向右回方向，车身调正后向前开到与2号杆3 0-50CM时停车。 3、第一退：先换到档,把方向往右打到底，然后到车，看左大灯与1号杆一齐时，开始回方向，车身调正后,继续到车靠近5
号杆就停车.这一进一退，车子基本上已经移到右边的库位一大半了。 4，第二进，第二退：原理同上。向前开时,自己选个车身参照物(比如挡风玻璃左下角的标志等.到车要左回头看后5号杆与后车窗玻璃中间位置重合时就要停车,(多练几次,主要保证左后轮在中线附近就可以停车了).停车后方向向右打到底,继续到,车身一正就完成了移库(从乙库移到甲库) 5、第三进（乙库位出去）。先把方向往左打一圈，再往前走车头基本上在1,2杆中间时把方向打正出去(参照中线大约45度出库)。稳住车速,右转回头看后挡风玻璃出现前中杆时,方向向左打到底,车身一正，停车. 6、左后到：和右后到方法差不多，不同的是,左转回头看到1,6杆重合前10-20CM就把方向向左打到底.保持从后车窗三角玻璃中间能看到2号杆,看不到就回一点方向,再次看到就继续打死方向.这时注视右后视镜,右后杆出现在右后视镜中间位置就停车.再注意两边车距,缓慢入库,只要车头入库就可以停车了. 7、出甲库(有的地方没有):挂1档,慢慢出库,右转回头一看到3号杆出现在右后车窗三角玻璃中间时马上将方向向右打死,慢慢调正车身,把车停在停止线以内。 特别注意: 1、不要挂错档位。 2、不要转错方向盘方向：贴库时右打轮，到库时左打轮。
形容词比较级和最高级 绝大多数形容词有三种形式,原级,比较级和最高级, 以表示形容词说明的性质在程度上的不同。形容词的原级: 形容词的原级形式就是词典中出现的形容词的原形。例如: poor tall great 形容词的比较级和最高级: 形容词的比较级和最高级形式是在形容词的原级形式的基础上变化的。分为规则变化和不规则变化。 规则变化如下: 1) 单音节形容词的比较级和最高级形式是在词尾加 -er 和 -est 构成。 great (原级) (比较级) (最高级) 2) 以 -e 结尾的单音节形容词的比较级和最高级是在词尾加 -r 和 -st 构成。 wide (原级) (比较级) (最高级) 3)少数以-y, -er, -ow, -ble结尾的双音节形容词的比较级和最高级是在词尾加-er 和-est 构成。 clever(原级) (比较级) (最高级) 4) 以 -y 结尾,但 -y 前是辅音字母的形容词的比较级和最高级是把 -y 去掉,加上 -ier 和-est 构成. happy (原形) (比较级) (最高级) 5) 以一个辅音字母结尾其前面的元音字母发短元音的形容词的比较级和最高级是双写该辅音字母然后再加-er和-est。 big (原级) (比较级) (最高级) 6) 双音节和多音节形容词的比较级和最高级需用more 和 most 加在形容词前面来构成。 beautiful (原级)(比较级) (比较级) difficult (原级) (最高级) (最高级) 常用的不规则变化的形容词的比较级和最高级: 原级------比较级------最高级 good------better------best many------more------most much------more------most bad------worse------worst far------farther, further------farthest, furthest old ----older,elder----older,eldest 形容词前如加 less 和 least 则表示"较不"和"最不" important 重要 less important 较不重要 least important 最不重要 形容词比较级的用法: 形容词的比较级用于两个人或事物的比较,其结构形式如下: 主语+谓语(系动词)+ 形容词比较级+than+ 对比成分。也就是, 含有形容词比较级的主句+than+从句。注意从句常常省去意义上和主句相同的部分, 而只剩下对比的成分。
形容词比较级、最高级变化表 一、形容词比较级、最高级变化规则 1.在形容词词尾加上“er” “est” 构成比较级、最高级： bright（明亮的）—brighter—brightest broad（广阔的）—broader—broadest cheap（便宜的）—cheaper—cheapest clean（干净的）—cleaner—cleanest 2.双写最后一个字母，再加上“er” “est” 构成比较级、最高级： big（大的）—bigger—biggest fat（胖的）—fatter—fattest hot（热的）—hotter—hottest red（红的）—redder—reddest 3.以不发音的字母e结尾的形容词，加上“r” “st” 构成比较级、最高级： able（能干的）—abler—ablest brave（勇敢的）—braver—bravest close（接近的）—closer—closest fine（好的，完美的）—finer—finest 4.以字母y结尾的形容词，把y改为i，再加上“er” “est” 构成比较级、最高级：busy（忙碌的）—busier—busiest dirty（脏的）—dirtier—dirtiest dry（干燥的）—drier—driest early（早的）—earlier—earliest 5.双音节、多音节形容词，在单词前面加上“more” “most” 构成比较级、最高级：afraid（害怕的）—more afraid—most afraid beautiful（美丽的）—more beautiful—most beautiful 6.不规则变化的形容词： bad（坏的）—worse—worst far（远的）—farther—farthest (far—further—furthest) good（好的）—better—best ill（病的）—worse—worst
c1科目二过连续科目二过连续过圆饼的过圆饼的过圆饼的技巧图解技巧图解 根据c1科目二过连续障碍考试要求，通过连续障碍时，车轮不得碰压圆饼，那么最好的路线就是每通过一个圆饼时都打正车轮，使车辆呈“Z”字形直线通过。搞清楚了车辆通过的路线，其实就两个关键，就是何时打轮何时回轮。 下面给大家介绍一种通过连续障碍的方法：通过连续障碍前，提前摆正车身，打正车轮，使左右前轮与1饼保持横向距离相等直行（圆饼直径为70cm，左右前轮横向间距为130cm，所以两前轮与圆饼两切边横向距离为30cm）， 作为初学的学员，有些不知道如何将车摆正，将第一个饼摆到两轮中间，在这里告诉大家一个方法：首先从车头看过去，看到150m 之外的地方，判断车身是否正了,以桑塔纳车型为例，从视野上看过去，左边凸起的“筋”延长至圆饼左切边，并与其保持30cm 的横向距离，这样就将1饼摆在两轮中间了（视野上看过去的距离与实际距离是不一样的）。 在骑越1饼的时候，当感觉1饼到了前排座位底下，也就是手刹底下，马上向右转半圈方向盘（打方向的时机），目标是2饼，从视野上看过去，当车头左前角尖与4饼右切边差不多对齐时，马上向左回半圈方向盘（回方向的时机），目的就是回正两前轮（注意，方向盘打多少回多少，回正端平，这样才能把轮回正），这样就把2饼摆在两前轮中间了，继续前行； 骑越2饼时，当感觉2饼到前排座位底下时，迅速向左打一圈方向盘，发现车头右前角尖差不多对正5饼左切边30cm 处迅速向右回一圈，端正方向盘，继续前行；
骑越3饼时，当感觉3饼到前排座位底下时，迅速向右转一圈再1/4方向盘（多打1/4方向的目的就是为了在短距离之内迅速将车移至目的位置），目的是4饼，发现车头左前角尖差不多对齐6饼右切边时，迅速往左回正方向盘（打多少回多少），继续前行； 骑越4饼时，当感觉4饼到前排坐位底下时，迅速向左转一圈再1/4方向盘，目的是5饼，发现车头右前角尖差不多对齐6饼左切边时迅速回正方向盘，继续前行； 骑越5饼时，当感觉5饼到前排座位底下时，迅速向右转一圈方向盘，这时候可以借助车头左“筋”，在视野上看过去，发现左“筋”与6饼左切边保持30cm时迅速往左回正方向盘（当车头盖住6饼时就凭感知能力判断30cm的位置，看到左“筋”等于看到左前轮），顺利通过。 值得注意的是，在骑越过程中，打方向要打的及时，回方向要回的利索，不要拖泥带水，不要看到车头盖住圆饼就心急打方向，要克服这个心理作用。练车不是做数学题，不是只按照公式做就能算出结果，还要靠自己在练习的时候多找感觉，调整打方向的时机。c1科目二过连续障碍技巧图解介绍的方法虽然好用，但重要的还是多练习，多思考，总结一套自己的好方法。
形 容 词 的 比 较 级 和 最 高 级 变 化 规 则 B.部分双音节与多音节的词比较级在原级之前加more, 最高级在原级之前 加most beautiful---more beautiful---most beautiful interesting--- difficult--- C.不规则变化的形容词: little / few（原形）- less （比较级）- least（最高级） good（原形）- better（比较级）- best（最高级） bad （原形）- worse（比较级）- worst（最高级） far （原形）-- further—furthest 例句： Tom is tall. John is tall. Bob is tall. I'm as tall as you. Tom is as tall as John.
Bob is taller than John. John is the tallest of the three. John is the tallest in his class. 写出以下各形容词的比较级和最高级： 1. nice ______________________ 2. fat ____________________ 3. slow _____________________ 4. dry ____________________ 5. happy ____________________ 6. wet ____________________ 7. much ____________________ 8. ill _____________________ 9. little _____________________ 10. bad ___________________ 11. thin ______________________ 12. far ____________________ 13. early _____________________ 14. careful_________________ 15. exciting ___________________ 16. busy __________________ 2. 根据句意，用所括号内所级形容词的比较等级形式填空： 1. Mr. Smith is _________ man in this office. (rich) 2. Winter is _________ season of the years. (cold) 4. It is much _______ today than yesterday. (hot) 5. She is a little ________ than her classmates. (careful) 6. ________ people came to the meeting than last time. (many) 7. Which book is ________, this one or that one? (easy) 8. My room is _______ than yours. (small) 9. Hainan is _______ from Beijing than Hunan. (far) 10. Skating is _______ than swimming. (exciting) 11. Jim is _______ than all the others. (honest) 12. Things are getting _______ and _______. (bad) 13. The higher you climb, the _______ it will be. (cold) 14. Now his life is becoming ________ and _______. (difficult) 用适当形式填空： 1. Bob is _________ ( young ) than Fred. but ___________ (tall) than Fred. 2. Almost all the students' faces are the same ,but Li Deming looks _______ (fat) than before after the summer holidays. 3.Which is _________ (heavy), a duck or a chicken? 5.-- How _________ (tall) is Sally? --She' s 1.55 metres ________ (tall). What about Xiaoling? -- She' s only 1.40 metres ________ (tall). She is much _______ (short) than Sally. She is also the _______ (short) girl in the class. 6. He is ______ (bad) at learning maths. He is much _______ (bad) at Chinese and he is the _________ (bad) at English. 7. Annie says Sally is the ________ (kind) person in the world. 8. He is one of the_________(friendly) people in the class, I think. 9. A dictionary is much _________ (expensive) than a story-book. 10. An orange ia a little ______ (big) than an apple, but much ________ (small) than a watermelon.
形容词 第一章比较级、最高级变化一览表 规则变化 1.单音节以及少数双音节的词尾加上“er”“est”构成比较级、最高级： bright（明亮的）—brighter—brightest broad（广阔的）—broader—broadest cheap（便宜的）—cheaper—cheapest clean（干净的）—cleaner—cleanest clever（聪明的）—cleverer—cleverest cold（寒冷的）—colder—coldest cool（凉的）—cooler—coolest dark（黑暗的）—darker—darkest dear（贵的）—dearer—dearest deep（深的）—deeper—deepest fast（迅速的）—faster—fastest few（少的）—fewer—fewest great（伟大的）—greater—greatest hard（困难的，硬的）—harder—hardest high（高的）—higher—highest kind（善良的）—kinder—kindest light（轻的）—lighter—lightest long（长的）—longer—longest loud（响亮的）—louder—loudest low（低的）—lower—lowest near（近的）—nearer—nearest new（新的）—newer—newest poor（穷的）—poorer—poorest quick（快的）—quicker—quickest quiet（安静的）—quieter—quietest rich（富裕的）—richer—richest short（短的）—shorter—shortest slow（慢的）—slower—slowest small（小的）—smaller—smallest smart（聪明的）—smarter—smartest soft（柔软的）—softer—softest strong（强壮的）—stronger—strongest sweet（甜的）—sweeter—sweetest tall（高的）-taller - tallest thick（厚的）—thicker—thickest warm（温暖的）—warmer—warmest weak（弱的）—weaker—weakest young（年轻的）—younger—youngest 2以一个元音加一个辅音字母结尾的单音节词（即重读闭音节词），双写结尾的辅音字母er， -est big（大的）—bigger—biggest fat（胖的）—fatter—fattest hot（热的）—hotter—hottest red（红的）—redder—reddest sad（伤心的）—sadder—saddest thin（瘦的）—thinner—thinnest wet（湿的）—wetter—wettest mad（疯的）—madder—maddest 特别提醒：new, few, slow, clean等词含有字母组合，且发的是长元音，不用双写。 3.以不发音的字母e结尾的形容词，加上“r”“st”构成比较级、最高级： able（能干的）—abler—ablest brave（勇敢的）—braver—bravest close（接近的）—closer—closest fine（好的，完美的）—finer—finest large（巨大的）—larger—largest late（迟的）—later—latest nice（好的）—nicer—nicest ripe（成熟的）—riper—ripest rude（粗鲁的）—ruder—rudest safe（安全的）—safer—safest strange（奇怪的）—stranger—strangest wide（宽广的）—wider—widest wise（睿智的，聪明的）—wiser—wisest white（白的）—whiter—whitest 4.“以辅音字母+y”结尾的词改y为i，再加-er, -est busy（忙碌的）—busier—busiest dirty（脏的）—dirtier—dirtiest
汽车方向盘打法技巧 很多刚开始学车的朋友都对如何打方向感到很困惑。这里以“三把方向法”为例，对如何打方向提出一些自己的认识。 三把方向法（用于变线、路边停车、移库等，倒车同理）： 第一把：向欲到达的新车道打方向。如转向盘右转x度，角速度为y度每秒。此时车轮偏右，车辆相对车道向右偏转； 第二把：车身中部接近新车道中心线时回方向。如转向盘左转2x度（若只转x度，车轮与车身方向一致，车辆停止右转，但仍会偏离新车道行驶）。转动转向盘的角速度约为2y度每秒（若为y度每秒，车辆会以弧线较多地驶过新车道中心线右侧）。此时车身开始回正，但车轮偏左； 第三把：车头即将正对新车道中心线时将车轮回正。如转向盘右转x度。转动转向盘角速度约为y度每秒。此时车轮与车身方向一致,车辆行驶在新车道中心线上； 听到靠边停车指令，开启右方向灯，首先减速，看右边前方有无路口，门口，还要注意消防栓，选择停车点．然后脚搁刹车，离合器踩到底，用刹车控制车速．看右边后视镜有无情况，无情况就推半把方向，眼睛看车头，车头的右三分之一处接触上街堰口，就回掉半把方向再拉四分之一方向，这样车头的右三分之一处沿着上街堰口走，看一下右后视镜，车身和上街堰平行了就把方向回正，轻踩刹车．这样车身与上街堰的距离是３０厘米左右．如果车不正就不要踩刹车，用车的惯性还可以进行微调． 转弯时打法：缓弯要早打慢打、早回慢回；急弯要晚打快打、早回快回。慢打则慢回；快打则快回。要有预见能力，在未转向前，双手就开始做好准备动作。还要考虑到提前量。 本人基本上是左手为主，单手扣腕旋转方式，右手附带辅助，这个基本上是个熟练活，多开多用就熟悉了。 好处：速打速回，打方向比较快 缺点:手心出汗时容易打滑，这个事小问题，给方向盘装个套就轻松解决了。 驾校三点倒桩法.懂得的说说.分别是那六个点.（桑塔纳） 倒桩么死点就看你怎么看的怎么学的我说的简单点吧若有不懂可以详细和我谈倒车起步给起航点线上挂上倒挡头放到二个座位之间看右边的大3角玻璃只要让中杆一直连结在三角学玻璃里只要浮此刻里面就开始动方向向右打要慢慢打看着中杆打方向你打方向的同时必需保证你在三角学玻璃里能看到中杆秋水晃一眼左车尾角看一下整个车尾是否快进去快进去的环境下直接打死如果已经打死就不需要了然后直接看倒车镜
比较级和最高级 一、比较级的用法: 当两个人或事物（A和B）进行比较时，我们需要用到形容词（副词）的原级或者比较级1．表达“A和B一样”，用as…as的结构。 公式: A+be动词+as+形容词原级+as…+B A+实义动词+as+副词原级+as…+B Eg I am as tall as you. He runs as fast as I. 我的房间和她的一样大。 他游得和我一样好。 2．表达“A不如B”用not as…as的结构。 公式: A+be动词的否定形式+as+形容词原级+as…+B A+助词的否定形式+动词+as+形容词原级+as…+B Eg I am not as tall as you. He doesn’t run as fast as I. 我的房间没有他的大。 我没有他游得好。 3. 表达“A大于B”用“比较级+than”的结构。 公式: A+be动词+形容词比较级+than+B… A+实义动词+副词比较级+than+B… Eg I am taller than you.我比你高。 He runs faster than I. 他跑得比我快。 我的房间比他的大。 我游得比他的好。 4.表示A 是...中最大的结构 公式：A+be动词+the +形容词最高级+范围 A+实义动词+the+形容词最高级+范围 I am the tallest in my class. He runs the fastest in my class. 我的房间是这里最大的。 我游得是我们班最好的。 二.形容词和副词的比较级和最高级的变化方法如下
(1) 符合规则的： (2)几个不规则的形容词和副词的比较级和最高级如下表： 原 级 比较级 最高级 good , well better best bad , ill worse worst many , much more most little less least far farther / further farthest / furthest 练习1：写出下列词的比较级和最高级 tall ﹍﹍ ﹍﹍ slow ﹍﹍ ﹍﹍ small ﹍﹍ ﹍﹍ fast ﹍﹍ ﹍﹍ smart ﹍﹍ ﹍﹍ few ﹍﹍ ﹍﹍ nice ﹍﹍ ﹍﹍ fine ﹍﹍ ﹍﹍ large ﹍﹍ ﹍﹍ late ﹍﹍ ﹍﹍ brave ﹍﹍ ﹍﹍ pretty ﹍﹍ ﹍﹍ easy ﹍﹍ ﹍﹍ funny ﹍﹍ ﹍﹍ happy ﹍﹍ ﹍﹍ lazy ﹍﹍ ﹍﹍ heavy ﹍﹍ ﹍﹍ dirty ﹍﹍ ﹍﹍ dry ﹍﹍ ﹍﹍ early ﹍﹍ ﹍﹍ 情 况 加 法 例 词 一 般 情 况 直接加 -er ; -est all-taller-tallest 以不发音e 结尾的词 去e 加 –er ; -est nice-nicer-nicest 以“辅音+y”结尾的词 变y 为i 再加-er ; -est dry-drier-driest heavy-heavier-heaviest 重读闭音节结尾的词 双写末尾辅音字母，再加-er ; -est thin-thinner-thinnest 多音节和部分双音节单词 在词前加 more ; most more delicious most delicious