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1.复杂（事物的种类、头绪等）多而杂。
——辞海In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory.在通常的用法中，复杂往往用于描述具有错综排列的很多部件的物体。
研究这些复杂的联系是复杂系统理论的主要的目标。
——维基百科事物的种类、头绪等多而杂;具有各种不同的,而且常是数量众多的部分、因素、概念、方面或影响是相互联系的,而这种相互联系又是难于分析、解答或理解。
——百度百科Neil Johnson describes complexity science as the study of the phenomena which emerge from a collection of interacting objects.[3]Neil Johnson将复杂科学描述成相互作用物体集合所产生现象的研究［3］。
从词源上，需要认识对象的数量过于巨大，和／或这些对象的性质存在差异且其组合关系又十分纠缠时，人们主观上形成一种缺乏统一性的不协调感，称为复杂。
复杂意指难于分割、分析或解决。
［4］2.复杂性在日常说法中，复杂或复杂性和简单相对立。
但在特定的场合，复杂的反面是各部分相互独立，而复杂化才与简单相对立。
日常生活中，“复杂（Complex）”经常与“复杂的（Complicated）”混用。
但在现今的系统科学中，它们一个是成千上万个相互连结着的“排烟管”，另一个则用来形容一些所谓高度“结合”在一起的解决方案。
(Lissack & Roos, 2000) 它是指，“复杂”（Complex）与“独立自主”相对的（译者注：在于突出系统各节点间错综复杂的联系），而“复杂的”（Complicated）才与“简单”相对（译者注：解决方案其内涵令人捉摸不清，让人产生一种复杂感，但是其结构却只是简单结合，并不复杂）。
小学上册英语第5单元期末试卷(含答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The goldfish has beautiful _______ (鳞片).2. A force acting in the opposite direction of motion is known as ______ (resistance).3.My sister enjoys __________ (做手工艺).4.The jellyfish floats in the _________ (海洋).5.The fish swims _____ (slowly/quickly) in the water.6. A saturated fat has no double ______.7. A _______ is a chart that organizes elements by their properties.8.What do you call the place where you learn about history?A. MuseumB. LibraryC. SchoolD. Park答案:A9.The __________ was an important period of change in the United States. (冷战)10.What is the name of the toy that you can build with blocks?A. PuzzleB. LegoC. DollD. Car答案: B11.The playground is _______ with children.12.The _____ (树木) provide shade on hot days.13.What is the name of the ocean that is the largest?A. Atlantic OceanB. Indian OceanC. Arctic OceanD. Pacific Ocean 答案: D14.I enjoy creating games with my toy ________ (玩具名称).15.Energy can be transformed from one form to _______.16.The cake is _____ (sweet/sour).17.My brother is a __________ (广告设计师).18.The chemical formula for palmitoleic acid is ______.19.What is the capital of the United States?A. New YorkB. Washington, D.C. C. ChicagoD. Los Angeles答案:B20.The tortoise carries its house on its _________. (背上)21.The process of evaporation causes liquids to turn into ______.22.The ______ contains a lot of water vapor.23.__________ are substances that can donate protons in a reaction.24.The ancient Romans built _______ to honor their leaders. (雕像)25.I _____ (love/hate) homework.26.The __________ (探险活动) is thrilling and educational.27.My favorite color is ________.28.The antelope runs very _________. (快)29.My toy ________ can zoom around the room.30.She is ________ (interested) in science.31.The stars are _____ (bright/dim) in the sky.32.The __________ is known for its stunning beaches.33.It is ___ (sunny) today.34. A rabbit's teeth never stop ______ (生长).35. A rabbit has powerful ______ (后腿) for jumping.36. A ______ (鸡) lays eggs every day.37.I _____ (want/wanted) to go to the park.38.What do you call a young goat?A. LambB. KidC. CalfD. Foal答案:B39.The hamster stores food in its ______ (脸颊).40. A solution is homogeneous, meaning it has a _______ composition.41.I like to help my mom ________ (整理) my room.42.Which fruit is known for having seeds on the outside?A. AppleB. StrawberryC. GrapeD. Pear答案:B.Strawberry43.I am proud of myself when I __________ because it shows __________.44.The process of photosynthesis occurs in the ______ (叶子).45.The __________ (希腊神话) includes many gods and heroes.46.We have math _____ today. (class/homework/friends)47. A dolphin can recognize itself in a ________________ (镜子).48.The __________ of an animal can vary greatly between species.49.The ________ has many petals and smells great.50.What is the color of the sky on a clear day?A. GreenB. BlueC. RedD. Yellow答案:B Blue51.The ______ is known for her photography.52.What is the first letter of the alphabet?A. BB. AC. CD. D答案:B53. A __________ is a mixture that can be separated by filtration.54.The ancient Egyptians used ________ for recording their history.55.The __________ is a large area of land used for agriculture. (农田)56.The ________ (生长周期) of a plant can vary.57. A ________ can swim in the ocean.58.The capital of the Maldives is ________ (马累).59.What is the capital of Ethiopia?A. Addis AbabaB. NairobiC. KampalaD. Khartoum答案:A. Addis Ababa60.Acids can donate ______ ions in solution.61.The _____ (猴子) loves to eat fruit.62.The snail carries its ______ (壳) on its back.63._____ (离子) in soil can affect plant health.64.The __________ (水域) is home to many fish.65.Water freezes at ______ degrees Celsius.66.My brother plays in a ____ (band) with his friends.67.The chemical symbol for bismuth is __________.68. A prism bends light to create a ______.69.Every year, we celebrate ________ (新年) with fireworks and family gatherings.70.I want to create a video game about my toy ____. (玩具名称)71.The _____ (大熊猫) is a rare animal that lives in China. 大熊猫是生活在中国的一种稀有动物。
briefings in functional genomics oxford -回复“Functional Genomics in Oxford: Unleashing the Potential of Genome Research”Introduction:Functional genomics is a rapidly evolving field of study that aims to understand the functions and interactions of genes in order to unravel the mysteries of life. The University of Oxford, with its esteemed reputation in scientific research, plays a pivotal role in advancing the frontiers of functional genomics. In this article, we will delve into the exciting world of functional genomics at Oxford, exploring the key focus areas, cutting-edge techniques, and significant contributions made by researchers in thisever-expanding field.1. Understanding Functional Genomics:Functional genomics encompasses the study of how the genome regulates biological processes and influences the phenotype of an organism. At Oxford, researchers employ various approaches, including computational biology, next-generation sequencing, andhigh-throughput screening, to enhance our understanding of gene function.2. Key Focus Areas at Oxford:a. Disease Research: Advances in functional genomics have paved the way for a deeper understanding of the genetic basis of diseases. Oxford researchers employ functional genomics techniques to unravel the complex mechanisms underlying diseases such as cancer, cardiovascular disorders, and neurodegenerative conditions, with the ultimate goal of developing targeted therapeutics.b. Epigenomics: The study of epigenetic modifications, such as DNA methylation and histone modifications, is a vibrant area of research at Oxford's functional genomics laboratories. By elucidating the role of epigenetics in gene expression and disease development, researchers are discovering novel therapeutic targets and potential biomarkers for early diagnosis.c. Gene Regulation: Oxford's functional genomics researchers investigate the intricate web of gene regulation mechanisms, including transcription factors, non-coding RNA, and chromatinstructure. The elucidation of these mechanisms enhances our knowledge of gene expression control, providing insights into normal development and disease progression.d. Functional Annotation of Genomes: Identifying the functions of genes encoded within a genome is a fundamental aim of functional genomics. Oxford researchers apply computational and experimental approaches to annotate gene functions, deciphering the roles of genes in various biological processes and shedding light on the evolutionary significance of gene function divergence.3. Cutting-Edge Techniques at Oxford:a. Next-Generation Sequencing (NGS): NGS technologies have revolutionized functional genomics research at Oxford. These high-throughput sequencing techniques allow for the characterization of entire genomes, transcriptomes, and epigenomes in a cost-effective and time-efficient manner. Researchers use NGS to unravel gene expression profiles, detect genetic variants, and investigate epigenetic alterations associated with diseases.b. CRISPR-Cas9 Genome Editing: Oxford researchers spearhead breakthroughs in CRISPR-Cas9-mediated genome editing, enabling precise manipulation of the genome to study gene function. This technique has expanded the possibilities of functional genomics research, offering unprecedented opportunities to elucidate the role of specific genes in disease mechanisms and therapeutic interventions.c. Functional Screens: High-throughput functional screens allow Oxford researchers to systematically identify genes involved in specific biological processes or diseases. These screens involve large-scale genetic perturbations, such as RNA interference (RNAi) or CRISPR knockout libraries, coupled with phenotypic analyses. By identifying genes essential for specific cellular functions, functional screens contribute to our understanding of gene function and potential therapeutic targets.4. Significant Contributions by Oxford Researchers:a. The Cancer Genome Atlas (TCGA): Oxford researchers were instrumental in the international collaboration that led to the creation of TCGA, a comprehensive catalog of genomic alterationsin various cancer types. TCGA has provided crucial insights into the genetic basis of cancer, paving the way for personalized medicine approaches and targeted therapies.b. ENCODE Project: As part of the ENCODE Project, Oxford researchers contributed to the functional annotation of the human genome. This project aimed to identify all functional elements within the genome, shedding light on gene regulation, non-coding RNA, and the three-dimensional architecture of the genome.c. Single-Cell Genomics: Oxford researchers have made significant contributions to the emerging field of single-cell genomics. By studying individual cells, researchers can unravel cellular heterogeneity, identify rare cell types, and investigate gene expression dynamics at unprecedented resolution. These insights have the potential to revolutionize our understanding of development, diseases, and therapeutic interventions.Conclusion:Functional genomics research at the University of Oxford continues to push the boundaries of our understanding of gene function andits impact on health and disease. Through their focused research areas, cutting-edge techniques, and noteworthy contributions, Oxford researchers play a crucial role in unraveling the mysteries of the genome. As the field of functional genomics continues to evolve, Oxford will undoubtedly remain at the forefront of groundbreaking discoveries with far-reaching implications for human health.。
1Popularity and Performance:A Large-Scale StudyPETER KRAFFT∗,JULIA ZHENG∗,and EREZ SHMUELI,Massachusetts Institute of TechnologyNICOL´AS DELLA PENNA,Australian National UniversityJOSHUA TENENBAUM and ALEX PENTLAND,Massachusetts Institute of Technology1.INTRODUCTIONSocial scientists have long sought to understand why certain people,items,or options become morepopular than others.One seemingly intuitive theory is that inherent value drives popularity.An alter-native theory claims that popularity is driven by the rich-get-richer effect of cumulative advantage—certain options become more popular not because they are higher quality but because they are alreadyrelatively popular.Realistically,it seems likely that popularity is driven by neither one of these forcesalone but rather both together.Recently researchers have begun using large-scale online experiments to study the effect of cumu-lative advantage in realistic scenarios[Salganik et al.2006],[Muchnik et al.2013],but there havebeen no large-scale studies of the combination of these two effects.We are interested in studying acase where decision-makers observe explicit signals of both the popularity and the quality of variousoptions.We derive a model for change in popularity as a function of past popularity and past perceivedquality.Our model implies that we should expect an interaction between these two forces—popularityshould amplify the effect of quality,so that the more popular an option is,the faster we expect it toincrease in popularity with better perceived quality.We use a data set from ,an online social investment platform,to support this hypothesis.2.MODELOur model describes the evolution of popularity of an individual action a(e.g.,choosing to buy a par-ticular brand or following a particular person).We assume afixed population of N agents that all havethe option to take action a at each of a series of discrete times.That is,at each time every agent decideswhether to take action a in that time step.We assume that agents want to make good choices,wherethe action being good is defined in some suitable domain-specific way.We also assume that at the end ofeach time step,all agents observe a single new signal of the action’s quality,as well as how many otheragents took the action in that step.We denote the number of agents taking the action at time t as n tand the signal of its quality at time t as q t.We assume agents attempt to evaluate whether the action isgood using Bayesian inference.In this case users are tasked with computing the posterior distributionP(good|q1,...,q t).Further we assume that agents choose whether to take the action at a particulartime step via probability matching,which means that each agent decides whether to take the actionat round t+1with probability P(good|q1,...,q t),i.e.agents match the probability that they take ac-tion a with the probability that the action is good.This assumption of probabilistic decision-makinghas precedent in cognitive science[Vul et al.2009],animal behavior[Prez-Escudero and de Polavieja2011],and economics[Anderson and Holt1997].∗Thefirst two authors contributed equally.Collective Intelligence2014.1:2•P.Krafft et al.Then,lettingαbe an arbitrary positive constant(used as a smoothing parameter),we haveP(good|q1,...,q t)=P(q t|good,q1,...,q t−1)P(q t|q1,...,q t−1)P(good|q1,...,q t−1)≈P(q t|good,q1,...,q t−1)P(q t|q1,...,q t−1)n t−1+αN+α,where the approximation follows from an interesting observation:Since users are probability match-ing,previous popularity actually approximates the posterior distribution from the last time step,and hence P(good|q1,...,q t−1)is given by the(smoothed)proportion of agents that chose to take the action in the last time step.Finally,noting that by the same argument future popularity will estimate P(good|q1,...,q t),assum-ingαis small relative to N,and letting f(q1,...,q t)=P(q t|good,q1,...,q t−1)P(q t|q1,...,q t−1),we see that the expected change in popularity in the next time step is given byn t+1−n t≈(f(q1,...,q t)−1)·n t+f(q1,...,q t)·α.When f is a monotonically increasing function of q t,this equation implies that we should expect a synergy between popularity and quality whereby increasing the popularity of an action should amplify the boost in popularity the action would get from a new signal of high quality.3.DATAThe data set we use to test this hypothesis was provided to us by the eToro company.eToro offers a website that incorporates several different trading platforms alongside a social trading network. Individuals can trade in the commodities,stock,and currencies markets.Furthermore,traders can conduct their own trades or view and copy trades made by other users,all using real money.The data set consists of transactions from June13,2011to November20,2013from their website,. One particularly interesting feature of the site is the ability that users have to mirror other users, automatically copying all of the trades they make.Importantly,users can choose who they want to mirror by viewing various measurements of performance as well as the current popularities of those traders.Thus the number of copiers a user has in the future,i.e.the future popularity of that user, could be influenced both by explicit signals of that user’s current popularity and of that user’s quality, indicated by eToro’s performance metrics.Although we do not have access to the signals that were actually displayed to the site’s users,we attempt to reconstruct proxies of these signals for our study.For popularity,we use a reconstructed version of the number of copiers each user has based on the transactions in our data.For quality,we use a rough measurement of recent performance:the performance of a user on a particular day is that user’s average expected daily return from closed trades in the last5business days(more specifically, the average over the subset of those days on which a user had any trading activity).In this work,we use100consecutive days of data,(ranging from September09,2011until December 29,2011rather than thefirst100to ensure we have good estimates of popularity),and we reserve the remainder of the data for validation in a planned extension of the current study.This subset includes the trading activity of24,587users.4.RESULTSTo support our model and our hypothesis,we approximate f by a linear function of performance.Our hypothesis then reduces to showing that there is a positive significant interaction between past popu-larity and past performance in a linear regression of change in popularity for each user.To investigate this interaction,we use our measurements of the popularity and performance of each user on each day. The regressions we examine include data points that consist of the performance of each active user on each day,the popularities of those users on those days,and the popularities of those users on the Collective Intelligence2014.Popularity and Quality:A Large-Scale Study•1:30.000.040.080.12Low Popularity ConditionBinned Past Performance M e a n D i f f e r e n c e i n P o p u l a r i t y <=−50−25025>=50Popularity 0Popularity 112345−40−2002040Average Change in Popularity (Middle Popularity Condition)Binned Past PopularityB i n n e d P a s t P e r f o r m a n ce 0.00.51.01.52.02.5Fig.1.Left:Plot of mean change in popularity against performance (for user-day pairs with popularity zero or one and with 95%confidence Gaussian error bars on the means and regression lines fitted from raw data).To visualize the results of the regression better,we rounded performance to the nearest multiple of 25and grouped values greater than 50or less than −50with those numbers,respectively .These graphical parameters roughly balanced the bin sizes on the x-axis.The difference in slope between the two lines in the third plot displays the hypothesized interaction effect.Right:Heat map of mean change in popularity as a function of previous popularity and previous performance for users with greater than zero copiers not in the top 100most-popular group.For this plot we binned popularities greater than 5with 5,and we rounded performance to the nearest 10and grouped values less than −40or greater than 40with those numbers,respectively .The nonlinear increase in the third dimensions moving up and to the right in this plot displays the hypothesized interaction.following days.For the present analysis,we look only at the subset of day-user pairs on which the user did not lose followers.We make this restriction because once a user has more copiers,it is naturally easier for that user to lose copiers,and this effect could alone give the interaction we hope to provide evidence for.Since we are also interested in replicating the widely observed cumulative advantage marginal effect of past popularity on future popularity ,we also attempt to control for a position bias in the interface of the eToro website.Since users can rank traders by eToro’s measurements of performance or by popularity ,traders might gain followers just by being displayed prominently on the website.A position bias should not artificially introduce an interaction since users can only sort by one signal at a time,but it could induce artificial marginal effects.To do this control,we further subset the data into two groups:user-day pairs on which the user had 0or 1past popularity (i.e.,zero copiers or one copier)and user-day pairs on which users had greater than 0past popularity but on which the user was not among the top 100most popular users.We use 100as the cutoff to be conservative.Since viewing the top 100traders requires scrolling several times,we would not expect the position bias to have an effect in either of these conditions.We attempt to provide evidence for our hypothesis within each of these conditions.Our hypothesized positive interaction effect and the marginal cumulative advantage effect of pop-ularity are supported in both conditions (p <10−5for all relevant regression coefficients).Figure 1displays our results.The plot on the left in Figure 1shows that having just one copier substantially increases the rate of increase in popularity as a function of performance over having no copiers.The plot on the right shows that a similar trend holds for greater values of popularity as well.We thus conclude that,for this range of popularity values,popularity may amplify the effect of performance in determining the magnitude of increase in future popularity .Perhaps the rich get richer for good reason.Collective Intelligence 2014.1:4•P.Krafft et al.AcknowledgementsThis research was partially sponsored by the Army Research Laboratory under Cooperative Agree-ment Number W911NF-09-2-0053.Views and conclusions in this document are those of the authors and should not be interpreted as representing the policies,either expressed or implied,of the sponsors. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.1122374.Any opinion,findings,and conclusions or recommendations ex-pressed in this material are those of the authors(s)and do not necessarily reflect the views of the National Science Foundation.REFERENCESLisa R.Anderson and Charles rmation cascades in the laboratory.The American Economic Review(1997). Lev Muchnik,Sinan Aral,and Sean Taylor.2013.Social Influence Bias:A Randomized Experiment.Science(2013).Alfonso Prez-Escudero and Gonzalo G.de Polavieja.2011.Collective Animal Behavior from Bayesian Estimation and Probability Matching.PLoS Computational Biology(2011).Matthew J.Salganik,Peter Sheridan Dodds,and Duncan J.Watts.2006.Experimental Study of Inequality and Unpredictability in an Artificial Cultural Market.(2006).Edward Vul,Noah D.Goodman,Thomas L.Griffiths,and Joshua B.Tenenbaum.2009.One and done?Optimal decisions from very few samples.In Proceedings of the31st Annual Conference of the Cognitive Science Society.Collective Intelligence2014.。
高超声速飞行器自抗扰姿态控制器设计秦昌茂;齐乃明;朱凯【期刊名称】《系统工程与电子技术》【年(卷),期】2011(33)7【摘要】针对高超声速飞行器无动力再入过程中具有强耦合、气动参数摄动及不确定性的非线性姿态模型,结合自抗扰控制中的扩张状态观测器(extended state observer,ESO)及非线性状态误差反馈律(nonlinear law state error feedback,NLSEF),分别设计了高超声速飞行器内环和外环自抗扰姿态控制器.将不确定性、耦合及参数摄动等干扰作为"总和干扰"利用扩张状态观测器进行估计并动态反馈补偿,再利用NLSEF抑制补偿残差.自抗扰控制器(active disturbance rejection control,ADRC)设计无需精确的飞行器被控模型,也无需精确的气动参数及摄动界限.仿真结果表明,控制系统能够克服干扰及气动参数大范围摄动的影响,在获取良好的动态品质和跟踪性能的同时,具有较强的鲁棒性.%For the hypersonic vehicle nonlinear attitude mode in re-entry process unpowered with a strong coupling, aerodynamic parameter perturbations and non-deterministic, combining extended state observer ( ESO) and nonlinear law state error feedback ( NLSEF) in the active disturbance rejection cotrol ( ADRC) , the hypersonic vehicle inner and outer ADRC attitude controller are designed respectively. Interferences, such as uncertainty, coupling and parameter perturbations, are regarded as "the sum of interference" , the extended state observer is used to estimate and implement dynamic feedback compensation, and then the NLSEF is used to inhibit thecompensating residual. ADRC controller is designed without a precise model of vehicle, and without precise perturbation boundaries of aerodynamic parameters. Simulation results show that the control system can overcome the impact of large-scale perturbations of interference and aerodynamic parameters, which has good dynamic qualities, tracking capabilities, and strong robustness.【总页数】4页(P1607-1610)【作者】秦昌茂;齐乃明;朱凯【作者单位】哈尔滨工业大学航天工程系,黑龙江,哈尔滨,150001;哈尔滨工业大学航天工程系,黑龙江,哈尔滨,150001;哈尔滨工业大学航天工程系,黑龙江,哈尔滨,150001【正文语种】中文【中图分类】V448【相关文献】1.高超声速飞行器自抗扰分数阶PID控制器设计 [J], 秦昌茂;齐乃明;吕瑞;朱凯2.高超声速飞行器自抗扰PID姿态控制 [J], 齐乃明;宋志国;秦昌茂3.高超声速飞行器改进自抗扰串级解耦控制器设计 [J], 齐乃明;秦昌茂;宋志国4.高超声速飞行器再入姿态改进自抗扰控制 [J], 周啟航;野邵文;向政委;齐乃明5.高超声速飞行器的自抗扰控制器设计 [J], 闫斌斌;闫杰因版权原因，仅展示原文概要，查看原文内容请购买。
大爆炸后的重子声学振荡English response:The baryon acoustic oscillations (BAO) are fluctuations in the distribution of visible baryonic matter (normal matter) in the universe. These fluctuations are caused by acoustic density waves in the primordial plasma of the early universe. The BAO signal is imprinted in the large-scale structure of the universe and can be used as a standard ruler to measure the expansion history of the universe.After the Big Bang, the universe was filled with a hot, dense plasma of electrons, protons, and photons. Small perturbations in the density of this plasma created sound waves that propagated through the plasma. When the universe was about 380,000 years old and had cooled enough for the protons and electrons to combine and form neutral hydrogen atoms, the sound waves were frozen in place. These sound waves left a characteristic scale in the distribution ofmatter, known as the baryon acoustic oscillation scale.This scale can be observed in the large-scaledistribution of galaxies and used to measure the expansion history of the universe. By studying the BAO signal in the cosmic microwave background radiation and the distributionof galaxies, scientists have been able to constrain the parameters of the standard cosmological model and shedlight on the nature of dark energy.中文回答：重子声学振荡（BAO）是宇宙中可见重子物质（正常物质）分布的波动。
Title:The Butterfly Effect:Small Changes in Nature,Large Impact on LifeIn the delicate dance of nature,every flap of a butterfly's wing holds the potential to stir winds that shape our world.The butterfly effect,a term coined from chaos theory,embodies the idea that minor perturbations in a system can lead to profound changes elsewhere.This concept,while rooted in the intricacies of weather patterns,extends its tendrils into the broader tapestry of environmental phenomena,reminding us of nature's intricate interdependence and vulnerability to small alterations.Nature is an elaborate orchestra,where each species,no matter how small,plays a distinctive note that contributes to the symphony of life.A subtle change,such as the introduction or removal of a species within an ecosystem,can initiate a cascade of effects that reverberate through ecological pathways.For example,the extinction of predatory birds like raptors once led to an explosion in the population of rodents,which in turn damaged crops and altered the structure of plant communities. These plants are homes and sources of food for other creatures;thus, their alteration affects diversity and the balance of life.Weather systems exemplify the butterfly effect's potency.The evaporation of water from a single lake or the albedo effect caused by melting snowcaps can modify local atmospheric conditions,potentially leading to changes in weather patterns thousands of miles away.Such modifications have far-reaching implications,influencing agricultural yields,natural resource availability,and even human health.The butterfly effect also finds resonance in the realm of climate change, where the burning of fossil fuels by human activities releases carbon dioxide and other gases into the atmosphere.These emissions,dwarfed in magnitude compared to the vastness of the celestial sphere, nevertheless culminate in global warming—a phenomenon with cataclysmic consequences.Ice caps melt,sea levels rise,and weather patterns become erratic,all testament to the power of small-scale actions to bring about colossal changes.Embracing the significance of the butterfly effect compels us to tread gently upon Earth.It underscores the necessity of sustainability and conservation,whereby our everyday actions—from waste managementto energy consumption—are imbued with the understanding that they have far-reaching impacts.By choosing renewable energy sources,we can mitigate the ripple effect of fossil fuels on climate.By preserving ecosystems,we maintain the complexity of life that buffers against unforeseen changes.In conclusion,the butterfly effect is not merely a chaotic metaphor but a stark reminder of the interconnectedness of nature.Each creature,each action,and each decision we make is a thread in the vast web of life, influencing processes beyond our immediate perception.As we navigate the future,let this humbling insight guide our actions,fostering a deeper respect for the delicate balance that sustains us all.For indeed,in the realm of nature,every choice we make has the power to transform our world,just as the gentle flap of a butterfly's wings can herald the approach of a tempest.。
蝴蝶效应为话题英文作文600字全文共2篇示例，仅供读者参考蝴蝶效应为话题英文作文600字1:The butterfly effect, a concept originating from chaos theory, suggests that small changes can lead to significant outcomes in complex systems. In the context of your article, exploring this phenomenon provides a fascinating lens through which to analyze various aspects of life, ranging from the natural world to social dynamics and beyond.Introduction:The butterfly effect, coined by mathematician and meteorologist Edward Lorenz, illustrates the interconnectedness of events and the amplification of small perturbations over time. Although initially applied to weather forecasting, its implications extend far beyond meteorology.Theoretical Background:To delve deeper, it's essential to understand the mathematical underpinnings of chaos theory. At its core, chaos theory deals with systems that are highly sensitive toinitial conditions, meaning that small changes in the starting state can lead to vastly different outcomes. This sensitivity to initial conditions is exemplified by the butterfly effect.Application in Nature:In nature, the butterfly effect manifests in various ways. For instance, the flapping of a butterfly's wings in Brazil could set off a chain reaction of events that ultimately result in a tornado in Texas. This concept underscores the complexity and unpredictability of natural systems.Social Dynamics:Beyond the realm of nature, the butterfly effect is evident in social dynamics. A single action or decision by an individual can have ripple effects that reverberate throughout society. Whether it's a groundbreaking scientific discovery, a political revolution, or a viral social media post, seemingly small events can catalyze significant change.Real-World Examples:Numerous real-world examples illustrate the butterfly effect in action. Consider the case of Rosa Parks, whose refusalto give up her seat on a segregated bus sparked the Montgomery Bus Boycott and catalyzed the Civil Rights Movement. Similarly, the assassination of Archduke Franz Ferdinand of Austria set off a chain of events that led to World War I.Implications:Understanding the butterfly effect has profound implications for various fields, including economics, psychology, and even personal decision-making. It highlights the interconnectedness of systems and the need for humility in predicting outcomes. Moreover, it underscores the importance of considering the long-term consequences of our actions.Conclusion:In conclusion, the butterfly effect serves as a powerful metaphor for the interconnectedness and complexity of the world we inhabit. By embracing this concept, we gain insight into the underlying dynamics shaping our lives and the world at large. As you continue to explore this topic in your article, delve deeper into specific examples and implications toprovide a comprehensive understanding for your readers.蝴蝶效应为话题英文作文600字2:Title: The Butterfly Effect: Unraveling the Complexity of Small ActionsIntroduction:The concept of the butterfly effect, derived from chaos theory, suggests that small actions can have significant and unpredictable impacts on complex systems. It originates from the metaphorical idea that the flapping of a butterfly's wings in one part of the world could potentially cause a tornado in another part. This phenomenon underscores the interconnectedness and sensitivity of systems to initial conditions. Through exploring the butterfly effect, we gain insight into the intricacies of cause and effect relationships, highlighting the importance of understanding and managing complexity in various domains.Body:1. Origins and Development of the Butterfly Effect:The term "butterfly effect" was coined bymathematician and meteorologist Edward Lorenz in the 1960s while studying weather patterns. Lorenz's research revealed that small changes in initial conditions could lead to drastically different outcomes over time. This idea challenged traditional linear thinking and paved the way for the study of chaos theory.2. Examples of the Butterfly Effect in Nature:Nature provides numerous examples of the butterfly effect in action. For instance, the movement of a single bee pollinating flowers can have far-reaching effects on ecosystems by influencing plant reproduction and the food chain. Similarly, the introduction of a non-native species to an environment can disrupt the balance of an entire ecosystem, demonstrating how small actions can trigger cascading consequences.3. The Butterfly Effect in Human Systems:Beyond the natural world, the butterfly effect manifests in human systems as well. In economics, minor fluctuations in market conditions or consumer behavior can lead to significant fluctuations in stock prices or economic growth.Similarly, the spread of ideas and innovations can be traced back to small-scale actions that snowball into transformative societal changes.4. Applications and Implications:Recognizing the butterfly effect has practical applications in various fields. For instance, in risk management, understanding how small events can escalate into crises enables proactive measures to mitigate potential impacts. In decision-making processes, considering the long-term repercussions of seemingly insignificant choices can lead to more informed and strategic actions.5. Managing Complexity in a Butterfly Effect World:Living in a world characterized by the butterfly effect necessitates a paradigm shift in how we approach complexity. Embracing uncertainty and adopting systems thinking allows us to navigate interconnected networks with greater resilience and adaptability. By acknowledging our interconnectedness and the potential ripple effects of our actions, we can strive for more sustainable and equitable outcomes.Conclusion:The butterfly effect serves as a poignant reminder of the profound interconnectedness and unpredictability of the world around us. It challenges us to reevaluate our understanding of causality and complexity, urging us to consider the ripple effects of our actions on both natural and human systems. By embracing the lessons of the butterfly effect, we can cultivate a deeper appreciation for the power of small actions and the significance of interconnectedness in shaping our collective future.。
蝴蝶效应在现实生活中的意义英语作文Title: The Significance of the Butterfly Effect in Real LifeThe butterfly effect, a concept originating from chaos theory, illustrates the profound impact small initial changes can have on complex systems over time. In real life, this principle manifests in various aspects, shaping events and outcomes in unexpected ways.One significant realm where the butterfly effect is evident is in weather patterns. A seemingly minor alteration in atmospheric conditions, such as the flapping of abutterfly's wings, can potentially trigger a chain reaction leading to significant weather events elsewhere. This phenomenon underscores the interconnectedness of Earth's climate systems, emphasizing how seemingly insignificant actions can culminate in drastic consequences.Moreover, in social dynamics, small actions or decisions by individuals can catalyze large-scale societal shifts. For instance, a single act of kindness may inspire others to pay it forward, ultimately fostering a culture of compassion within a community. Conversely, a seemingly inconsequential choice, such as a careless remark, can escalate intoconflicts or misunderstandings with far-reaching ramifications.In economics, the butterfly effect is observable in market dynamics. A slight fluctuation in consumer behavior or investor sentiment can cascade through financial networks, influencing stock prices, consumer spending patterns, and overall market stability. This sensitivity to initial conditions underscores the unpredictability of economic systems and the importance of monitoring even the smallest perturbations.Furthermore, the butterfly effect resonates strongly in the realm of technology and innovation. A seemingly minor breakthrough in one field of science or technology cantrigger a cascade of innovations across multiple disciplines, leading to transformative advancements with wide-ranging implications for society. This interconnectedness underscores the collaborative nature of scientific progress and the importance of interdisciplinary research.On a personal level, the butterfly effect highlights the significance of individual agency and responsibility. Our everyday choices, no matter how seemingly inconsequential, contribute to shaping our own lives and the lives of those around us. Whether it be choosing to pursue a new hobby, cultivating healthy habits, or nurturing relationships, each decision sets off a chain of events that can profoundly impact our future trajectories.Moreover, the butterfly effect serves as a reminder of the interconnectedness of all living beings and ecosystems on Earth. A small change in one part of the world can have cascading effects on biodiversity, ecosystems, and theoverall health of the planet. This underscores the importance of environmental stewardship and collective action in addressing pressing global challenges such as climate change and habitat loss.In conclusion, the butterfly effect underscores the interconnectedness, complexity, and inherent unpredictability of the world we inhabit. From weather patterns to social dynamics, economics, technology, and personal choices, small initial changes can have profound and far-reaching consequences. Understanding and appreciating the significance of the butterfly effect empowers us to navigate complexity with mindfulness, responsibility, and a sense of interconnectedness with the world around us.。
a rXiv:as tr o-ph/95144v113Jan1995LANCS-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 (Ω0<1)universe,cosmologists retain only sub-curvature modes (defined 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 field ,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 fluctuation of a scalar field there will be no super-curvature modes in nature.Finally the effect on the cmb of any super-curvature contribution is considered,which generalizes to Ω0<1the analysis given by Grishchuk and Zeldovich in 1978.A formula is given,which is used to estimate the effect.In contrast with the case Ω0=1,the effect contributes to all multipoles,not just to the quadrupole.It is important to find out whether it has the same l dependenceas the data,by evaluating the formula numerically.1IntroductionOn grounds of simplicity,the present energy density Ω0of 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 conflict 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 Ω0=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 Ω0=1is the most natural,because any other value of Ωis time dependent.The preference for Ω0=1is sharpened if,as is widely believed,the hot big bang is preceded by an era of inflation.In that case Ωhas its present value at the epoch when the present Hubble scale leaves the horizon,and for a generic choice of the inflaton potential this indeed implies that Ω0is very close to 1more or less independently ofthe initial value ofΩ.It is also easier for inflation to explain the homogeneity and isotropy of the observable universe ifΩ0=1.On the other hand it is certainly not the case that Ω0=1is an unambiguous prediction of inflation[3,4].The literature on theΩ0<1cosmology is small compared with the enormous output on the caseΩ0=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 effect 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 randomfield.Such afield 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 randomfield 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.IfΩ0=1space isflat 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 afinite region of the universe can be evolved forward in time.Secondly,it gives the most general homogeneous Gaussian randomfield.Instead of the Fourier expansion one can use the entirely equivalent expansion in spherical polar coordinates.IfΩ0<1,the curvature of space defines 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 significantly 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 befinite and any function defined in afinite 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 thecurvature 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 randomfield 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 vacuumfluctuation of the inflatonfield,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 randomfield, one is lead to ask if a failure of the assumption could be detected by observing the cmb anisotropy.We note that forΩ0=1this question has already been discussed by Grishchuk and Zeldovich[10],and we extend their discussion to the caseΩ0<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 randomfield 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 caseΩ<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 caseΩ0<1,and as we have mentioned it is out of touch with the relevant pure mathematics literature where the theory of randomfields is discussed.Thefirst 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 sufficient 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 withΩ<1.In Section3the standard procedure is described, and in the next section it is extended to the super-curvature modes.Inflation 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 scalesIgnoring 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 Einsteinfield equation,the time dependence of a is governed by the Friedmann equation which may be written1−Ω=−K2r ph=Ω−10−1.Even the smallest conceivable valueΩ0≃0.1gives r ph=3.6,so effect 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 isa0r ph=(1−Ω0)−1/2H−10r ph(3) ForΩ0=1it is2H−10,and even forΩ0=0.1it is only3.8H−10.Thus it is not very much bigger than the Hubble distance H−10.3Sub-curvature modesWe are concerned with thefirst 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 differential 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 keeponly 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 define the quantityq2=k2−1(4) 3.1The spherical expansionSpherical coordinates are defined by the line elementd l2=a2[d r2+sinh2r(dθ2+sin2θdφ2)](5) In the region r≪1curvature is negligible and this becomes theflat-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 of a sphere are related by V=A/2. In contrast with theflat-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 differential equation.As in theflat-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 formf(r,θ,φ,t)= ∞0dq lm f klm(t)Z klm(r,θ,φ)(6)whereZ 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)12iq−1sinh rd23These expressions correct some misprints in[19,4].The un-normalised radial functions˜Πkl satisfy a recurrence relation[20]˜Πk,l+2=− (l+1)2+q2 ˜Πkl+(2l+3)coth r˜Πk,l+1(15) and thefirst three functions are˜Πk0=1q (16)˜Πk1=1q (17)˜Πk2=1q (18)The caseΩ=1corresponds to q→∞with qrfixed,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 defined 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 propertiesWe are interested in the stochastic properties of the perturbations,atfixed time.To define 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 defined 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.(Forflat 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 afinite 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 different locations.In this approach one defines 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 arefixed.The correlation function isdefined 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 inflat 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−Ω0)−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 different locations within the smooth patch of the universe that we inhabit,there are two other possibilities. One is that the ensemble corresponds to different smooth patches,which are indeed supposed to exist both in‘chaotic’[24]and bubble nucleation[25,26,27,28]scenarios of inflation. 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 vacuumfluctuation of the inflatonfield,which we consider later.3.3Gaussian perturbationsIt is generally assumed that cosmological perturbations are Gaussian,in the regime where they are evolving linearly.A Gaussian perturbation is normally defined 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 coefficient times a given function,the co-efficient 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 coefficient having an independent Gaussian distribution.Its stochastic properties are completely determined by the mean squares f2n of the coefficients.(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 isf(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 afixed time.The time dependence is trivial if we expand in eigenfunctions of the Laplacian,because each coefficient 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 coefficients 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-efficients 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 define the spectrum of a generic perturbation f by[4]f∗klm f k′l′m′ =2π2k3P f(k)δ(k−k′)δll′δmm′(24) The correlation function is given byξf= ∞0d q2π2k P f(k)sin(qr)k =q d qkrd kk P f(k)(30)Theflat-space limit isξf(0)≡ f2 = ∞0d kξf(0)<rIn order for f2 to be well defined,the spectrum must have appropriate behaviour at q=∞and0.As q→∞one needs P→0.As q→0one needs P→0in theflat case,but only q2P f(k)→0in the curved case.Note that in the curved case the limit q→0does not correspond to infinite 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 toflat space).For future reference,we note that most other authors have used a different definition of the spectrum.This is usually denoted by P f,and it is related to our P f byP f(k)=q(q2+1)2π2 ∞0d qq2P f(k)sin(qr)sinh rd22sinh r sinh(|q|r)4One of us(DHL)is indebted to R.Gott and P.J.E.Peebles for pointing out thisfact.˜Πk1=1|q|(40)˜Πk2=1|q|(41)At large r the super-curvature modes go like exp[−(1−|q|)r].Because the volume element is d V=sinh2r sinθdrdθdϕthe 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 anyfinite 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 defined 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 define the corresponding spectrum by analogy with Eq.(23),f klm f∗k′l′m′ =2π2k P f(k)sinh(|q|r)k P f(k)(45) Unified expressions including all modesThe 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.Unified expressions including all modes on an equal footing would use k in the mode expansion,so defining new coefficients˜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′ =2π2k 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 flat-space case,where the contribution from a mode with k ≪1gives a correlation length of order 1/k .The difference can be understood in terms of the different 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 givesrd r ≃−k 2r V (r )T =w .e +∞ l =2+l m =−la 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 nowon.If the perturbations in the universe are Gaussian,the real and imaginary part of eachmultipole 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 identified 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 imaginarypart 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 different hypotheses about the C l.Any hypothesis can be made consistent with observation by supposing that the region around us is sufficiently atypical.The surface of last scattering of the cmb is practically at the particle horizon,whose coordinate distance isη0with sinh2η0/2=Ω−10−1.An angleθsubtends at this surface a coordinate distance d given by[23]θ=12(a0H0Ω0d)(55)Spatial curvature is negligible when d≪1,corresponding toθ≪30(1−Ω0)−1/2Ω0degrees(56) A structure with angular sizeθradians is dominated by multipoles withl∼1/θ(57) one expects that spatial curvature will be negligible for the multipolesl≫2√Ω0(58)This is the regime l≫20ifΩ0=0.1,and the regime l≫6ifΩ0=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 effect,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 caseΩ0=1[46],and recently it has been advocated also for the case Ω0<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 significant lower bound onΩ0[47].5.1The curvature perturbationThe curvature perturbation is conveniently characterised by a quantity R,which is defined in terms of the perturbation in the curvature scalar by 54(k 2+3)R klm /a 2=δR (3)klm(59)In the limit Ω→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 Ω(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 Ωbreaks away from 1.After that it has the time dependence R klm =F ˆRklm where ˆR klm is the early time constant value and F =5sinh 2η−3ηsinh η+4cosh η−4k 2+3δρklm5R klm (63)For Ω0=1this reduces to a 2H 2ρ=2k ≫.0075The quantity R was called φm by Bardeen who first 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 Ω=1,it is the ζof [50],and three times the ζof [51].。
