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## **Meet the editor**

Professor Abraão Jessé Capistrano de Souza is a graduate of Physics awarded by Federal University of Pará, Brazil (2004) and obtained his master's degree in Physics (2006) and doctorate in Theoretical Physics from Universidade de Brasília, Brazil (2009). His research focused on problems in mathematical physics, quantum fields, astrophysics, cosmology, and gravitation. Also,

he is performing studies in rocket science and robotics focused on undergraduate engineering projects as well as activities in teaching physics and astronomy to a broader audience through institutional extension projects. He is currently a full professor at Federal University of Latin American Integration and collaborator researcher at Casimiro Montenegro Filho Astronomy Center (Technological Park of Itaipu).

## Contents

#### Chapter 10 **Semi-Analytic Techniques for Solving Quasi-Normal Modes 197** Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell

## Preface

Chapter 7 **The Impact of Baryons on the Large-Scale Structure of the**

Chapter 8 **Cosmological Consequences of a Quantum Theory of Mass**

Chapter 9 **Deformed Phase Space in Cosmology and Black Holes 177**

Chun-Hung Chen, Hing-Tong Cho and Alan S. Cornell

E.A. Mena-Barboza, L.F. Escamilla-Herrera, J.C. López-Domínguez

Weiguang Cui and Youcai Zhang

**Universe 137**

**VI** Contents

**and Gravity 159** Brian Albert Robson

and J. Torres-Arenas

Chapter 10 **Semi-Analytic Techniques for Solving Quasi-Normal Modes 197**

The modern cosmology has been turned into an outstanding field of active research through the years. Historically, since its origin as a search of understanding the universe came from a mythological, religious, or philosophical point of view, the cosmology depended from es‐ sentially what culture, society, and/or what time we are referring to. It was not considered strictly speaking as a *natural science* rather than a philosophical thinking about the world we are living in. The breakthrough occurred in the sixteenth century with Galileo Galilei and his enhanced version of telescope and the finding of new ways of not only thinking about the natural world but new ways *to see* the world. Accordingly, the scenario radically changed in the beginning of the twentieth century. To get this far, it is important to note that this development has been long in the making. This is a fine example of the scientific *way of thinking* and its *modus faciendi* as a human construction of knowledge. More importantly, any advance comes from a collective work and collaborative spirit since our generation is em‐ bedded in the greatest challenging issues of scientific history. For instance, we cite seminal works from Henrietta Leavitt with the observations of variable stars, Albert Einstein's gen‐ eral relativity, Friedman's metric launching the basics of the standard cosmological model of an expanding universe, Vesto Slipher with the measurements of radial velocities of galaxies and the use of spectroscopy for astronomical observations that led to the finding of galactic redshift, Edwin Hubble's expanding universe, and Fritz Zwicky's anomalies in velocities of the clusters of galaxies that were not predicted by Newtonian theory of gravity. Today, such anomaly is called *the dark matter problem*. This was reinforced by Vera Rubin's observations in spiral galaxies in the 1970s also showing a huge discrepancy of Newton's law to observa‐ tions and further developments in both theoretical and observational methods from genera‐ tions of scientists.

Particularly, in the last two decades, we have an intense growth of scientific interest in cos‐ mology due to the improvement of observational instruments of astronomical science, which motivated theoretical advances from the necessity to deal with the resulting huge amount of scientific data that should be taken into account, analyzed, and properly justified. Accordingly, cosmology as a discipline turned to be a scientific ground-based branch of physics. Today, we have more scientific data in modern cosmology (and also astronomy and astrophysics) than we could get rid of it, which makes the present days an exciting era for scientific knowledge.

In 1998, we have the discovery that our universe is not only expanding but expanding in an accelerated way. The currently simpler explanation to this effect is that the universe is fulfil‐ led with a negative pressure fluid (regarded as fluctuations of the quantum vacuum energy density) producing some sort of *dark energy* that drives the universe to speed up. Of course, this is just one of the explanations for the accelerated regime of the universe, which has been

confirmed again by the recent Planck mission (European space agency) in their last report in 2015. Accordingly, more effort is currently made to understand the physics of the so-called dark sector. Intriguingly enough, roughly 4% of the universe is barely known, and the rest is not properly explained by the standard particle physics or cosmology, and new fresh modes of perception are needed.

More recently, we had an incredible technological advance from the discovery of gravita‐ tional waves in the Laser Interferometer Gravitational-Wave Observatory (LIGO) confirm‐ ing Einstein's prediction. It has given to scientific community a brand new open arena for physics and several areas, particularly on the understanding of black hole physics, another challenging problem on gravitational physics and astrophysics.

The title of this book could be no other. The chapters presented discuss different aspects of cosmology pointed out by some experts in different areas. This book is divided into three sections with related chapters. The first section is devoted to cultural aspects of cosmology in which, as a human construction, it influenced other human activities such as arts, philoso‐ phy, and politics during the human history. The subsequent chapters are intended to dis‐ cuss technical scientific advances in the field with capital themes today as the problems of the physics of dark sector (dark matter and dark energy problems), black holes, galaxies, large structure formation, and particles. In fact, it shows our endless searching for better un‐ derstanding of the universe, and it remembers the human odyssey that is a journey of knowledge as a legacy for next generations. It has been a great opportunity to work with InTech's editorial team and such an honor to read all the chapters proposed by professional researchers in the field.

> **Abraão Jessé Capistrano de Souza,** Federal University of Latin American Integration, Casimiro Montenegro Filho Astronomy Center (Technological Park of Itaipu), Brazil

## **Cosmology and Culture**

confirmed again by the recent Planck mission (European space agency) in their last report in 2015. Accordingly, more effort is currently made to understand the physics of the so-called dark sector. Intriguingly enough, roughly 4% of the universe is barely known, and the rest is not properly explained by the standard particle physics or cosmology, and new fresh modes

More recently, we had an incredible technological advance from the discovery of gravita‐ tional waves in the Laser Interferometer Gravitational-Wave Observatory (LIGO) confirm‐ ing Einstein's prediction. It has given to scientific community a brand new open arena for physics and several areas, particularly on the understanding of black hole physics, another

The title of this book could be no other. The chapters presented discuss different aspects of cosmology pointed out by some experts in different areas. This book is divided into three sections with related chapters. The first section is devoted to cultural aspects of cosmology in which, as a human construction, it influenced other human activities such as arts, philoso‐ phy, and politics during the human history. The subsequent chapters are intended to dis‐ cuss technical scientific advances in the field with capital themes today as the problems of the physics of dark sector (dark matter and dark energy problems), black holes, galaxies, large structure formation, and particles. In fact, it shows our endless searching for better un‐ derstanding of the universe, and it remembers the human odyssey that is a journey of knowledge as a legacy for next generations. It has been a great opportunity to work with InTech's editorial team and such an honor to read all the chapters proposed by professional

**Abraão Jessé Capistrano de Souza,**

(Technological Park of Itaipu),

Brazil

Federal University of Latin American Integration, Casimiro Montenegro Filho Astronomy Center

challenging problem on gravitational physics and astrophysics.

of perception are needed.

VIII Preface

researchers in the field.

## **The Importance of Cosmology in Culture: Contexts and Consequences**

Nicholas Campion

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/67976

#### **Abstract**

Scientific cosmology is the study of the universe through astronomy and physics. However, cosmology also has a significant cultural impact. People construct anthropological cosmologies (notions about the way the world works), drawing in scientific theories in order to construct models for activities in disciplines, such as politics and psychology. In addition, the arts (literature, film and painting, for example) comment on cosmological ideas and use them to develop plot lines and content. This chapter illustrates examples of such work, arguing that scientific cosmology should be understood as a significant cultural influence.

**Keywords:** cosmology, culture, politics, psychology, literature, film, space travel

#### **1. Introduction**

Modern scientific cosmology is valuable in itself for what it reveals about the nature of the cosmos we inhabit [1]. It is a demonstration of the power of modern science to transform our understanding of who we are and where we came from. However, most cosmologists focus on scientific questions and are not fully aware of the impact of cosmological theories on culture, including politics and the arts. This chapter introduces this wider context on the basis that both scientists and the public should be aware of the broader importance of their work and its influence on the way we think. Cosmologists often rely on the fascination the subject brings: as Rowe observed in his textbook way back in 1968, 'In the fields of astronomy and cosmology we live in a period of excitement' [2]. Cosmology therefore both impacts culture and is described and represented by it. This chapter explores some ways in which this happens. As Muriel Rukeyser wrote, 'The universe is made of stories, not of atoms' [3]; see also Impey [4].

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

If we select four fundamental causes of changes in our perceptions of the world in the last century, then they would be first relativity, second quantum mechanics, third the expanding universe and fourth, the space programme. The first three date from a fairly narrow time band, if we date special relativity from 1905, general relativity from 1915, that the universe is expanding and is much bigger than previous thought from Edwin Hubble's publications from 1924 to around 1930 and quantum mechanics from Niels Bohr and Werner Heisenberg's formulation of the Copenhagen interpretation in 1925–1927 [5]. This epic revision of scientific knowledge of underlying structures of the universe was therefore concentrated into just a quarter of a century. The dramatic period of the human space programme was concentrated into just over 8 years from the first human space flight in 1961 to the Moon landing in 1969.

All have fundamentally altered the way that we think about life here on Earth. Often these changes are taken for granted. For example, mobile phone technology, dependent as it is on satellite networks, is transforming not only the social lives of teenagers in the west, but also the economic muscle of poor farmers across the third world. Meanwhile, super‐fast quantum computing makes use of phenomena such as entanglement and is driving the development of artificial intelligence, and hence of robotics. The implications for society over the next few decades are potentially enormous. The most important conclusion to be drawn from this combination of revolutionary changes is the role of the observer: as the basis of differing perspectives of time and space in relativity, an influence on the world (at least, at the sub‐atomic level) in quantum mechanics, and the witness for the first time, of the spherical earth, hanging in space, in photographs taken by Apollo astronauts in 1968. Such ideas and experiences have decisively underpinned modern ideas that one person's complete individual experience or perception is as equally valid as anyone else's. Einstein is held particularly responsible for these ideas [6, 7] as a result of popular equations between relativity on the one hand, and cultural relativism (the idea that no one culture is superior or inferior to another) on the other. Moral relativism (the idea that no one culture is morally superior or inferior to another) is controversial and widely rejected, but cultural relativism does have beneficial scholarly consequences. This is especially the case in the new field of cultural anthropology in which academic rigour requires that in order to better understand other cultures, researchers must abandon any idea that one culture is superior or inferior to another.

#### **2. Defining cosmology**

The term cosmology can be traced to the 1730s, although its appearance in a scientific sense dates from only after the Second World War [8]. The *logos*, which is the root of 'logy', means 'an account', so that, as a preliminary working definition, cosmology is simply 'an account of the cosmos'. The primary Latin equivalent of the Greek *Kosmos* is *Universus*, from *Unus verto*, or 'changing into one', thereby suggesting unity. We can divide the definitions of cosmology into two: the scientific and the anthropological. The scientific are perhaps the more familiar, but even here there is variation. Scientific definitions range from the narrow, such as 'the study of the universe' [9], to the broad ('the science, theory or study of the universe as an orderly system, and of the laws that govern it; in particular, a branch of astronomy that deals with the structure and evolution of the universe') [10]. The idea that cosmology is synonymous with astronomy is widespread [2, 11]. Yet the relationship between astronomy and cosmology is not settled, and Hawley and Holcomb [9] argue that cosmology is a separate discipline, which combines features of both astronomy and physics. There are other places where disciplines overlap. For example, astronomy has to include geophysics precisely because the Earth itself is in space [12].

Anthropological cosmologies are based on the proposition that ideas about the cosmos are integral part of human cultural and social systems. For example, the archaeologist Timothy Darvill talks of a cosmology as being, 'The world view and belief system of a community based upon their understanding of order in the universe' [13]. George Gumerman and Miranda Warburton argued that '… to truly comprehend a culture we must have some sense of its cosmology – the group's conception of themselves in relation to the heavens' [14]. And without diminishing the scientific status of modern cosmology, ideas do not come from theory and experiment alone, but can be inspired by wider cultural influences, as Holton [15] illustrates in relation to Einstein's reading and education before he formulated the theory of special relativity. Heisenberg [16] actually argued that science and art are parallel attempts to describe the world, and may both be part of a wider cultural picture. Bell [17] then talks of 'complex subsystems of cosmic exchange' which underpin mundane systems of behaviour, such as socio‐economic exchange, but are designed to reinforce individual and social existence within the cosmos.

#### **3. Defining cosmology**

If we select four fundamental causes of changes in our perceptions of the world in the last century, then they would be first relativity, second quantum mechanics, third the expanding universe and fourth, the space programme. The first three date from a fairly narrow time band, if we date special relativity from 1905, general relativity from 1915, that the universe is expanding and is much bigger than previous thought from Edwin Hubble's publications from 1924 to around 1930 and quantum mechanics from Niels Bohr and Werner Heisenberg's formulation of the Copenhagen interpretation in 1925–1927 [5]. This epic revision of scientific knowledge of underlying structures of the universe was therefore concentrated into just a quarter of a century. The dramatic period of the human space programme was concentrated into just over 8 years from the first human space flight in 1961 to the Moon landing in 1969.

All have fundamentally altered the way that we think about life here on Earth. Often these changes are taken for granted. For example, mobile phone technology, dependent as it is on satellite networks, is transforming not only the social lives of teenagers in the west, but also the economic muscle of poor farmers across the third world. Meanwhile, super‐fast quantum computing makes use of phenomena such as entanglement and is driving the development of artificial intelligence, and hence of robotics. The implications for society over the next few decades are potentially enormous. The most important conclusion to be drawn from this combination of revolutionary changes is the role of the observer: as the basis of differing perspectives of time and space in relativity, an influence on the world (at least, at the sub‐atomic level) in quantum mechanics, and the witness for the first time, of the spherical earth, hanging in space, in photographs taken by Apollo astronauts in 1968. Such ideas and experiences have decisively underpinned modern ideas that one person's complete individual experience or perception is as equally valid as anyone else's. Einstein is held particularly responsible for these ideas [6, 7] as a result of popular equations between relativity on the one hand, and cultural relativism (the idea that no one culture is superior or inferior to another) on the other. Moral relativism (the idea that no one culture is morally superior or inferior to another) is controversial and widely rejected, but cultural relativism does have beneficial scholarly consequences. This is especially the case in the new field of cultural anthropology in which academic rigour requires that in order to better understand other cultures, researchers must

The term cosmology can be traced to the 1730s, although its appearance in a scientific sense dates from only after the Second World War [8]. The *logos*, which is the root of 'logy', means 'an account', so that, as a preliminary working definition, cosmology is simply 'an account of the cosmos'. The primary Latin equivalent of the Greek *Kosmos* is *Universus*, from *Unus verto*, or 'changing into one', thereby suggesting unity. We can divide the definitions of cosmology into two: the scientific and the anthropological. The scientific are perhaps the more familiar, but even here there is variation. Scientific definitions range from the narrow, such as 'the study of the universe' [9], to the broad ('the science, theory or study of the universe as an orderly system, and of the laws that govern it; in particular, a branch of astronomy that deals with the

abandon any idea that one culture is superior or inferior to another.

**2. Defining cosmology**

4 Trends in Modern Cosmology

The philosopher Terry Eagleton concluded, '"Culture" is said to be one of the two or three most complex words in the English language' [18]. A good working definition, though, is offered by the anthropologist Clifford Geertz, who argued that culture is 'an historically transmitted pattern of meanings embodied in symbols, a system of inherited conceptions expressed in symbolic forms by means of which men communicate, perpetuate and develop their knowledge of and attitudes toward life' [19]. The influence of cosmology on culture then becomes a matter of exploring its impact on political and religious ideas, and its use in the arts, perhaps mainly in literature, painting and film. Michael Rowan‐Robinson wrote that 'the intellectual horizon of the human race at any time has always been inextricably bound up with the scale of the universe… there can be little doubt that a people's perceived scale of the universe must play a fundamental role in its culture and consciousness' [20]. He could equally have said that culture and consciousness are bound up with conceptions of the universe as a whole. It was Einstein himself who made the case for the cosmologist intervening in culture. In 1936 he wrote, 'the physicist cannot simply surrender to the philosopher the critical contemplation of the theoretical foundations' of the universe, and 'the critical thinking of the physicist cannot possibly be restricted to examination of his own field. He cannot proceed without considering critically a much more difficult problem, the problem of analysing the nature of everyday thinking' [21]. Mostly the cosmologist does not intervene directly in modern culture. Instead other people interpret and represent cosmological ideas.

### **4. Politics**

Cosmology has had a substantial impact on political theory and continues to do so. We can therefore talk about 'political cosmology' [22]. The earliest recorded societies used the stars or planets to represent their rulers or guide their actions, functions which tended to overlap with religion. The Egyptians and Inca saw their rulers embodied in the Sun, the Chinese in the Pole Star, and in Babylon the chief god, Marduk, was represented by Jupiter. On a practical level, most pre‐modern cultures structured their religious sanctuaries, and sometimes their urban communities, according to their conceptions of the cosmos. Hence we may speak of 'cosmopolises' or 'cosmograms'. A 'cosmic state' is one in which the entire state is organized to embody the structure of the cosmos.

The significant step into modelling politics on the cosmos as an organised system was taken by the Greek philosopher Plato (ca. 428/7?–348/7). Plato's cosmos emanated out of a single consciousness which existed in a realm of unchanging eternity. The material world was then an imperfect representation of the world of Ideas (Plato spoke of the physical world being embedded in the world‐soul) and was governed by time, the mathematically regulated, harmonious rhythms measurable by the motions of the planets. Plato's mathematical universe provides a rationale for scientific speculation to the present day [23, 24].

Plato advocated an education emphasising such subjects as music, mathematics and geometry, and a political system based on rule by philosophers, all designed to harmonise society with the cosmos, for the common good. This concept of the perfect society underpins the entire history of utopianism down to the present day. There were two main consequences of Plato's system, with consequences down to the present day. First, the perfect ruler was envisioned as the Philosopher King, whose right to rule was justified by his wisdom and understanding of cosmic principles. Second, the system tended to be authoritarian because, being founded on cosmic principle, it could brook no opposition. Plato's ideas were revived in Renaissance Europe and were to become extremely influential. The notion of human history as the progressive unfolding of the world soul towards a final, perfect condition was central to the ideas of Georg Friedrich Hegel (1770–1831). Hegel's influence on Karl Marx (1818–1883) led the twentieth century philosopher of science, Popper [25], to see Plato's thought as the foundation of modern totalitarianism: where Plato influenced Marx, via Hegel, was in the notion that history has an inescapable trajectory, founded in the structure of the cosmos itself. For revolutionary Marxists, such as Lenin, Stalin and Mao Tse‐tung, it was then inconceivable that anyone could oppose their rule, for to do so was to oppose the cosmos.

Separate to the platonic strand in European political cosmology, the astronomical discoveries from Copernicus onwards all helped shape western politics. Copernicus' argument that the Sun, not the Earth, is the centre of the universe (or the solar system as we would now say) was attached to the ancient idea that the Sun is associated with kings. It was then used to support claims that, just as the entire universe orbits the Sun, so the whole of society orbits the king [26]. Propaganda in support of absolute monarchy then reached its height in the iconography of the French king Louis XIV. Such authoritarian ideas were directly countered by what I have called Political Newtonianism [22]. This held that, just as Newton had argued that one law governed the whole universe, so the same principle must apply to terrestrial affairs and one law must govern the whole of human society. In principle, then, all people were equal, and there was no justification for monarchy. Such ideas were influential among both the American revolutionaries of 1776 [27] and the French in 1789.

Newtonianism was taken one step further by Auguste Comte (1798–1857), the founder of sociology. Adding Galileo and Kepler to Newton as sources of authority, Comte [28] argued that if the entire universe was a mathematically regulated mechanism, so human society must also be governed by the same principle. If planets moved in mathematically determined patterns, Comte reasoned, so must people. By collecting and analysing data on human behaviour, Comte concluded, the same laws that controlled the wider universe should be discovered in human affairs. And in turn the state, governed by experts who were the modern equivalent of Plato's philosophers, could be managed for the good of all. This remains the foundation of twenty‐first century sociology.

There have been a few attempts, for example, to identify Einsteinian relativity either as a form of political discourse, or to draw political implications from it. As far as the former is concerned, I refer to the French feminist and social theorist Luce Irigaray who has identified the theory of relativity as a political rather than scientific formula [29]. Sokal and Bricmont [28], meanwhile, noted how the notion of relativity in time and space was used by postmodern theorists in order to advocate cultural relativity on the grounds that, if the universe has no single centre, neither does culture.

The anthropologist Falzon [30] has defended multi‐sited ethnography against the charge of lack of depth by arguing that it takes into account shifting perceptions of space and time. The anthropologist Marcus [31] refers to 'space‐time compression', in which the essential difference between space and time contracts in light of the recognition that both are socially produced as a result of what Falzon calls 'a product of interrelations'; Marcus derived his understanding from special relativity, and so directly from his understanding of Einstein. Elsewhere, Einstein's call for humanity in general to take on the implications of the new cosmology has been used to advocate a collaborative global order in which international problems are solved thought global institutions rather than war [32].

#### **5. Psychology**

**4. Politics**

6 Trends in Modern Cosmology

to embody the structure of the cosmos.

Cosmology has had a substantial impact on political theory and continues to do so. We can therefore talk about 'political cosmology' [22]. The earliest recorded societies used the stars or planets to represent their rulers or guide their actions, functions which tended to overlap with religion. The Egyptians and Inca saw their rulers embodied in the Sun, the Chinese in the Pole Star, and in Babylon the chief god, Marduk, was represented by Jupiter. On a practical level, most pre‐modern cultures structured their religious sanctuaries, and sometimes their urban communities, according to their conceptions of the cosmos. Hence we may speak of 'cosmopolises' or 'cosmograms'. A 'cosmic state' is one in which the entire state is organized

The significant step into modelling politics on the cosmos as an organised system was taken by the Greek philosopher Plato (ca. 428/7?–348/7). Plato's cosmos emanated out of a single consciousness which existed in a realm of unchanging eternity. The material world was then an imperfect representation of the world of Ideas (Plato spoke of the physical world being embedded in the world‐soul) and was governed by time, the mathematically regulated, harmonious rhythms measurable by the motions of the planets. Plato's mathematical universe

Plato advocated an education emphasising such subjects as music, mathematics and geometry, and a political system based on rule by philosophers, all designed to harmonise society with the cosmos, for the common good. This concept of the perfect society underpins the entire history of utopianism down to the present day. There were two main consequences of Plato's system, with consequences down to the present day. First, the perfect ruler was envisioned as the Philosopher King, whose right to rule was justified by his wisdom and understanding of cosmic principles. Second, the system tended to be authoritarian because, being founded on cosmic principle, it could brook no opposition. Plato's ideas were revived in Renaissance Europe and were to become extremely influential. The notion of human history as the progressive unfolding of the world soul towards a final, perfect condition was central to the ideas of Georg Friedrich Hegel (1770–1831). Hegel's influence on Karl Marx (1818–1883) led the twentieth century philosopher of science, Popper [25], to see Plato's thought as the foundation of modern totalitarianism: where Plato influenced Marx, via Hegel, was in the notion that history has an inescapable trajectory, founded in the structure of the cosmos itself. For revolutionary Marxists, such as Lenin, Stalin and Mao Tse‐tung, it was then inconceivable

provides a rationale for scientific speculation to the present day [23, 24].

that anyone could oppose their rule, for to do so was to oppose the cosmos.

Separate to the platonic strand in European political cosmology, the astronomical discoveries from Copernicus onwards all helped shape western politics. Copernicus' argument that the Sun, not the Earth, is the centre of the universe (or the solar system as we would now say) was attached to the ancient idea that the Sun is associated with kings. It was then used to support claims that, just as the entire universe orbits the Sun, so the whole of society orbits the king [26]. Propaganda in support of absolute monarchy then reached its height in the iconography of the French king Louis XIV. Such authoritarian ideas were directly countered by what I have called Political Newtonianism [22]. This held that, just as Newton had argued that one law The consequences of Newtonianism (the belief that the entire universe is mathematically regulated) permeate western thought wherever there has been a search for a universal law based on supposedly hard data. Psychology is a prime example. When the word 'psychology' was coined around 1800, it was thought that, since Newtonian science explained everything that exists and occurs in the material world, there could and should be just one science explaining what exists and occurs in the psychological world. As Gilbert Ryle wrote, '"Psychology" was supposed to be the title of the empirical study of "mental phenomena"', a counterpart to Newtonian celestial mechanics [33]. Concepts such as normality and deviation have dominated some of the major schools of western psychology, their roots in Newtonian cosmology's devotion to predictable order unrecognised and forgotten. The measurement and mathematical analysis of the human mind then became the basis of much psychiatry and academic psychology. Even the notion developed by Sigmund Freud (1856–1939), that psychic—or psychological—material can be 'repressed', as if by some kind of downward pressure, is derived from Newtonian mechanics through the influential German physicist Hermann von Helmholtz (1821–1894) [34]. The goal of Freudian psychoanalysis is for the patient to become aware of such repressed material through the so‐called talking cure, the conversations which take place during sessions with the psychoanalyst, thus releasing in downward pressure.

Freud's student C.G. Jung opened a radically different strand of thought in modern psychology which is highly influential in many schools of psychotherapy and counselling practiced in society as a whole, although usually outside the academic system. Jung revived the Platonic theory that everything in the world is a manifestation of an original pure idea or archetype. Jung's system, known as analytical psychology, adheres to a kind of archetypal philosophy in which all psychological types correspond to an archetype, such as the eternal youth (the puer aeternus), anima (female principle) or senex (wise old man), which exist in the collective unconscious, Jung's update of the Platonic world‐soul. The aim of Jung's therapeutic model is for the individual to become truly themselves by recognising the role of the archetypes in their lives and, in effect, understanding their true connections to the cosmos. The idea that one can become one who truly is also relates back to Aristotelian cosmology in which it was thought that the four elements (fire, earth, air and water) all try to find their natural place in the world. This, Aristotle thought, was why flames go up to the sky, where fire belongs, and water falls to the ground, because that is where it finds its natural home. In Aristotelian politics, kings are at the top of society and peasants at the bottom, because that is the natural state of affairs; in Aristotelian psychology every individual then has a natural way of being. Jung, though, was equally concerned with the latest science, and formed a collaboration and friendship with the quantum physicist Wolfgang Pauli (1900–1958) [35]. Together they formulated the concept of synchronicity by which meaningful events are connected because they take place at the same time, without any causal connection [36]. Newtonian psychology—the belief that all mental states can be measured—survives in university departments and psychiatry. But in the wider world, where increasing numbers of people seek counselling and psychotherapy, Einstein is taken as the inspiration for the argument that therapists and analysts must be 'less concerned with the basic nature of time and more with the human experience of it' [37].

#### **6. Literature and film**

One of the major genres of writing in western culture goes under the name 'celestial journey literature', derived from ancient texts on the soul's journey. The soul's journey was secularised in Dante's (1265–1321) 'Divine Comedy' [38]. Inspired by Plato's myth of Er, Dante is guided by the poet Virgil and his love, Beatrice, through the spheres of Hell, Purgatory and Paradise (Hell and Purgatory are structured in spheres analogous to the spheres on which the planets and stars orbit). The last great example of the celestial journey of the soul or a dream world was Johannes Kepler's 'Somnium', an account of lunar astronomy written in 1608 but published in 1634, and sometimes referred to as the first work of science fiction. The genre took a decisive step forward in Francis Godwin's 'Man in the Moon' [39], published in 1638. Like Dante, Godwin used his story to describe the structure of the cosmos, now, after Johannes Kepler and Galileo, rejecting the planetary spheres and challenging the Aristotelian idea that all things have their natural place. Godwin departed from the old idea of a journey of the soul or a dream world. Instead, his hero, Gonzales, flies to the Moon carried by giant geese. From Godwin onwards the celestial journey becomes physical, arriving at the Moon rocket in Jules Verne's 1865 novel, 'From the Earth to the Moon'. Verne's book was one of the inspirations for what may be the first space film, Georges Méliès' 'Le Voyage dans la Lune'—in English 'A Trip to the Moon' after which astronomy and cosmology have a regular presence in film culture [40]. In Méliès' film, the astronomers encounter the inhabitants of the Moon, known as Selenites, in what is clearly a parable for European colonialism: the film was released midway through the so‐called 'Scramble for Africa', the final face of the European take‐over of Africa from the 1880s to 1914. From Méliès on, the major celestial journey novels have often been filmed. Perhaps the most famous is Stanley Kubrick's film of Arthur C. Clarke's '2001: A Space Odyssey'. Clarke's metaphysical story is vividly portrayed first through the transition from ape to human, and then, in the final scene, the transformation of the dying astronaut into the star child. Accompanied by the stirring music of Wagner's 'Also sprach Zarathustra', Kubrick created a vivid evocation of both ancient beliefs in the soul's ascent to the stars and modern ideas that human destiny may take us to realms beyond our current imagination.

devotion to predictable order unrecognised and forgotten. The measurement and mathematical analysis of the human mind then became the basis of much psychiatry and academic psychology. Even the notion developed by Sigmund Freud (1856–1939), that psychic—or psychological—material can be 'repressed', as if by some kind of downward pressure, is derived from Newtonian mechanics through the influential German physicist Hermann von Helmholtz (1821–1894) [34]. The goal of Freudian psychoanalysis is for the patient to become aware of such repressed material through the so‐called talking cure, the conversations which take place during sessions with the psychoanalyst, thus releasing in downward pressure.

Freud's student C.G. Jung opened a radically different strand of thought in modern psychology which is highly influential in many schools of psychotherapy and counselling practiced in society as a whole, although usually outside the academic system. Jung revived the Platonic theory that everything in the world is a manifestation of an original pure idea or archetype. Jung's system, known as analytical psychology, adheres to a kind of archetypal philosophy in which all psychological types correspond to an archetype, such as the eternal youth (the puer aeternus), anima (female principle) or senex (wise old man), which exist in the collective unconscious, Jung's update of the Platonic world‐soul. The aim of Jung's therapeutic model is for the individual to become truly themselves by recognising the role of the archetypes in their lives and, in effect, understanding their true connections to the cosmos. The idea that one can become one who truly is also relates back to Aristotelian cosmology in which it was thought that the four elements (fire, earth, air and water) all try to find their natural place in the world. This, Aristotle thought, was why flames go up to the sky, where fire belongs, and water falls to the ground, because that is where it finds its natural home. In Aristotelian politics, kings are at the top of society and peasants at the bottom, because that is the natural state of affairs; in Aristotelian psychology every individual then has a natural way of being. Jung, though, was equally concerned with the latest science, and formed a collaboration and friendship with the quantum physicist Wolfgang Pauli (1900–1958) [35]. Together they formulated the concept of synchronicity by which meaningful events are connected because they take place at the same time, without any causal connection [36]. Newtonian psychology—the belief that all mental states can be measured—survives in university departments and psychiatry. But in the wider world, where increasing numbers of people seek counselling and psychotherapy, Einstein is taken as the inspiration for the argument that therapists and analysts must be 'less concerned

with the basic nature of time and more with the human experience of it' [37].

One of the major genres of writing in western culture goes under the name 'celestial journey literature', derived from ancient texts on the soul's journey. The soul's journey was secularised in Dante's (1265–1321) 'Divine Comedy' [38]. Inspired by Plato's myth of Er, Dante is guided by the poet Virgil and his love, Beatrice, through the spheres of Hell, Purgatory and Paradise (Hell and Purgatory are structured in spheres analogous to the spheres on which the planets and stars orbit). The last great example of the celestial journey of the soul or a dream world was Johannes Kepler's 'Somnium', an account of lunar astronomy written in 1608 but published in 1634,

**6. Literature and film**

8 Trends in Modern Cosmology

While celestial journey films can be enjoyed as simple adventures, they often contain deeper meanings. The 1951 movie 'The Day the Earth Stood Still' exploited the current public interest in Flying Saucers. Released at a time when the Cold War was reaching its height with the conflict in Korea, the story featured a wise alien who arrived from space in order to reveal to humanity the error of its ways. The rejection of the alien's words of wisdom presented a gloomy view of humanity as incapable of solving its problems. Cosmology, through film, then becomes a means of commenting on societal change. 'Star Trek', the biggest celestial journey TV franchise of them all, was launched in 1966. 'Star Trek' was altogether more sophisticated than 'Lost in Space' and was entirely more optimistic. It is set in a utopian future in which there is one world government, collaborating with other worlds through United Federation of Planets, and money has been abolished. In the crew's adventures, the European voyages of the fifteenth and sixteenth centuries are replayed in a universe of an infinite number of galaxies, except that now alien cultures are to be respected and preserved rather than conquered and destroyed. The values espoused by the Federation were American: freedom from tyranny, freedom of expression and respect for minorities. Compared to 'Star Trek', the blockbuster film franchise, 'Star Wars', launched in 1976 and still going strong forty years later, projects into space a simpler version of the endless struggle for freedom against an evil empire, very much an update of anti‐Nazi war films. In all such cases the cosmos is seen as a blank slate, a tabula rasa, on which human concerns are imposed.

There is a constant strand of literary comment on cosmology in the nineteenth century. Edgar Allan Poe (1809–1849), often known as the author of the first detective novel, wrote a remarkable work which he called 'Eureka' [41], deliberately suggesting an imaginative breakthrough in the understanding of the cosmos. Poe realised that in a Newtonian universe the stars are likely to collapse in on each other, and that therefore the universe must be evolving [42]. Thomas Hardy (1840–1928) was as fascinated by cosmology as Poe, but unlike him lived to see the publication of Darwin's 'Origin of Species' in 1959 (when he was just 19). For Hardy evolution was a reality. He combined his encyclopaedic knowledge of myth, astronomy and cosmology into a 'moral astrophysics' [43] which provided the background for the individual conflicts and tragedies in novels such as 'Far from the Madding Crowd' (1874) and 'Tess of the d'Urbervilles' (1891/1892). Virginia Woolf (1882–1941) used Edwin Hubble's discovery of the expanding universe in the 1920s as a metaphor for personal and political insecurity in the 1930s. In Woolf's view, as we all live together on the same delicate, vulnerable planet in a vastness of space, it is incumbent upon us to live together rather than fight [44]. The Marxist playwright Bertolt Brecht (1898–1956) looked back to an earlier cosmology but equally wanted to illustrate a modern point in his 1938 play, 'Galileo'. His portrayal of Galileo as the heroic intellectual defending Copernicus, struggling against an obscurantist Inquisition (inaccurate because many in the senior Catholic hierarchy were Copernicans), was an allegory of the revolutionary struggles of the 1930s.

One of the other familiar tropes derived from modern cosmology is time travel, a topic rarely dealt with in 'Star Trek', in spite of the regular use of faster‐than‐light travel. There is now a considerable literature which draws on Gerald Feinberg's 1967 paper 'Possibility of faster‐ than‐light particles', which proposed the existence of the tachyon [45]. This hypothetical particle, the tachyon, might as Martin Rees [46], says alter the order of events, if a signal from a tachyon arrived before it was sent. The genre's earliest notable example was Wells' novel 'The Time Machine' (1895) [47]. Wells coined the phrase time machine to describe a time‐travelling vehicle which moved because the fourth dimension was of time rather than space. Wells was a utopian socialist and his main preoccupation was to explore varieties of human society, considering whether progress inevitably resulted in human improvement: it's clear that he didn't think that this was the case, and that ignorance and superstition could easily flourish in the future. Neither does 'Dr Who', the most successful TV time travel franchise, explain how it is possible to travel to the distant past or future. The time machine, the TARDIS (short for Time and Relative Dimensions in Space), is bigger inside than outside and references Einstein by referencing relativity in its name. The Doctor himself is increasingly represented as a lonely figure, destined to exist in perpetual sadness caused by the death or departure of his companions.

The concept of alienation is developed by Alan Lightman in his 'Einstein's Dreams' [48], a journal set in 1905—the 'annus mirabilis' when Einstein developed the theory of Special Relativity. Lightman's character experiences the alienation of a world in which any particular point in space‐time is delicate, temporary and liable to vanish, an 'exile in time' [48].  Raindrops hang motionless in air. Pendulums of clocks float mid‐swing' [49]. At the centre of space‐time nothing moves. The concept that all time exists simultaneously actually has a long lineage. It is central to Plato's cosmology, occurs in the Bible (Ecclesiastes 3.15), and was elaborated by St. Augustine (V.9) in the fifth century [50]. He described a universal paradox whereby even if a future event in our individual lives already exists, it depends on an act of our free‐will in order to take place. T.S. Elliot, impressed by Einstein, combined the lessons of relativity with Plato, Ecclesiastes and Augustine. Following Einstein's English visit in 1921, the year he won the Nobel Prize, Elliot wrote, 'Einstein the Great has visited England (and) has taken his place in the newspapers with the comet, the sunspots, the poisonous xxx‐jellyfish and octopus at Margate, and other natural phenomena' [51].

In two poems composed in the 1930s, and published in 1941–1942, Elliot considered the conundrum of time for the human condition. In 'Burnt Norton' he wrote that 'All time is unredeemable', for if the past exists in the future and the future exists in the past, all possibilities are eternally present, and in 'East Coker' he wrote the famous line 'In my beginning is my end' [52]. Elliot's speculations on time were shared by Priestly in such metaphysical plays as 'Time and the Conways' (1937) and, perhaps his most famous work, 'An Inspector Calls' (1947), in which a detective from the future extracts confessions of guilt for a poor girl's suicide from a comfortable middle class family. Priestley's immediate inspiration was Dunne's [53] work on time, which drew on Einstein (a cautiously supportive note was included by Arthur Eddington in the appendix to the third edition) in order to explain why the future could be predicted by precognition.

The popular end of such speculation is best represented by the collected works of Philip K. Dick (1928–1982). Like Elliot, Dick was inspired by ancient philosophy and modern science, especially the conclusions of quantum mechanics as expressed in Heisenberg's uncertainty principle and Erwin Schrȍdinger's famous thought‐experiment with the cat (1935), ideas responsible for modern multi‐verse theory. If one cannot tell both where a particle is and where it is travelling to, whether it is even a particle at all (or a wave), and how far the act of observing it has altered its state, how can one ever trust what appears to be real. For example, in the 'The Cosmic Puppets' (1957), an ordinary suburban couple return to their hometown after a gap of several years to find that everyone and everything has changed, and nobody recognises them. The novel then shifts into a traditional religious mode, located in the Zoroastrian (Persian) struggle between the good god Ormazd and his evil rival Ahriman. Eventually Ormazd triumphs, the illusory world created by Ahriman is removed, and reality returns. In Dick's award‐winning counter‐factual history, 'The Man in the High Castle' (1962), the ability of the observer to act on—and change—the material world is described via the lead characters' use of the Chinese oracle, the I Ching, to alter the future.

Dick's intensity is absent from the most whimsical of recent cosmological fiction, that of Italo Calvino (1923–1985). Calvino took cosmological ideas and exaggerated them until they were absurd. His short story, 'The Form of Space' (2002), points out that if one fell in curved space, one would logically fall for ever, while 'the Distance of the Moon' imagines a time in the distant past when the Moon was closer to the Earth, close enough for people to jump up to it and gather such delicacies as Moon‐milk.

#### **7. The visual arts**

likely to collapse in on each other, and that therefore the universe must be evolving [42]. Thomas Hardy (1840–1928) was as fascinated by cosmology as Poe, but unlike him lived to see the publication of Darwin's 'Origin of Species' in 1959 (when he was just 19). For Hardy evolution was a reality. He combined his encyclopaedic knowledge of myth, astronomy and cosmology into a 'moral astrophysics' [43] which provided the background for the individual conflicts and tragedies in novels such as 'Far from the Madding Crowd' (1874) and 'Tess of the d'Urbervilles' (1891/1892). Virginia Woolf (1882–1941) used Edwin Hubble's discovery of the expanding universe in the 1920s as a metaphor for personal and political insecurity in the 1930s. In Woolf's view, as we all live together on the same delicate, vulnerable planet in a vastness of space, it is incumbent upon us to live together rather than fight [44]. The Marxist playwright Bertolt Brecht (1898–1956) looked back to an earlier cosmology but equally wanted to illustrate a modern point in his 1938 play, 'Galileo'. His portrayal of Galileo as the heroic intellectual defending Copernicus, struggling against an obscurantist Inquisition (inaccurate because many in the senior Catholic hierarchy were Copernicans), was an allegory of the

One of the other familiar tropes derived from modern cosmology is time travel, a topic rarely dealt with in 'Star Trek', in spite of the regular use of faster‐than‐light travel. There is now a considerable literature which draws on Gerald Feinberg's 1967 paper 'Possibility of faster‐ than‐light particles', which proposed the existence of the tachyon [45]. This hypothetical particle, the tachyon, might as Martin Rees [46], says alter the order of events, if a signal from a tachyon arrived before it was sent. The genre's earliest notable example was Wells' novel 'The Time Machine' (1895) [47]. Wells coined the phrase time machine to describe a time‐travelling vehicle which moved because the fourth dimension was of time rather than space. Wells was a utopian socialist and his main preoccupation was to explore varieties of human society, considering whether progress inevitably resulted in human improvement: it's clear that he didn't think that this was the case, and that ignorance and superstition could easily flourish in the future. Neither does 'Dr Who', the most successful TV time travel franchise, explain how it is possible to travel to the distant past or future. The time machine, the TARDIS (short for Time and Relative Dimensions in Space), is bigger inside than outside and references Einstein by referencing relativity in its name. The Doctor himself is increasingly represented as a lonely figure, destined to exist in perpetual sadness caused by the death or departure of

The concept of alienation is developed by Alan Lightman in his 'Einstein's Dreams' [48], a journal set in 1905—the 'annus mirabilis' when Einstein developed the theory of Special Relativity. Lightman's character experiences the alienation of a world in which any particular point in space‐time is delicate, temporary and liable to vanish, an 'exile in time' [48].  Raindrops hang motionless in air. Pendulums of clocks float mid‐swing' [49]. At the centre of space‐time nothing moves. The concept that all time exists simultaneously actually has a long lineage. It is central to Plato's cosmology, occurs in the Bible (Ecclesiastes 3.15), and was elaborated by St. Augustine (V.9) in the fifth century [50]. He described a universal paradox whereby even if a future event in our individual lives already exists, it depends on an act of our free‐will in order to take place. T.S. Elliot, impressed by Einstein, combined the lessons of relativity with

revolutionary struggles of the 1930s.

10 Trends in Modern Cosmology

his companions.

Representations of the sky, stars or cosmos in visual form date back to the Stone Age and are familiar throughout the ancient world. They may be symbolic, as in Egyptian astronomical‐ceilings, or take on human form, as in Roman images of planetary deities. Later they might be decorative, as in Renaissance star maps, or attempt accuracy, as in modern star maps, or be entirely abstract, as in twentieth century surrealism. The Sun and Moon make regular appearances in western painting, as one would expect. The cosmological statements, though, are often simple. Often the Sun and Moon are poetic additions, symbolising time or heaven in medieval and Renaissance art, casting light or embodying the power of nature, and even serve as political satire in the nineteenth century [54]. Cosmological changes encouraged the new spirit: the philosopher Berlin [55] argued that the astronomical revolution's abandonment of the old crystalline planetary spheres in favour of a universe without boundaries encouraged the emergence of the chaos and adventurism of Romantic art. Painting began the shift towards the abstract with the extraordinary work of Joseph Turner, who drew on both esoteric wisdom and the latest science in his portrayal of light [56, 57]. Perhaps the most famous example of nineteenth century astronomical art is van Gogh's 1889 masterpiece 'The Starry Night', a painting partly inspired by the pre‐dawn rising of Venus, but easily interpreted as representing the swirling chaos of van Gogh's inner world.

The relationship between modern art movements and science is complicated by many artists' multiple affiliations. For example, many notable early twentieth century painters were followers of Theosophy, a spiritual teaching highly indebted to Plato, Renaissance alchemy and Freudian psychoanalysis, all of which could deal with unseen realities and the interdependence of all things in the cosmos. It is therefore not easy to distinguish scientific influences on twentieth art from mystical or magic ones and it is up to art historians to interpret [58]. However, it is clear that the new physics encouraged the move towards radical, abstract forms of expression. Heisenberg's uncertainty principle appears to have supported the playful, apparently chaotic, practices of Dada, in which nothing is quite as it seems. When Marcel Duchamp (1887–1968) displayed a urinal in 1917 he was making a radical statement that, if art is not what one imagines it is, neither is anything else.

André Breton (1896–1966), the poet and author of the Surrealist Manifesto, singled out Einstein (along with Freud) as significant in 1922. J.W. Dunne's adaptation of Einstein to precognition and psychology was popular with the surrealists, as it as with Priestley. That the observer stands at the centre of time and space, as popular conceptions of relativity and quantum mechanics assume, underpins the playful juxtaposition of images and ideas, sense and nonsense, which runs through the entire history of modern abstract and conceptual art. When Joan Miro paints a picture such as 'Dog barking at the Moon' (1926), he is alluding to ideas that the Moon makes one mad— lunatic—as well as departing significantly from naturalism, but also raising a smile.

A separate strand of painting drew on mathematical conceptions of the universe, as did Duchamp's earlier painting 'Nude Descending a Staircase' (1912), or Man Ray's (1890–1976) geometrical models, such as 'Polyèdres' (1934–1936). The distinctive angular lines of Picasso's painting were also decisively influenced by the idea of multi‐dimensional mathematics as explained by Miller [59]. By contrast, the whirling lines in Max Ernst's '…sur le plan de la Physique' (1943) evoke the spinning of atoms. And for Klee [60], writing in 1920, movement in painting was essential because everything in the universe is characterised by motion. Perhaps the most famous portrayals of the new physics are Salvador Dali's (1904–1989) paintings of bent clocks as representations of distorted space‐time, as in 'The Persistence of the Disintegration of Memory' (1952).

#### **8. Space flight**

astronomical‐ceilings, or take on human form, as in Roman images of planetary deities. Later they might be decorative, as in Renaissance star maps, or attempt accuracy, as in modern star maps, or be entirely abstract, as in twentieth century surrealism. The Sun and Moon make regular appearances in western painting, as one would expect. The cosmological statements, though, are often simple. Often the Sun and Moon are poetic additions, symbolising time or heaven in medieval and Renaissance art, casting light or embodying the power of nature, and even serve as political satire in the nineteenth century [54]. Cosmological changes encouraged the new spirit: the philosopher Berlin [55] argued that the astronomical revolution's abandonment of the old crystalline planetary spheres in favour of a universe without boundaries encouraged the emergence of the chaos and adventurism of Romantic art. Painting began the shift towards the abstract with the extraordinary work of Joseph Turner, who drew on both esoteric wisdom and the latest science in his portrayal of light [56, 57]. Perhaps the most famous example of nineteenth century astronomical art is van Gogh's 1889 masterpiece 'The Starry Night', a painting partly inspired by the pre‐dawn rising of Venus, but easily interpreted as representing the

The relationship between modern art movements and science is complicated by many artists' multiple affiliations. For example, many notable early twentieth century painters were followers of Theosophy, a spiritual teaching highly indebted to Plato, Renaissance alchemy and Freudian psychoanalysis, all of which could deal with unseen realities and the interdependence of all things in the cosmos. It is therefore not easy to distinguish scientific influences on twentieth art from mystical or magic ones and it is up to art historians to interpret [58]. However, it is clear that the new physics encouraged the move towards radical, abstract forms of expression. Heisenberg's uncertainty principle appears to have supported the playful, apparently chaotic, practices of Dada, in which nothing is quite as it seems. When Marcel Duchamp (1887–1968) displayed a urinal in 1917 he was making a radical statement that, if art is not what one imagines it is, neither is anything else.

André Breton (1896–1966), the poet and author of the Surrealist Manifesto, singled out Einstein (along with Freud) as significant in 1922. J.W. Dunne's adaptation of Einstein to precognition and psychology was popular with the surrealists, as it as with Priestley. That the observer stands at the centre of time and space, as popular conceptions of relativity and quantum mechanics assume, underpins the playful juxtaposition of images and ideas, sense and nonsense, which runs through the entire history of modern abstract and conceptual art. When Joan Miro paints a picture such as 'Dog barking at the Moon' (1926), he is alluding to ideas that the Moon makes one mad— lunatic—as well as departing significantly from naturalism,

A separate strand of painting drew on mathematical conceptions of the universe, as did Duchamp's earlier painting 'Nude Descending a Staircase' (1912), or Man Ray's (1890–1976) geometrical models, such as 'Polyèdres' (1934–1936). The distinctive angular lines of Picasso's painting were also decisively influenced by the idea of multi‐dimensional mathematics as explained by Miller [59]. By contrast, the whirling lines in Max Ernst's '…sur le plan de la Physique' (1943) evoke the spinning of atoms. And for Klee [60], writing in 1920, movement in painting was essential because everything in the universe is characterised by motion.

swirling chaos of van Gogh's inner world.

12 Trends in Modern Cosmology

but also raising a smile.

It is well understood that the photographs of the whole Earth taken by the Apollo astronauts encouraged concepts of the global village, a world devoid of racial divisions, religious schism and political boundaries [61, 62]. The first major use of the Apollo photographs was on the cover of the first Whole Earth Catalogue in 1968, placed there by the editor, political activist Stewart Brand. Subsequently, the photographs became an inspiration for the emerging environmental movement.

Since 1968/1969 we have been able to look down on our sky from space. The euphoric consequences of this experience, still enjoyed by a few hundred people, was named the 'Overview Effect' by Frank White in 1987 [63]. Interviewed on BBC Radio 4's iPM programme on 25 May 2013, the astronaut Geoff Hoffman described his own experience of the effect [64]. He recalled the strange sensation of looking down at the Earth, watching the terrestrial sky from above instead of from below, witnessing the flash of lightning storms and streaks of light as meteors plunged into the atmosphere. He saw the world as one, drawing salutary ecological lessons from the visible deforestation of tropical areas. Inspired by the ethereal nature of the Earth's halo, Hoffman hesitated to use the word 'spiritual', put to him by his interviewer in a leading question, but was happy to describe the condition he experienced on his mission as being a 'state of grace', words which he said had been suggested to him by a Jesuit priest. Shamans, Pharaohs and Platonic souls may have seen the Earth in their imaginations, but astronauts experience it physically.

The 'Overview Effect' has been institutionalised in the Overview Institute, whose purpose is to utilise the Effect for the common good. The Institute's apocalyptic and utopian agenda draws a direct connection between the experience of space travel and the need to save the Earth: 'We live at a critical moment in human history. The challenges of climate change, food, water and energy shortages as well as the increasing disparity between the developed and developing nations are testing our will to unite, while differences in religions, cultures, and politics continue to keep us apart. The creation of a 'global village' through satellite TV and the Internet is still struggling to connect the world into one community. At this critical moment, our greatest need is for a global vision of planetary unity and purpose for humanity as a whole' [65]. In this sense, the institute completes the earlier visions of Virginia Woolf and Stewart Brand.

#### **9. Conclusion**

Modern scientific cosmology needs to be valued not just for what it tells us about the universe, but for how what it tells us informs the ways that people think and behave in wider culture. A number of themes emerge, including the vastness of space as a metaphor for loneliness and insecurity, and the new physics as a source of freedom and adventure. Scientific cosmology's wider significance needs to be more widely acknowledged, for modern society still benefits from 'complex subsystems of cosmic exchange' between scientists —cosmologists—and the general public.

#### **Author details**

Nicholas Campion

Address all correspondence to<EMAIL_ADDRESS>
University of Wales Trinity Saint David, Swansea, Wales, UK

#### **References**

number of themes emerge, including the vastness of space as a metaphor for loneliness and insecurity, and the new physics as a source of freedom and adventure. Scientific cosmology's wider significance needs to be more widely acknowledged, for modern society still benefits from 'complex subsystems of cosmic exchange' between scientists —cosmologists—and the general public.

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**Dark Matter Phenomenology**

## **Constraining the Parameters of a Model for Cold Dark Matter**

### Abdessamad Abada and Salah Nasri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.69044

#### Abstract

This chapter aims at reviewing how modeling cold dark matter as weakly interacting massive particles (WIMPs) gets increasingly constrained as models have to face stringent cosmological and phenomenological experimental results as well as internal theoretical requirements like those coming from a renormalization-group analysis. The review is based on the work done on a two-singlet extension of the Standard Model of elementary particles. We conclude that the model stays viable in physically meaningful regions that soon will be probed by direct-detection experiments.

PACS numbers: 95.35.þd; 98.80.�k; 12.15.�y; 11.30.Qc

Keywords: cold dark matter, light WIMP, extension of Standard Model, rare decays, RGE

#### 1. Introduction

Dark matter accounts for about 26.5% of the total mass-energy density of the Universe [1], but we still do not know what it is. It is called dark because it is not accounted by the visible matter, the conventional baryons and leptons, which take about 4.9% of the total mass-energy density [1]. As it clearly interacts through gravity, some argue that it could still be baryonic, in the form of massive astrophysical compact halo objects (MACHOs) which emit dim or no light [2] or some sort of huge gravitational objects like galaxy-sized black holes. Indeed, such high concentrations of matter would bend passing light, the so-called gravitational lensing phenomenon, including microlensing, in ways we can detect. But the amount of dark matter we know of would produce gravitational lensing with a significantly higher number of occurrences than what observation accounts for.

Neutrinos have long been thought of composing the dark matter around us. However, Standard Model neutrinos are light, and so too fast-moving (hot) to compose the (cold) dark matter

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

structures we see. But sterile neutrinos, non-Standard Model particles, can be heavier, and so could be dark matter candidates. This possibility has been reignited with the recent detection of an X-ray emission line at an energy of 3:55 keV coming from galaxy clusters, the Andromeda galaxy, the Galactic Center and the Draco dwarf spheroidal galaxy. This line is consistent with the decay of a 7:1 keV sterile neutrino [3].

In fact, there is by now quasi-consensus that dark matter ought to be understood outside the realm of conventional matter. One other scenario is that of (pseudo)scalar particles of tiny mass � <sup>10</sup>�<sup>22</sup> eV, the so-called ultralight axions that could account for the dark matter content of the Universe. This is supported by high-resolution cosmological simulations [4]. Axions originated in quantum chromodynamics, the theory of quarks and gluons, in relation to the axial anomaly in this theory and the strong Charge Conjugation Parity Symmetry Violation (CP violation) problem. But like anything else related to dark matter, they elude detection. The Axion Dark Matter Experiment (ADMX) may bring in answers in the near future [5].

But maybe the most popular candidate for dark matter is an electrically neutral and colorless weakly interacting massive particle (WIMP). Such a particle originated in supersymmetric (SUSY) extensions of the StandardModel. The most obvious such a candidate is the neutralino, a neutralRodd supersymmetric particle. Indeed, neutralinos are only produced or destroyed in pairs, thus constituting the lightest SUSYparticles. However, alas, as rich, attractive and beautiful as SUSY can be, supersymmetric particles continue to elude detection at the Large Hadron Collider (LHC), at least in Run 1 experiments with a center-of-mass energy ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV [6]. Run 2 experiments with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV are currently under way, targeting a final luminosity of about 100 fb�<sup>1</sup> , and so are tested in more involved and less stringent formulations of supersymmetry [7].

It must be stressed that until now, we have not detected dark matter, at least not in a conclusive manner. Indeed, we know dark matter is there only because of its gravitational interactions, and this is why and how we believe it contributes about a quarter of the mass energy of the known Universe. But we still do not know whether dark matter really interacts with ordinary matter. We believe it does, even if very weakly. We believe these interactions can yield signals with enough strength so that we can detect dark matter or produce it in collisions of Standard Model particles [8].

We must also understand that a detection process relies primarily on a theory or a model. A theory like supersymmetry, which originated in the realm of elementary particle physics, is devised as an extension to the Standard Model that is based on a yet-to-be-detected symmetry between fermionic and bosonic states [9]. Its DM connection came only later. In fact, in the rather long period between the Higgs mechanism proposal [10] and the detection of the Higgs particle [11], various extensions of the Standard Model were proposed in order to alleviate some of its shortcomings, the so-called "Beyond the Standard Model" (BSM) Physics [12]. A number of these BSM models bear in them extra fields, meaning extra particles with specific properties. Until today, such particles have never been detected. With time and change in focus, the most stable of these hypothetical particles have then been proposed as candidates for dark matter, many in the form of WIMPs. The advantage of such a paradigm is clear: the calculational techniques that built strength in the realm of particle physics were ready at the service of dark matter search with little extra effort in development. But the experimental framework was also ready. Such a state of affairs could partly explain the popularity of WIMP physics, compared to other possible scenarios for dark matter.

structures we see. But sterile neutrinos, non-Standard Model particles, can be heavier, and so could be dark matter candidates. This possibility has been reignited with the recent detection of an X-ray emission line at an energy of 3:55 keV coming from galaxy clusters, the Andromeda galaxy, the Galactic Center and the Draco dwarf spheroidal galaxy. This line is consistent

In fact, there is by now quasi-consensus that dark matter ought to be understood outside the realm of conventional matter. One other scenario is that of (pseudo)scalar particles of tiny mass � <sup>10</sup>�<sup>22</sup> eV, the so-called ultralight axions that could account for the dark matter content of the Universe. This is supported by high-resolution cosmological simulations [4]. Axions originated in quantum chromodynamics, the theory of quarks and gluons, in relation to the axial anomaly in this theory and the strong Charge Conjugation Parity Symmetry Violation (CP violation) problem. But like anything else related to dark matter, they elude detection. The Axion Dark

But maybe the most popular candidate for dark matter is an electrically neutral and colorless weakly interacting massive particle (WIMP). Such a particle originated in supersymmetric (SUSY) extensions of the StandardModel. The most obvious such a candidate is the neutralino, a neutralRodd supersymmetric particle. Indeed, neutralinos are only produced or destroyed in pairs, thus constituting the lightest SUSYparticles. However, alas, as rich, attractive and beautiful as SUSY can be, supersymmetric particles continue to elude detection at the Large Hadron Collider (LHC), at

It must be stressed that until now, we have not detected dark matter, at least not in a conclusive manner. Indeed, we know dark matter is there only because of its gravitational interactions, and this is why and how we believe it contributes about a quarter of the mass energy of the known Universe. But we still do not know whether dark matter really interacts with ordinary matter. We believe it does, even if very weakly. We believe these interactions can yield signals with enough strength so that we can detect dark matter or produce it in collisions of Standard

We must also understand that a detection process relies primarily on a theory or a model. A theory like supersymmetry, which originated in the realm of elementary particle physics, is devised as an extension to the Standard Model that is based on a yet-to-be-detected symmetry between fermionic and bosonic states [9]. Its DM connection came only later. In fact, in the rather long period between the Higgs mechanism proposal [10] and the detection of the Higgs particle [11], various extensions of the Standard Model were proposed in order to alleviate some of its shortcomings, the so-called "Beyond the Standard Model" (BSM) Physics [12]. A number of these BSM models bear in them extra fields, meaning extra particles with specific properties. Until today, such particles have never been detected. With time and change in focus, the most stable of these hypothetical particles have then been proposed as candidates for dark matter, many in the form of WIMPs. The advantage of such a paradigm is clear: the calculational techniques that built strength in the realm of particle physics were ready at the

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV are currently under way, targeting a final luminosity of about 100 fb�<sup>1</sup>

tested in more involved and less stringent formulations of supersymmetry [7].

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV [6]. Run 2 experiments with

, and so are

Matter Experiment (ADMX) may bring in answers in the near future [5].

least in Run 1 experiments with a center-of-mass energy ffiffi

ffiffi

Model particles [8].

with the decay of a 7:1 keV sterile neutrino [3].

22 Trends in Modern Cosmology

Accordingly, many experiments have been devised specifically to detect dark matter. Each, of course, must be based on a specific scheme that is based on a specific scenario. There are experiments that try to detect dark matter directly, through missing energy momentum after a WIMP collides directly with an ordinary nucleus. The low-background DAMA (NaI) and then DAMA/LIBRA (NaI[Ti]) experiments at Gran Sasso in Italy [13] add a twist to this by trying to detect dark matter in the galactic halo via its suggested model-independent flux annual modulation [14]. The CoGeNT experiment [15] in Soudan (Minnesota, USA) also tries to detect this annual modulation, but in the region where the WIMP mass is ≲10 GeV. The CDMS I (Stanford, USA) [16], then CDMS II (Soudan, USA) [17], and now the superCDMS (Soudan, USA, then SNOLAB, Sudbury, Canada) [18] perform direct detection, measuring ionization and phonon signals resulting from a WIMP-nucleus collision, sensitive in the lowmass region. The XENON10 [19], then XENON100 [20], then the coming XENON1t [21], all in Gran Sasso, Italy, use liquid Xenon as a detecting medium for WIMP-nucleon and WIMPelectron collisions. There is also the Large Underground Xenon (LUX) experiment (South Dakota, USA) [22], as a direct-detection experiment, and its more sensitive successor LZ experiment [23]. The CRESST experiment [24], followed by CRESST II [25], both at Gran Sasso, Italy, also try to detect dark matter directly with low mass. We also have the series of EDEL-WEISS experiments [26] (Modane, France), which target low-mass WIMPs. The list is exhaustive, and could not be accounted here due to space constraints.

The above experiments are terrestrial, with instruments buried underground to reduce noise. But there are other experiments which are space borne that carry out indirect detection in cosmic rays. There is the Fermi Gamma-Ray Space Telescope (Fermi-LAT), which has found excess of gamma rays in the galactic center that cannot be explained by conventional sources and which is compatible with the presence of dark matter [27]. Fermi-LAT uses what we call indirect methods, namely, collecting gamma-ray signals and removing from these those emitted by all possible known sources. Another space-borne experiment is the Alpha Magnetic Spectrometer (AMS) experiment at the international space station [28], collecting and analyzing signals from cosmic rays. In addition, the Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics (PAMELA) experiment [29] is a particle identifier that uses a permanent magnet spectrometer for space cosmic-ray direct measurements.

A third prong in the dark matter search enterprise is to produce it in particle colliders like the LHC [8]. There is an added difficulty here, which is that we do not know in which mass range we should look into. It could well be that the present center-of-mass energy that is available, 13 TeV, may not be sufficient. Nevertheless, the search for dark matter at the LHC is intense. One reason is that, experimentally, this is feasible now: small amounts of missing energy and transverse momentum can be detected now. Note that the present detectors are not built to detect dark matter directly. Rather, the latter would appear as a missing energy or missing momentum. For example, we now look at events in which a Z boson and a missing transverse momentum are produced in a proton-proton collision at ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV. The <sup>Z</sup> boson decays into two charged leptons, a recognizable signature, and a possible missing transverse momentum, which would indicate the production of dark matter in the process. A similar search, conducted previously by the CMS Collaboration and based on data collected with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV (Run 1), found no evidence of new physics and hence set limits on dark matter production. A recent search performed by the ATLAS Collaboration with ffiffi <sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV with an integrated luminosity of 3:2 fb�<sup>1</sup> also reported no evidence [30].

What should be clear by now is that interpreting signals as dark matter necessitates modeling. On the other hand, any model needs experimental results to restrict the range of its free parameters, to fine-tune these parameters, and, ultimately, in many cases, to be eliminated. The aim of this chapter is to shed light on the main steps a phenomenologist takes when building a model for dark matter, then testing the model against experimental results. It is an attempt to look into the modeling process itself, from the "cradle to the grave," so to speak. The discussion is based on a model proposed in Ref. [31] for cold dark matter, exposed to particle-physics phenomenology in [32], and further restricted by internal consistency in Ref. [33]. We will see how gradually the parameters of the model are constrained, and how the region of viability is reached. To carry out the discussion smoothly, we have chosen a model which is simple enough to avoid confusion created by the often involved details of the calculations and could-be-complexity of the model itself, but at the same time rich enough to be able to accommodate a vast range of experimental results. The material presented in this chapter is drawn from the works just cited.

This chapter is organized as follows. After this Introduction, Section 2 motivates and then presents the model based on WIMP physics, namely, a two-singlet extension of the Standard Model of elementary particles. We will try to avoid lengthy arguments and focus on the essentials. Section 3 shows how the measured amount of dark matter relic-density constrains the value of the dark matter annihilation cross-section, a constraint any model has to satisfy. We then discuss how the two-singlet extension fits into this, and add to it a perturbativity ingredient. Section 4 takes the two-singlet model into the arena of particle phenomenology and sees how it copes with rare meson decays. Section 5 goes back to the fundamentals and runs a renormalization-group analysis to inquire into the sustainability of the model. Section 6 puts all these constraints together and determines the regions of viability of the model. Section 7 is left for concluding remarks.

#### 2. A model for dark matter: motivation and parametrization

As mentioned in the Introduction, the most popular candidate for dark matter is an electrically neutral colorless weakly interacting massive particle (WIMP), and the neutralino, the lightest supersymmetric particle, is a robust fit for this role. However, as explained in Ref. [31] and references therein, it is hard to argue in favor of a neutralino when it comes to light cold dark matter, say, a WIMP mass of up to 10 GeV. In addition, up to now, we have not detected supersymmetric signatures at the LHC [34].

Therefore, with no prior hints as to what the internal structure of the WIMP might be, one adopts a bottom-up approach, in which one extends the Standard Model by adding to it the simplest of fields, one real spinless scalar, which will be the WIMP. This field must be a Standard Model gauge singlet so that we avoid any "direct contact" with any of the Standard Model particles. It is allowed to interact with visible particles only via the Higgs field. It is made stable against annihilation by enforcing upon it the simplest of symmetries, a discrete Z<sup>2</sup> symmetry that does not break spontaneously. This construction is called the minimal extension to the Standard Model. In view of its cosmological implication, the minimal extension has first been proposed in Ref. [35] and has been extensively studied and explored in Ref. [36]. However, this model is shown in Ref. [37] to be inadequate if we want the WIMP to be light.

into two charged leptons, a recognizable signature, and a possible missing transverse momentum, which would indicate the production of dark matter in the process. A similar search,

(Run 1), found no evidence of new physics and hence set limits on dark matter production. A

What should be clear by now is that interpreting signals as dark matter necessitates modeling. On the other hand, any model needs experimental results to restrict the range of its free parameters, to fine-tune these parameters, and, ultimately, in many cases, to be eliminated. The aim of this chapter is to shed light on the main steps a phenomenologist takes when building a model for dark matter, then testing the model against experimental results. It is an attempt to look into the modeling process itself, from the "cradle to the grave," so to speak. The discussion is based on a model proposed in Ref. [31] for cold dark matter, exposed to particle-physics phenomenology in [32], and further restricted by internal consistency in Ref. [33]. We will see how gradually the parameters of the model are constrained, and how the region of viability is reached. To carry out the discussion smoothly, we have chosen a model which is simple enough to avoid confusion created by the often involved details of the calculations and could-be-complexity of the model itself, but at the same time rich enough to be able to accommodate a vast range of experimental results. The material presented in this

This chapter is organized as follows. After this Introduction, Section 2 motivates and then presents the model based on WIMP physics, namely, a two-singlet extension of the Standard Model of elementary particles. We will try to avoid lengthy arguments and focus on the essentials. Section 3 shows how the measured amount of dark matter relic-density constrains the value of the dark matter annihilation cross-section, a constraint any model has to satisfy. We then discuss how the two-singlet extension fits into this, and add to it a perturbativity ingredient. Section 4 takes the two-singlet model into the arena of particle phenomenology and sees how it copes with rare meson decays. Section 5 goes back to the fundamentals and runs a renormalization-group analysis to inquire into the sustainability of the model. Section 6 puts all these constraints together and determines the regions of viability of the model. Section 7 is left for concluding remarks.

As mentioned in the Introduction, the most popular candidate for dark matter is an electrically neutral colorless weakly interacting massive particle (WIMP), and the neutralino, the lightest supersymmetric particle, is a robust fit for this role. However, as explained in Ref. [31] and references therein, it is hard to argue in favor of a neutralino when it comes to light cold dark matter, say, a WIMP mass of up to 10 GeV. In addition, up to now, we have not detected

Therefore, with no prior hints as to what the internal structure of the WIMP might be, one adopts a bottom-up approach, in which one extends the Standard Model by adding to it the

2. A model for dark matter: motivation and parametrization

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 8 TeV

<sup>s</sup> <sup>p</sup> <sup>¼</sup> 13 TeV with an integrated

conducted previously by the CMS Collaboration and based on data collected with ffiffi

recent search performed by the ATLAS Collaboration with ffiffi

luminosity of 3:2 fb�<sup>1</sup> also reported no evidence [30].

24 Trends in Modern Cosmology

chapter is drawn from the works just cited.

supersymmetric signatures at the LHC [34].

In the logic of this bottom-up approach, adding another real scalar seems the natural step forward. This field will also be endowed with a Z<sup>2</sup> symmetry, but this one we will break spontaneously, and the reason is to open new channels for dark matter annihilation, which implies an increase in the corresponding annihilation cross-section, which in turn would allow smaller WIMP masses, something we want to achieve. Needless to say that this auxiliary field must also be a Standard Model gauge singlet.

Therefore, we extend the Standard Model by adding two real, spinless and Z2-symmetric fields: the dark matter field S<sup>0</sup> for which the Z<sup>2</sup> symmetry is unbroken and an auxiliary field for which it is spontaneously broken. Both fields are Standard Model gauge singlets and hence can interact with "visible" particles only via the Higgs doublet, taken in the unitary gauge. We must also assume all processes calculable in perturbation theory. The details of the spontaneous breaking of the electroweak gauge symmetry and the additional auxiliary Z<sup>2</sup> symmetry are left aside [31].

The potential function that involves the physical scalar Higgs field h, the dark matter field S0, and the physical auxiliary scalar field S<sup>1</sup> is as follows:

$$\begin{split} \mathcal{U} &= \frac{1}{2} m\_0^2 S\_0^2 + \frac{1}{2} m\_h^2 h^2 + \frac{1}{2} m\_1^2 S\_1^2 \\ &+ \frac{\lambda\_0^{(3)}}{2} S\_0^2 h + \frac{\eta\_{01}^{(3)}}{2} S\_0^2 S\_1 + \frac{\lambda^{(3)}}{6} h^3 + \frac{\eta\_1^{(3)}}{6} S\_1^3 + \frac{\lambda\_1^{(3)}}{2} h^2 S\_1 + \frac{\lambda\_2^{(3)}}{2} h S\_1^2 \\ &+ \frac{\eta\_0}{24} S\_0^4 + \frac{\lambda^{(4)}}{24} h^4 + \frac{\eta\_1^{(4)}}{24} S\_1^4 + \frac{\lambda\_0^{(4)}}{4} S\_0^2 h^2 + \frac{\eta\_{01}^{(4)}}{4} S\_0^2 S\_1^2 + \frac{\lambda\_{01}^{(4)}}{2} S\_0^2 h S\_1 \\ &+ \frac{\lambda\_1^{(4)}}{6} h^3 S\_1 + \frac{\lambda\_2^{(4)}}{4} h^2 S\_1^2 + \frac{\lambda\_3^{(4)}}{6} h S\_1^3. \end{split} (1)$$

The quantities m0, mh, and m<sup>1</sup> are the masses of the corresponding fields S0, h, and S1, respectively, and all the other parameters are real coupling constants. Also, the part of the Standard Model Lagrangian that is relevant to Dark matter annihilation is given in terms of the physical fields h and S<sup>1</sup> by the following potential function:

$$\begin{split} \mathcal{U}\_{\rm SM} &= \sum\_{f} \left( \lambda\_{\rm l\overline{\mu}} h \overline{\!\!\!f} + \lambda\_{\rm l\overline{\tau}} \mathcal{S}\_{\rm l} \overline{\!\!\!f} \right) + \lambda\_{\rm l\overline{\mu}}^{(3)} h \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(3)} \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} \\ &+ \lambda\_{\rm l\overline{\mu}}^{(3)} h \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\tau}}^{(3)} \mathcal{S}\_{1} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}}^{(4)} h^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(4)} \mathcal{S}\_{1}^{2} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} \\ &+ \lambda\_{\rm l\overline{\mu}} h \mathcal{S}\_{1} \mathcal{W}\_{\mu}^{-} \mathcal{W}^{+\mu} + \lambda\_{\rm l\overline{\mu}}^{(4)} h^{2} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}}^{(4)} \mathcal{S}\_{1}^{2} \left( \mathcal{Z}\_{\mu} \right)^{2} + \lambda\_{\rm l\overline{\mu}} h \mathcal{S}\_{1} \left( \mathcal{Z}\_{\mu} \right)^{2}. \end{split} \tag{2}$$

The coupling constants in the above expression are given by the following relations, in which the quantities mf , mw, and mz are the masses of the fermion f , the W, and the Z gauge bosons, respectively:

$$\begin{aligned} \lambda\_{\text{hf}} &= -\frac{m\_f}{v} \cos \Theta; \quad \lambda\_{1f} = \frac{m\_f}{v} \sin \Theta; \\ \lambda\_{\text{hz}}^{(3)} &= 2 \frac{m\_w^2}{v} \cos \Theta; \quad \lambda\_{1w}^{(3)} = -2 \frac{m\_w^2}{v} \sin \Theta; \\ \lambda\_{\text{hz}}^{(3)} &= \frac{m\_z^2}{v} \cos \Theta; \quad \lambda\_{1z}^{(3)} = -\frac{m\_z^2}{v} \sin \Theta; \\ \lambda\_{\text{hw}}^{(4)} &= \frac{m\_w^2}{v^2} \cos^2 \Theta; \quad \lambda\_{1w}^{(4)} = \frac{m\_w^2}{v^2} \sin^2 \Theta; \quad \lambda\_{h1w} = -\frac{m\_w^2}{v^2} \sin 2\Theta; \\ \lambda\_{\text{hz}}^{(4)} &= \frac{m\_z^2}{2v^2} \cos^2 \Theta; \quad \lambda\_{1z}^{(4)} = \frac{m\_z^2}{2v^2} \sin^2 \Theta; \quad \lambda\_{h1z} = -\frac{m\_z^2}{2v^2} \sin 2\Theta. \end{aligned} \tag{3}$$

The angle θ is the mixing angle between the fields h and S<sup>1</sup> [31]. The quantities v and v1, both positive, are the vacuum expectation values of the Higgs and auxiliary fields, respectively.

This model has nine free parameters to start with, three mass parameters and six coupling constants [31]. As already mentioned, perturbativity is assumed, which means all the original coupling constants are small. The dark matter self-coupling constant η<sup>0</sup> in Eq. (1) will not enter the lowest-order calculations we will consider, and so this parameter stays free for the time being and we are left with eight parameters. The spontaneous breaking of the electroweak and Z<sup>2</sup> symmetries for the Higgs and auxiliary fields, respectively, introduces the two vacuum expectation values v and v1. The value of v is fixed experimentally to be 246 GeV [38] and for the present discussion, we fix the value of v<sup>1</sup> at the order of the electroweak scale, say, 100 GeV. In addition, the Higgs mass is now known [11], mh ¼ 125 GeV. Hence, five free parameters remain. Three of these are chosen to be the two physical masses m<sup>0</sup> (dark matter) and m<sup>1</sup> (S<sup>1</sup> field), plus the mixing angle θ between S<sup>1</sup> and h. The two last parameters we choose are the two physical mutual coupling constants λð Þ<sup>4</sup> <sup>0</sup> (dark matter—Higgs) and <sup>η</sup>ð Þ<sup>4</sup> <sup>01</sup> (dark matter—S<sup>1</sup> particle), see Eq. (1).

#### 3. Constraints from cosmology and perturbativity

Any model of dark matter has to comply with astrophysical observations. Indeed, dark matter is believed to have been produced in the early Universe. A most popular paradigm for this production is the so-called "freeze-out scenario" by which dark matter, thought of as a set of elementary particles, interacts with ordinary matter, weakly but with enough strength to generate common thermal equilibrium at high temperature. However, as the cosmos is cooling down, at some temperature Tf , the rate of expansion of the Universe becomes higher than the rate of dark matter particle annihilation, which forces dark matter to decouple from ordinary matter, and hence a "freeze-out"—Tf is thus called the freeze-out temperature. The DM relic density ΩDM is essentially the one we measure today [1]:

$$
\Omega\_{\rm DM} \overline{h}^2 = 0.1199 \pm 0.0022 \approx 0.12,\tag{4}
$$

where <sup>h</sup> is Hubble constant in units of 100 km � <sup>s</sup>�<sup>1</sup> � Mpc�<sup>1</sup> .

The coupling constants in the above expression are given by the following relations, in which the quantities mf , mw, and mz are the masses of the fermion f , the W, and the Z gauge bosons,

sin θ;

m2 w v

sin θ;

sin θ;

<sup>θ</sup>; <sup>λ</sup>h1<sup>w</sup> ¼ � <sup>m</sup><sup>2</sup>

<sup>θ</sup>; <sup>λ</sup>h1<sup>z</sup> ¼ � <sup>m</sup><sup>2</sup>

<sup>0</sup> (dark matter—Higgs) and <sup>η</sup>ð Þ<sup>4</sup>

w <sup>v</sup><sup>2</sup> sin 2θ; ð3Þ

<sup>01</sup> (dark matter—S<sup>1</sup>

z <sup>2</sup>v<sup>2</sup> sin 2θ:

v

<sup>1</sup><sup>w</sup> ¼ �2

<sup>1</sup><sup>z</sup> ¼ � <sup>m</sup><sup>2</sup> z v

> <sup>1</sup><sup>w</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> w <sup>v</sup><sup>2</sup> sin <sup>2</sup>

<sup>1</sup><sup>z</sup> <sup>¼</sup> <sup>m</sup><sup>2</sup> z <sup>2</sup>v<sup>2</sup> sin <sup>2</sup>

The angle θ is the mixing angle between the fields h and S<sup>1</sup> [31]. The quantities v and v1, both positive, are the vacuum expectation values of the Higgs and auxiliary fields, respectively.

This model has nine free parameters to start with, three mass parameters and six coupling constants [31]. As already mentioned, perturbativity is assumed, which means all the original coupling constants are small. The dark matter self-coupling constant η<sup>0</sup> in Eq. (1) will not enter the lowest-order calculations we will consider, and so this parameter stays free for the time being and we are left with eight parameters. The spontaneous breaking of the electroweak and Z<sup>2</sup> symmetries for the Higgs and auxiliary fields, respectively, introduces the two vacuum expectation values v and v1. The value of v is fixed experimentally to be 246 GeV [38] and for the present discussion, we fix the value of v<sup>1</sup> at the order of the electroweak scale, say, 100 GeV. In addition, the Higgs mass is now known [11], mh ¼ 125 GeV. Hence, five free parameters remain. Three of these are chosen to be the two physical masses m<sup>0</sup> (dark matter) and m<sup>1</sup> (S<sup>1</sup> field), plus the mixing angle θ between S<sup>1</sup> and h. The two last parameters we choose are the

Any model of dark matter has to comply with astrophysical observations. Indeed, dark matter is believed to have been produced in the early Universe. A most popular paradigm for this production is the so-called "freeze-out scenario" by which dark matter, thought of as a set of elementary particles, interacts with ordinary matter, weakly but with enough strength to generate common thermal equilibrium at high temperature. However, as the cosmos is cooling down, at some temperature Tf , the rate of expansion of the Universe becomes higher than the rate of dark matter particle annihilation, which forces dark matter to decouple from ordinary matter, and hence a "freeze-out"—Tf is thus called the freeze-out temperature.  Constraints from cosmology and perturbativity

density ΩDM is essentially the one we measure today [1]:

particle), see Eq. (1).

In a model where dark matter is seen as WIMPs that can annihilate into ordinary elementary particles, the relic density ΩDM can be related to the annihilation DM cross-section σann. Indeed, in the framework of the standard cosmological model, one can derive the following relation [39]:

$$
\Omega\_{\rm DM} \overline{h}^2 \simeq \frac{1.07 \times 10^9 \text{x}\_{\circ}}{\sqrt{\mathcal{g}\_\*} m\_{\rm Pl} \langle \sigma\_{\rm ann} v \rangle \,\text{GeV}}; \quad \mathbf{x}\_{\circ} \simeq \ln \frac{0.0038 m\_{\rm Pl} m\_0 \langle \sigma\_{\rm ann} v \rangle}{\sqrt{\mathcal{g}\_\* \overline{\chi}\_f}}. \tag{5}
$$

The quantity <sup>m</sup>Pl <sup>¼</sup> <sup>1</sup>:<sup>22</sup> � 1019GeV is the Planck mass, <sup>m</sup><sup>0</sup> is the dark matter mass, xf <sup>¼</sup> <sup>m</sup>0=Tf , and g<sup>∗</sup> is the number of relativistic degrees of freedom with a mass less than Tf . The quantity 〈σannv〉 is the thermally averaged annihilation cross-section of a pair of two dark matter particles multiplied by their relative speed in their center-of-mass reference frame. Solving (4) with the current value (5) for ΩDM with xf between 19.2 and 21.6 [40], we obtain the following constraint on the annihilation cross-section:

$$
\langle \sigma\_{\rm ann} v \rangle \simeq 2 \times 10^{-9} \text{GeV}.\tag{6}
$$

This is one major constraint any WIMP model like the one we discuss here has to satisfy. Indeed, the quantity 〈σannv〉 is calculable in perturbation theory, and so, the implementation of (6) will induce an admittedly complicated but important relation between the free parameters of the model, hence reducing their space of freedom, reducing their number by one. Also, the constraint induced by (6) can be used to examine aspects of the theory like perturbativity. To implement perturbativity in the present two-singlet model, we use (6) to obtain the mutual coupling constant ηð Þ<sup>4</sup> <sup>01</sup> (coupling between the DM field S<sup>0</sup> and auxiliary field S1) in terms of the dark matter mass m<sup>0</sup> for given values of λð Þ<sup>4</sup> <sup>0</sup> (coupling between S<sup>0</sup> and Higgs) and study its behavior to tell which dark matter mass regions are consistent with perturbativity. It should be mentioned that once the two mutual coupling constants λð Þ<sup>4</sup> <sup>0</sup> and <sup>η</sup>ð Þ<sup>4</sup> <sup>01</sup> are small, all the other physical coupling constants will be small.

The quantity 〈σannv〉 is calculated in perturbation theory using all possible annihilation channels the model allows for [31]. As the model has many parameters, the behavior of the mutual coupling constant ηð Þ<sup>4</sup> <sup>01</sup> is bound to be rich. Sampling is therefore necessary. In this review, we briefly comment on the behavior of ηð Þ<sup>4</sup> <sup>01</sup> for two sets of the parameters (θ, m1, <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ). A more substantial discussion can be found in Ref. [31].

The first set of parameters is a small mixing angle <sup>θ</sup><sup>¼</sup> 10o, a weak mutual <sup>S</sup>0-Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> <sup>¼</sup> <sup>0</sup>:01, and a <sup>S</sup>1-mass <sup>m</sup><sup>1</sup> <sup>¼</sup> 10 GeV. The corresponding behavior of <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>0</sup> is shown in Figure 1. The range of m<sup>0</sup> displayed is from 0:1 to 200 GeV. In this regime, the first feature we see is that the relic-density constraint on dark matter annihilation forbids WIMP

Figure 1. η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>1</sup> for very light S1, small mixing, and very small WIMP-Higgs coupling.

masses m0≲1:3 GeV. Furthermore, just about m0≃1:3 GeV, the c-quark threshold, the S<sup>0</sup> � S<sup>1</sup> mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> starts at about 0:8, a value, while perturbative, that is roughly 80-fold larger than the mutual S<sup>0</sup> Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> . Then as the DM mass increases, η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> decreases, steeply first, more slowly as we cross the τ mass toward the b mass. Just before m1=2, the coupling η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> hops onto another solution branch that is just emerging from negative territory, gets back to the first one at precisely m1=2 as this latter carries now smaller values, and then jumps up again onto the second branch as the first crosses the m<sup>0</sup> axis down. It goes up this branch with a moderate slope until m<sup>0</sup> becomes equal to m1, a value at which the S<sup>1</sup> annihilation channel opens. Just beyond m1, there is a sudden fall to a value η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> ≃0:0046 that is about half the value of λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , and <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> stays flat till m0≃45 GeV where it starts increasing, sharply after 60 GeV. In the mass interval m<sup>0</sup> ≃ 66–79 GeV, there is a "desert" with no positive real solutions to the relic-density constraint, hence no viable dark matter candidate exists. Beyond m0≃79 GeV, the mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> keeps increasing monotonously, with a small notch at the W mass and a less noticeable one at the Z mass. As it increases, its values remain perturbative.

 The behavior of the <sup>S</sup><sup>0</sup> � <sup>S</sup><sup>1</sup> mutual coupling constant <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus the DM mass m<sup>0</sup> is displayed in Figure 2. Here too, no viable DM masses exist below roughly 1:4 GeV, at which value η<sup>ð</sup>4<sup>Þ</sup> 01 starts at 1:95. It decreases with a sharp change of slope at the b-quark threshold, then makes a sudden dive at about 5 GeV, a change of branch at m1=2 down till about 12 GeV where it jumps up back onto the previous branch just before going to cross into negative territory. It drops

Figure 2. η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> versus m<sup>0</sup> for small mixing, moderate m1, and WIMP-Higgs coupling.

sharply at m<sup>0</sup> ¼ m<sup>1</sup> and then increases slowly until m<sup>0</sup> ≃43:3 GeV. Then, no viable WIMP masses exist, a desert. As we see, for this set of parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1), the model constrains the dark matter mass inside the interval 1:6 GeV ≲ m<sup>0</sup> ≲ 43:3 GeV, with perturbative coupling constants.

With the same mixing angle <sup>θ</sup> <sup>¼</sup> 10o and mutual coupling constant <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:2, larger masses m<sup>1</sup> yield roughly the same behavior, but with values of η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> that could be nonperturbative. For example, when <sup>m</sup><sup>1</sup> <sup>¼</sup> 60 GeV, the mutual coupling <sup>η</sup><sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> starts very high (≃85) at m<sup>0</sup> ≃ 1:5 GeV, and then decreases rapidly. There is a usual change of branches and a desert starting at about 49 GeV, a behavior that is peculiar in a way because the desert starts at a mass m<sup>0</sup> < m1, that is, before the opening of the S<sup>1</sup> annihilation channel. In other words, the dark matter is annihilating into the light fermions only and the model is perturbatively viable in the range of 20–49 GeV.

#### 4. Constraints from direct detection

masses m0≲1:3 GeV. Furthermore, just about m0≃1:3 GeV, the c-quark threshold, the S<sup>0</sup> � S<sup>1</sup>

<sup>01</sup> versus m<sup>1</sup> for very light S1, small mixing, and very small WIMP-Higgs coupling.

<sup>01</sup> decreases, steeply first, more slowly as we cross the τ mass toward the b mass. Just before

territory, gets back to the first one at precisely m1=2 as this latter carries now smaller values, and then jumps up again onto the second branch as the first crosses the m<sup>0</sup> axis down. It goes up this branch with a moderate slope until m<sup>0</sup> becomes equal to m1, a value at which the S<sup>1</sup>

after 60 GeV. In the mass interval m<sup>0</sup> ≃ 66–79 GeV, there is a "desert" with no positive real solutions to the relic-density constraint, hence no viable dark matter candidate exists. Beyond

notch at the W mass and a less noticeable one at the Z mass. As it increases, its values remain

The second set of parameters we feature is still a small Higgs <sup>S</sup><sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> 10o, an

in Figure 2. Here too, no viable DM masses exist below roughly 1:4 GeV, at which value η<sup>ð</sup>4<sup>Þ</sup>

starts at 1:95. It decreases with a sharp change of slope at the b-quark threshold, then makes a sudden dive at about 5 GeV, a change of branch at m1=2 down till about 12 GeV where it jumps up back onto the previous branch just before going to cross into negative territory. It drops

annihilation channel opens. Just beyond m1, there is a sudden fall to a value η<sup>ð</sup>4<sup>Þ</sup>

<sup>01</sup> starts at about 0:8, a value, while perturbative, that is roughly

<sup>01</sup> stays flat till m0≃45 GeV where it starts increasing, sharply

<sup>01</sup> keeps increasing monotonously, with a small

<sup>0</sup> ¼ 0:2, and a moderate S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV.

<sup>01</sup> versus the DM mass m<sup>0</sup> is displayed

<sup>01</sup> hops onto another solution branch that is just emerging from negative

<sup>0</sup> . Then as the DM mass increases,

<sup>01</sup> ≃0:0046 that is

01

mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup>

m1=2, the coupling η<sup>ð</sup>4<sup>Þ</sup>

about half the value of λ<sup>ð</sup>4<sup>Þ</sup>

perturbative.

η<sup>ð</sup>4<sup>Þ</sup>

Figure 1. η<sup>ð</sup>4<sup>Þ</sup>

28 Trends in Modern Cosmology

80-fold larger than the mutual S<sup>0</sup> Higgs coupling constant λ<sup>ð</sup>4<sup>Þ</sup>

<sup>0</sup> , and <sup>η</sup><sup>ð</sup>4<sup>Þ</sup>

m0≃79 GeV, the mutual coupling constant η<sup>ð</sup>4<sup>Þ</sup>

increased <sup>S</sup>0-Higgs mutual coupling constant <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup>

The behavior of the <sup>S</sup><sup>0</sup> � <sup>S</sup><sup>1</sup> mutual coupling constant <sup>η</sup><sup>ð</sup>4<sup>Þ</sup>

Perhaps the most known constraints on a WIMP model are those coming from direct-detection experiments like the many we have cited in the introductory section. In such experiments, the signal sought for would typically come from the elastic scattering of a WIMP off a nonrelativistic nucleon target. However, as mentioned in the Introduction, until now, none of these direct-detection experiments have yielded an unambiguous dark matter signal. Rather, with increasing precision from one generation to the next, these experiments put increasingly stringent exclusion bounds on the dark matter-nucleon elastic-scattering total cross-section σdet in terms of the dark matter mass m0, and because of these constraints, many models can get excluded.

Therefore, a theoretical dark matter model like the two-singlet extension we discuss here has to satisfy these bounds to remain viable. For this purpose, we calculate σdet as a function of m<sup>0</sup> for different values of the parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1Þ and compare its behavior against the experimental bounds. The calculation is carried out with sufficient details in Ref. [31], and the total cross-section for non-relativistic S0-nucleon elastic scattering is given by

$$\sigma\_{\rm det} = \frac{m\_N^2 \left( m\_N - \frac{7}{9} m\_B \right)^2}{4 \pi (m\_N + m\_0)^2 v^2} \left[ \frac{\lambda\_0^{(3)} \cos \theta}{m\_h^2} - \frac{\eta\_{01}^{(3)} \sin \theta}{m\_1^2} \right]^2. \tag{7}$$

In this relation, mN is the nucleon mass and mB is the baryon mass in the chiral limit. The mutual coupling constants λ<sup>ð</sup>3<sup>Þ</sup> <sup>0</sup> and <sup>η</sup><sup>ð</sup>3<sup>Þ</sup> <sup>01</sup> are defined in Eq. (1). The relic-density constraint on the dark matter annihilation cross-section (6) has to be imposed throughout. In addition, we require now that the coupling constants be perturbative, and we do this by imposing the additional requirement 0 ≤ η<sup>ð</sup>4<sup>Þ</sup> <sup>01</sup> ≤ 1.

Generically, as m<sup>0</sup> increases, the detection cross-section σdet starts from high values, slopes down to minima that depend on the parameters, and then picks up moderately. There are features and action at the usual mass thresholds, with varying sizes and shapes. Regions coming from the relic-density constraint and new ones originating from the additional perturbativity requirement are excluded.

For the purpose of illustration, we choose three indicative sets of values for the parameters (θ, λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> , m1). We start first with a Higgs-S<sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> <sup>10</sup>o, a weak mutual <sup>S</sup>0-Higgs coupling λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:01, and an S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV. The behavior of σdet versus m<sup>0</sup> is shown in Figure 3. There, we see that for the two mass intervals 20–65 GeV and 75–100 GeV, plus an almost singled-out dip at m<sup>0</sup> ¼ m1=2, the elastic scattering cross-section is below the sensitivity of SuperCDMS. However, XENON1T should probe all these masses, except m<sup>0</sup> ≃58 and 85 GeV.

Increasing m<sup>1</sup> has the effect of closing possibilities for very light dark matter and thinning the intervals as it drives the predicted masses to larger values. Indeed, in Figure 4, where

Figure 3. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-Higgs coupling.

σdet ¼

mutual coupling constants λ<sup>ð</sup>3<sup>Þ</sup>

30 Trends in Modern Cosmology

additional requirement 0 ≤ η<sup>ð</sup>4<sup>Þ</sup>

(θ, λ<sup>ð</sup>4<sup>Þ</sup>

85 GeV.

Higgs coupling.

coupling λ<sup>ð</sup>4<sup>Þ</sup>

perturbativity requirement are excluded.

m2

<sup>N</sup> mN � <sup>7</sup>

4πð Þ mN þ m<sup>0</sup>

<sup>0</sup> and <sup>η</sup><sup>ð</sup>3<sup>Þ</sup>

<sup>01</sup> ≤ 1.

9 mB � �<sup>2</sup>

> 2 v2

In this relation, mN is the nucleon mass and mB is the baryon mass in the chiral limit. The

the dark matter annihilation cross-section (6) has to be imposed throughout. In addition, we require now that the coupling constants be perturbative, and we do this by imposing the

Generically, as m<sup>0</sup> increases, the detection cross-section σdet starts from high values, slopes down to minima that depend on the parameters, and then picks up moderately. There are features and action at the usual mass thresholds, with varying sizes and shapes. Regions coming from the relic-density constraint and new ones originating from the additional

For the purpose of illustration, we choose three indicative sets of values for the parameters

Figure 3. There, we see that for the two mass intervals 20–65 GeV and 75–100 GeV, plus an almost singled-out dip at m<sup>0</sup> ¼ m1=2, the elastic scattering cross-section is below the sensitivity of SuperCDMS. However, XENON1T should probe all these masses, except m<sup>0</sup> ≃58 and

Increasing m<sup>1</sup> has the effect of closing possibilities for very light dark matter and thinning the intervals as it drives the predicted masses to larger values. Indeed, in Figure 4, where

Figure 3. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-

<sup>0</sup> , m1). We start first with a Higgs-S<sup>1</sup> mixing angle <sup>θ</sup><sup>¼</sup> <sup>10</sup>o, a weak mutual <sup>S</sup>0-Higgs

<sup>0</sup> ¼ 0:01, and an S<sup>1</sup> mass m<sup>1</sup> ¼ 20 GeV. The behavior of σdet versus m<sup>0</sup> is shown in

λð Þ<sup>3</sup> <sup>0</sup> cos θ m<sup>2</sup> h

� <sup>η</sup>ð Þ<sup>3</sup> <sup>01</sup> sin θ m<sup>2</sup> 1

<sup>01</sup> are defined in Eq. (1). The relic-density constraint on

: ð7Þ

" #<sup>2</sup>

Figure 4. Elastic N � S<sup>0</sup> scattering cross-section as a function of m<sup>0</sup> for moderate m1, small mixing, and small WIMP-Higgs coupling.

m<sup>1</sup> ¼ 40 GeV, in addition to the dip at m1=2 that crosses SuperCDMS but not XENON1T, we see acceptable masses in the ranges of 40–65 GeV and 78 GeV up. The intervals narrow as we descend, surviving XENON1T only as spiked dips at 62 GeV and around 95 GeV.

On the other hand, a larger mutual coupling constant λ<sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> has the general effect of squeezing the acceptable intervals of m<sup>0</sup> by pushing the values of σdet up, and it may even happen that at some point, the model has no predictability. This case is shown in Figure 5, where <sup>θ</sup><sup>¼</sup> 10o, <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> ¼ 0:4, and <sup>m</sup><sup>1</sup> <sup>¼</sup> 60 GeV. In this example, the effects of increasing the values of both <sup>λ</sup><sup>ð</sup>4<sup>Þ</sup> <sup>0</sup> and m1. As we see, the model cannot even escape Cryogenic Dark Matter Search II (CDMSII).

Figure 5. Elastic cross-section σdet versus m<sup>0</sup> for heavy S1, small mixing, and relatively large WIMP-Higgs coupling.

#### 5. Constraints from particle phenomenology

If a dark matter model based on WIMP physics is not killed already by the constraints coming from cosmology, perturbativity, and direct detection, it has to undergo the tests of particle phenomenology. To see how this works, we discuss here the constraints on our two-singlet model that come from a small selection of low-energy processes, namely, the rare decays of ϒ mesons. The forthcoming discussion is based on work done in Ref. [32]. There, the interested reader will find a fuller account of this study, together with relation to Higgs phenomenology. Note that the dark matter relic-density constraint in Eq. (6) and the perturbativity requirement 0 < ηð Þ<sup>4</sup> <sup>01</sup> < 1 are implemented systematically. Also, as in Ref. [32], we will restrict the discussion to light cold dark matter.

We therefore look at the constraints that come from the decay of the meson ϒ in the state nS (n ¼ 1; 3) into one photon γ and one particle S1. For m1≲8 GeV, the branching ratio for this process is given by the relation:

$$\mathrm{Br}(\mathbf{Y}\_{n\mathbb{S}} \rightarrow \boldsymbol{\mathcal{Y}} + \boldsymbol{\mathcal{S}}\_{1}) = \frac{G\boldsymbol{\pi}m\_{b}^{2}\sin^{2}\theta}{\sqrt{2}\pi\alpha} \,\mathrm{x}\_{\boldsymbol{n}}\left(1 - \frac{4\alpha\_{\mathrm{s}}}{3\pi}f(\mathbf{x}\_{\boldsymbol{n}})\right) \mathrm{Br}^{(\mu)}\,\boldsymbol{\Theta}(\boldsymbol{m}\_{\mathbf{Y}\_{\boldsymbol{n}\mathbb{S}}} - \boldsymbol{m}\_{1}).\tag{8}$$

In the above expression, xn � <sup>1</sup> � <sup>m</sup><sup>2</sup> 1=m<sup>2</sup> <sup>ϒ</sup>ns � � with the mass of <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> given by <sup>m</sup><sup>ϒ</sup>1ð3Þ<sup>S</sup> <sup>¼</sup> <sup>9</sup>:46ð10:355<sup>Þ</sup> GeV, the branching ratio Brðμ<sup>Þ</sup> � Br <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> ! <sup>μ</sup>þμ� � � <sup>¼</sup> <sup>2</sup>:48ð2:18Þ � <sup>10</sup>�<sup>2</sup> [41], <sup>α</sup> is the QCD coupling constant, α<sup>s</sup> ¼ 0:184 the QCD coupling constant at the scale m<sup>ϒ</sup>nS , the quantity GF is the Fermi coupling constant, and mb is the b-quark mass [38]. The function fðxÞ incorporates the effect of QCD radiative corrections given in [42] and the step function is denoted by Θð Þx . However, a rough estimate of the lifetime of S<sup>1</sup> indicates that the latter is likely to decay inside a typical particle detector, which means we should take into account its most dominant decay products. We first have a process by which S<sup>1</sup> decays into a pair of pions, with the following decay rate:

$$\begin{split} \Gamma(S\_1 \to \pi\pi) &\simeq \frac{G\_F m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left[ \frac{m\_1^2}{27} \left( 1 + \frac{11m\_n^2}{2m\_1^2} \right)^2 \right. \\ &\quad \times \left( 1 - \frac{4m\_n^2}{m\_1^2} \right)^{\frac{1}{2}} \Theta(m\_1 - 2m\_n)(2m\_K - m\_1) \right] \\ &\quad + 3 \left( M\_u^2 + M\_d^2 \right) \left( 1 - \frac{4m\_n^2}{m\_1^2} \right)^{\frac{3}{2}} \Theta(m\_1 - 2m\_K) \, . \end{split} \tag{9}$$

Here, m<sup>π</sup> is the pion mass and mK is the kaon mass. Also, chiral perturbation theory is used below the kaon pair production threshold [43, 44], and the spectator-quark model above up to roughly 3 GeV, with the dressed u and d quark masses Mu ¼ Md ≃ 0:05 GeV. Note that this rate includes all pions, charged and neutral. Above the 2mK threshold, there is the production of both a pair of kaons and η particles. The decay rate for K production is

Constraining the Parameters of a Model for Cold Dark Matter http://dx.doi.org/10.5772/intechopen.69044 33

$$\Gamma(\mathbb{S}\_1 \to \mathbb{K} \mathbb{K}) \simeq \frac{9}{13} \frac{\mathfrak{Z} G\_{\mathbb{F}} M\_s^2 m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(1 - \frac{4m\_K^2}{m\_1^2}\right)^{\frac{3}{2}} \Theta(m\_1 - 2m\_\mathbb{K}).\tag{10}$$

In the above rate, Ms ≃ 0:45 GeV is the s-quark mass in the spectator-quark model [45, 46]. For η production, replace mK by m<sup>η</sup> and 9=13 by 4=13.

5. Constraints from particle phenomenology

0 < ηð Þ<sup>4</sup>

to light cold dark matter.

32 Trends in Modern Cosmology

process is given by the relation:

BrðϒnS ! <sup>γ</sup> <sup>þ</sup> <sup>S</sup>1Þ ¼ GFm<sup>2</sup>

In the above expression, xn � <sup>1</sup> � <sup>m</sup><sup>2</sup>

pions, with the following decay rate:

If a dark matter model based on WIMP physics is not killed already by the constraints coming from cosmology, perturbativity, and direct detection, it has to undergo the tests of particle phenomenology. To see how this works, we discuss here the constraints on our two-singlet model that come from a small selection of low-energy processes, namely, the rare decays of ϒ mesons. The forthcoming discussion is based on work done in Ref. [32]. There, the interested reader will find a fuller account of this study, together with relation to Higgs phenomenology. Note that the dark matter relic-density constraint in Eq. (6) and the perturbativity requirement

<sup>01</sup> < 1 are implemented systematically. Also, as in Ref. [32], we will restrict the discussion

xn <sup>1</sup> � <sup>4</sup>α<sup>s</sup>

<sup>3</sup><sup>π</sup> <sup>f</sup>ðxn<sup>Þ</sup> � �

Brðμ<sup>Þ</sup> <sup>Θ</sup>ð Þ <sup>m</sup><sup>ϒ</sup>nS � <sup>m</sup><sup>1</sup> : <sup>ð</sup>8<sup>Þ</sup>

#

ð9Þ

Θð Þ� m<sup>1</sup> � 2mK :

with the mass of ϒ<sup>1</sup>ð3Þ<sup>S</sup> given by m<sup>ϒ</sup>1ð3Þ<sup>S</sup> ¼

We therefore look at the constraints that come from the decay of the meson ϒ in the state nS (n ¼ 1; 3) into one photon γ and one particle S1. For m1≲8 GeV, the branching ratio for this

<sup>9</sup>:46ð10:355<sup>Þ</sup> GeV, the branching ratio Brðμ<sup>Þ</sup> � Br <sup>ϒ</sup><sup>1</sup>ð3Þ<sup>S</sup> ! <sup>μ</sup>þμ� � � <sup>¼</sup> <sup>2</sup>:48ð2:18Þ � <sup>10</sup>�<sup>2</sup> [41], <sup>α</sup> is the QCD coupling constant, α<sup>s</sup> ¼ 0:184 the QCD coupling constant at the scale m<sup>ϒ</sup>nS , the quantity GF is the Fermi coupling constant, and mb is the b-quark mass [38]. The function fðxÞ incorporates the effect of QCD radiative corrections given in [42] and the step function is denoted by Θð Þx . However, a rough estimate of the lifetime of S<sup>1</sup> indicates that the latter is likely to decay inside a typical particle detector, which means we should take into account its most dominant decay products. We first have a process by which S<sup>1</sup> decays into a pair of

sin <sup>2</sup>

π m2 1

<sup>u</sup> <sup>þ</sup> <sup>M</sup><sup>2</sup> d � � <sup>1</sup> � <sup>4</sup>m<sup>2</sup>

Here, m<sup>π</sup> is the pion mass and mK is the kaon mass. Also, chiral perturbation theory is used below the kaon pair production threshold [43, 44], and the spectator-quark model above up to roughly 3 GeV, with the dressed u and d quark masses Mu ¼ Md ≃ 0:05 GeV. Note that this rate includes all pions, charged and neutral. Above the 2mK threshold, there is the production of

<sup>θ</sup> <sup>m</sup><sup>2</sup> 1 27

2

1 þ

� �<sup>2</sup> "

11m<sup>2</sup> π 2m<sup>2</sup> 1

Θð Þ m<sup>1</sup> � 2m<sup>π</sup> ð Þ 2mK � m<sup>1</sup>

2

π m<sup>2</sup> 1

� �<sup>3</sup>

<sup>b</sup> sin <sup>2</sup> θ

� �

1=m<sup>2</sup> ϒns

ffiffiffi 2 <sup>p</sup> πα

<sup>Γ</sup>ð Þ <sup>S</sup><sup>1</sup> ! ππ <sup>≃</sup> GFm<sup>1</sup>

both a pair of kaons and η particles. The decay rate for K production is

4 ffiffiffi 2 <sup>p</sup> <sup>π</sup>

<sup>þ</sup> <sup>3</sup> <sup>M</sup><sup>2</sup>

� <sup>1</sup> � <sup>4</sup>m<sup>2</sup>

� �<sup>1</sup>

The particle S<sup>1</sup> also decays into c and b quarks (mainly c). Including the radiative QCD corrections, the corresponding decay rates are given by

$$\Gamma(\mathbb{S}\_1 \to q\overline{q}) \simeq \frac{3G\_F \overline{m}^{\theta^2} m\_1}{4\sqrt{2}\pi} \sin^2 \theta \left(\frac{1 - 4\overline{m}\_q^2}{m\_h^2}\right)^{\frac{3}{2}} \left(1 + 5.67 \frac{\overline{a}\_s}{\pi}\right) \Theta\left(m\_1 - 2\overline{m}\_q\right). \tag{11}$$