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Modern Cosmology

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Hi, you have reached the site of mystery and intrigue, take a tour by taking a slice out of the space-time loaf and make it one worth toasting! Whether you do or don’t may not be a question of free will because free will may be the biggest illusion of them all, but I’m not willing to comment.

Cheers Pete

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We’re New Here – Modern Cosmology

Stereolab cross-pollinate with Brazilian band Mombojó to grow something new

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Stereolab cross-pollinate with Brazilian band Mombojó to grow something new, in our JUNE 2023 issue of Uncut, available to buy here .

It’s clear from more than just their name that Modern Cosmology believe the universe has a plan for them. Full of serendipitous turns and lengthy gaps in the action, their tale begins in 2012 in São Paulo, Brazil, where Stereolab’s Laetitia Sadier was performing a solo concert. In attendance were several members of Mombojó , an act from Recife known for playfully merging Brazilian folk, samba and psychedelia with the alt-pop sounds of their favourite UK and US bands, Stereolab in particular. “We grew up on them,” guitarist Marcelo Machado (aka Um Cara Massa ) tells Uncut . “We’d been eating their sounds and textures from the time we were 15.”

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Starstruck but fuelled by liquid courage, Machado’s bandmate Felipe S passed Sadier a Mombojó CD that night. Months later, she listened to the gift and loved what she heard. As she describes it now, her Facebook message to them essentially read as follows: “If you need some backing vocals, I’m your gal.”

Mombojó’s 2014 album Alexandre included their inaugural collaboration, but the bonds deepened when Sadier joined her five new friends for a two-week residency in Recife. By then the singer was ready to make a stronger commitment: “When it was still ‘ Laetitia with Mombojó ’, I found that a little horrifying, actually. I thought if we were in the realm of creation, we might as well create a band – that felt like a more honest way to behave.”

The newly minted Modern Cosmology’s first release was 2016’s  “Summer Long”, an intoxicating EP to anyone who appreciates Stereolab’s dabblings in bossa nova and has wondered what a space-age Os Mutantes might sound like. Seven years later, they return with What Will You Grow Now? , a full-length LP that’s among the most inventive work either party has ever made, amply demonstrating the “freedom and joy” Sadier found in Mombojó’s amiable company.

Though Stereolab’s reactivation and other projects kept Sadier far from Brazil, they connected online to work on songs started in Recife and others that grew out of Mombojó jams (such as the free-flowing one recorded on a boat that yielded the album’s title track). The project also provided some solace to the participants amid the turbulence that surrounded its creation. “This project bracketed a complex time in the world,” says bassist Missionário José . “The Trump administration was a pain to everybody, and here we had the Bolsonaro administration. We began to make this album before they started to win the elections, and now it’s coming out after all that has… not quite ended, but at least halted for now.”

A vibrant statement of resistance and resilience, the music also retains its sense of spontaneity despite its protracted recording process. “When they sent me the backing tracks, I adored the fact that it felt so open and unfinished,” says Sadier . “I told them we should keep this vibe, because I feel that in music, things can be polished and sealed and finished. I loved that open-ended quality.” Mombojó are equally appreciative of their partner. “She has a talent of flipping ideas upside down,” says José . “We think a song is going in a certain direction but when she adds her vocals and lyrics, now there’s a song we couldn’t have imagined.”

The Modern Cosmology team express their eagerness to happen upon more of these happy accidents, and to finally perform live together again; Brazil’s the likeliest site of shows for now, but time will tell. Either way, they’re thrilled to see this project blossom. “The seeds found the right ground and got the right amount of water and light,” says Sadier , “and they have been growing steadily ever since.”

What Will You Grow Now? is out on May 5 via Duophonic

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Article contents

The emergence of modern cosmology.

• Helge Kragh Helge Kragh Niels Bohr Institute
• https://doi.org/10.1093/acrefore/9780190871994.013.127
• Published online: 13 December 2023

The term modern cosmology primarily refers to the developments concerned with the expansion of the universe, its origin billions of years ago, and the concept of dark matter. Similar to the history of any other area of science, the history of cosmology is rich in wrong theories and false trials. According to the simplest version of Brandon Carter’s anthropic principle, carbon-based life could not have originated in a universe evolving just slightly differently from the one observed. The present debate concerning the anthropic principle and its consequences is in some ways strikingly similar to the cosmological controversy of the past between the steady-state theory and relativistic evolution theories.

• modern cosmology
• dark matter
• Brandon Carter
• anthropic principle
• steady-state theory
• relativistic evolution theories

To the minds of modern astronomers and physicists engaged in the quest of understanding the universe at large, the term modern cosmology probably refers to developments within the 2010s to 2020s. However, these developments, as exemplified by new concepts such as dark matter and the accelerating universe, depended crucially on much earlier progress. In a broader and more historical perspective, “modern cosmology” rested on two associated insights going back to the 1930s. It was in this period that scientists discovered the expansion of the universe, and the first hypotheses of a finite-age universe were tentatively proposed. The mind-boggling idea of a big bang in which literally everything came into being was at first resisted and only widely accepted in the mid-1960s. If one insists on a specific year marking the birth of modern cosmology, 1965 is a good choice.

To give the necessary background, this article starts earlier—namely, with charting the state of the art of cosmology from about 1890 to 1925 . During this period, Hubble’s discovery of galaxies beyond the Milky Way was an observational climax and Einstein’s cosmological model based on general relativity was a theoretical climax. General relativity came to be seen as the only theoretical framework for cosmological models, but in the 1950s, this was not yet uniformly accepted. The steady-state model did not rest on the foundation of general relativity theory, and it denied that the universe had ever come into existence. Although the model turned out to be wrong, it was a fruitful mistake that stimulated new thinking and new observational programs. Like the history of any other area of science, the history of cosmology is rich in wrong theories and false trails. These belong to its history no less than do the theories and approaches that are recognized today as correct or progressive.

This article deals only superficially with the advances in instrument technologies that were particularly important in the second part of the 20th century and today even more so with the advent of so-called high-precision cosmology. Also, other aspects of the historical development are missing or just mentioned, such as the philosophical consequences of the modern picture of the universe. Fortunately, several scholarly works describe the development more comprehensively, and which the reader may wish to consult for further details and a broader scope (e.g., Kragh & Longair, 2019 ; Longair, 2006 ; North, 1994 ; Peebles, 2020 ).

1. Fin-de-Siècle Cosmologies

Astronomers and physicists in the 19th century rarely dealt with questions of cosmology and cosmogony, which were widely considered to belong to philosophical speculations ( Merleau-Ponty, 1983 ). Yet there was an interest in problems related to the structure and evolution of the universe at large, such as whether cosmic space and its material content were finite or infinite. What became known as Olbers’s paradox was formulated by the German astronomer Heinrich Wilhelm Olbers in 1826 but has a history that goes much farther back in time ( Harrison, 1987 ; Jaki, 1969 ). The essence of this so-called paradox is that in an infinite universe uniformly populated with stars, the sky at night should be as bright as on a sunny day or even brighter. The explanation favored by Olbers and most other astronomers at the time was to assume that space is filled with a light-absorbing medium. It was generally agreed that space is infinite, and in the 19th century , astronomers did not generally regard the darkness of the night sky as a serious problem. It only became so in the early part of the 20th century when it turned out that interstellar space was much more transparent than Olbers and his contemporaries had assumed.

The question of the size of the stellar universe also turned up in connection with the paradigmatic belief that in the end, it was governed by Newton’s law of gravitation. In works of 1895 and 1896 , the prominent German astronomer Hugo von Seeliger proved that an infinite universe with stars distributed approximately uniformly could not be brought into agreement with Newton’s law ( Norton, 1999 ). Seeliger did not conclude that the universe was therefore finite—which would lead to other problems—but instead suggested a slight modification of the law at very large distances. While Newton’s law states that the force between two masses is attractive and varies inversely with the square of their distance, Seeliger added an exponential attenuation factor corresponding to a long-range repulsive force. In this way, he could save the cherished infinite stellar universe and still remain on the safe ground of classical mechanics.

The problem highlighted by Seeliger came to be known as the “gravitational paradox,” a kind of mechanical analog to Olbers’s optical paradox. Without relying on Seeliger’s ad hoc hypothesis of a repulsive cosmic force, the problem was also considered by other astronomers in the early 20th century . The paradox or problem assumed a homogeneous distribution of matter, an assumption that some astronomers questioned. Carl Charlier, a leading Swedish astronomer, developed in 1908 a so-called hierarchic model of the universe that allowed it to be finite and yet in accordance with Newtonian mechanics ( Jaki, 1969 ; Norton, 1999 ). Thus, by making more or less artificial assumptions, it was possible to argue in both favor and disfavor of a finite stellar universe. However, most astronomers and physicists agreed that space itself was infinite if not necessarily filled throughout with matter.

As regards the temporal extension of the universe, the consensus view was either to ignore the issue or to defend an eternal universe as the only reasonable possibility. As Charlier stated in a paper of 1896 , “A finite time is a contradiction. . . . An infinite time may be difficult to conceive, but it is not contradictory” ( Charlier, 1896 , p. 481).

The discussions mentioned so far took it for granted that space was flat or Euclidean. Although non-Euclidean geometries with curved space had been known since the 1830s, it took a generation or more before these ideas made an impact on astronomers’ conception of space. As a few mathematicians realized in the 1860s, there are three and only three geometries with constant curvature: flat space (zero curvature), spherical space (positive curvature), and hyperbolic space (negative curvature). In terms of the dimensionless curvature constant k , the three possibilities were given by k = 0, k = +1, and k = −1, respectively. Which of these possibilities corresponded to the real physical space in which people live?

The first astronomer to consider the question in a cosmological context was the German astrophysicist Karl Friedrich Zöllner, who in a little-noticed work of 1872 focused on the case of constant positive curvature ( k = +1) first considered by the mathematician Bernard Riemann in the 1850s. As Zöllner pointed out, if cosmic space were positively curved in analogy with the two-dimensional space of the surface of a sphere, then space and all its material content would be finite and yet there would be no limit to the universe ( Kragh, 2012b ). Since there were only a finite number of stars in such a model universe, it offered a novel solution to Olbers’s paradox. However, neither Zöllner himself nor other scientists followed up on his brilliant argument, which left no trace in the further development of cosmology.

Although a few astronomers in the late 19th century considered the possibility of a non-Euclidean universe, it was more as an academic exercise than because they believed that the universe might actually be curved. Leading scientists such as the American astronomer Simon Newcomb and the French mathematician and physicist Henri Poincaré maintained that although space might be curved, there were neither observational nor theoretical reasons to believe that it was. According to Poincaré’s conventionalist view of science, the geometry of space was nothing but a convention and not something that could be determined objectively.

The first astronomer since Zöllner who seriously examined the geometrical structure of space was young Karl Schwarzschild, who in 1900 systematically compared astronomical observations with expectations based on the hypothesis of curved space ( Kragh, 2012a ). However, the data available to Schwarzschild did not allow a definite conclusion except that they indicated a lower bound for the radius of curvature corresponding to either a spherical (closed) or a hyperbolic (open) universe. In the first case, he concluded that the radius of the universe must be greater than 1,600 light years. Although Schwarzschild’s careful investigation was known to contemporary astronomers, it did not disturb the consensus view that in all likelihood, cosmic space was flat and consequently infinite. The situation only changed when Newton’s theory of gravitation was replaced by Einstein’s theory of general relativity.

2. Galactic and Extragalactic Astronomy

By 1900 , the question of the location of the nebulae relative to the Milky Way system had moved to the forefront of what in retrospect can be termed observational cosmology . The crucial question was whether the nebulae, such as the Andromeda, were located within the Milky Way or whether they were large extragalactic structures apart from it and of a size and nature comparable to the Milky Way ( Smith, 2019 ). According to the astronomer and astronomy writer Agnes Clerke (1890 , p. 368) , “No competent thinker, with the whole of the available evidence before him, can now, it is safe to say, maintain any single nebula to be a star system of coordinate rank with the Milky Way.” And yet there were astronomers, even competent thinkers, who defended the latter “island universe” hypothesis going back to a work of 1755 written by the famous philosopher Immanuel Kant. All the same, the majority probably shared Clerke’s view and its corollary that the Milky Way system was essentially identical to the material universe. Among the reasons for the uncertainty was not only lack of knowledge about the distances to the nebulae but also that the size of the Milky Way was a matter of dispute.

Due to works of Henrietta Leavitt, Ejnar Hertzsprung, and Harlow Shapley, by 1918 , astronomers realized that if they could find just a single variable star of the type known as Cepheids in a nebula, it would be enough to determine its distance. All they had to do was to measure the luminosity period of the star. Unfortunately, no such Cepheid had been identified in either the Andromeda nebula or other nebulae.

During the same period in which the Cepheid method was developed by Shapley and others, Vesto Slipher at the Lowell Observatory in Arizona studied the spectra of the Andromeda and other spiral nebulae. Interpreting the measured shifts in wavelengths ∆ λ as Doppler shifts implied that they were due to radial velocities v in accordance with Doppler’s formula ∆ λ / λ = v / c , where c denotes the speed of light. Slipher came to the surprising conclusion that most of the nebulae receded from the Earth with radial velocities up to 1,000 km/s or more ( Way & Hunter, 2013 ). Although he refrained from interpreting the nebular redshifts cosmologically, his observations contributed to a revival of the island universe theory. After all, they suggested that the fast-moving spiral nebulae could not be gravitationally bound to the Milky Way. However, Slipher’s data constituted no proof, and in the absence of knowledge of the distances to the nebulae, the redshift phenomenon was circumstantial evidence only. By the late 1920s, nebular redshifts would become the observational basis for the expanding universe, but around 1920 , no one thought along such lines.

The opposing views of the universe were discussed in public at a meeting in Washington, DC on April 26, 1920 , subsequently known as the “Great Debate” ( Smith, 1982 ). The discussants were Heber Curtis, who argued for the island universe, and Shapley, who argued against it and for an immensely large Milky Way. Shapley believed at the time that the galactic Milky Way system consisting of stars and nebulae had a diameter of no less than 300,000 light years and that this was incompatible with the island universe model. Although the Washington debate was enlightening, from a scientific point of view, it was disappointing insofar that it did not result in any consensus. No new observations were reported that unambiguously spoke for one of the rival theories and against the other.

For yet another few years, the situation remained a stalemate if now with increased support for the island universe theory from Slipher’s nebular redshifts and also from observations of novae in spiral nebulae. The stalemate ended drastically in the autumn of 1923 , when Edwin Hubble, working at the Mount Wilson Observatory, made an important discovery. In the course of searching for novae in spiral nebulae, he noticed on a photographic plate of the Andromeda nebula a variable starlike object that he first thought was a nova. By examining the object more closely, he realized that it was not a nova but a Cepheid variable ( Berendzen et al., 1984 ). The luminosity and period of the Cepheid told him that Andromeda was about 930,000 light years away from the Earth and thus could not possibly be a member of the Milky Way system even should it have the size that Shapley surmised.

Hubble’s celebrated discovery, which was only announced in early 1925 , effectively settled the Great Debate in favor of the island universe theory, which was soon accepted by the astronomical community. The few opponents did not succeed in reversing the consensus view that the spiral nebulae were extragalactic worlds with a scale and structure comparable to that of the Milky Way. Now the spiral nebulae became extragalactic objects or just galaxies, and the universe at large came to be seen as a vast congregation of galaxies floating in a possibly infinite cosmic space. The advance that led to the new picture of the universe was observational in the classical astronomical tradition with no significant contribution from modern physical theory.

It is worth noting that the word Galaxy (with capital G) was traditionally a synonym of Milky Way, meaning that the term Milky Way Galaxy was redundant. Hubble never spoke of galaxies but consistently used nebulae or, to denote those outside the Milky Way, extra-galactic nebulae . His widely read semipopular book published in 1936 carried the title The Realm of the Nebulae . The presently used term galaxy was advocated by Shapley, but for a long time, it was unpopular and won broad acceptance only after Hubble’s death in 1953 .

Armed with knowledge of the distances of more and more spiral galaxies (to use modern terminology), Hubble began in 1928 a new research project with the aim of relating the distances ( r ) to their redshifts ∆ λ / λ . There were theoretical reasons for expecting some kind of simple relationship, and on March 15, 1929 , Hubble reported in the Proceedings of the National Academy of Sciences that the two quantities were nearly proportional ( Nussbaumer & Bieri, 2009 , pp. 114–120; Smith, 1982 , pp. 180–186). He took most of the redshifts from Slipher’s work and, like other astronomers, translated them by means of Doppler’s formula into velocities ( v ). More precisely, he found that v = Hr with the constant of proportionality H (the Hubble constant or parameter) being around 500 km/s/Mpc. The symbol Mpc stands for megaparsec, a distance unit dating from 1913 and approximately equal to 3.3 million light years.

Hubble’s 1929 paper is famous and often identified with the discovery of the expanding universe, but this is to read too much in it ( Hubble, 1929 ). Nowhere in his paper did he conclude that the galaxies are actually receding from us or otherwise suggest that the universe is in a state of expansion. The velocities appearing in the linear law were really redshifts that could be, but did not need to be, understood as real recessional velocities. Hubble cautiously referred to them as “apparent velocities.” For more than a year, his paper created but little attention, and it was far from hailed as a revolutionary contribution to cosmology, an observational proof that the universe is expanding. That only came 1 or 2 years later and then because of progress in theoretical cosmology.

3. Relativistic Cosmological Models

In February 1917 , Albert Einstein read a paper to the Prussian Academy of Sciences in which he suggested a modification of the field equations of general relativity theory introduced 2 years earlier. The new equations became known as his “cosmological field equations” because they were aimed to describe the universe as a whole. According to Einstein, there was only one physically realistic solution to the equations corresponding to a particular cosmological model. This original Einstein model was characterized by being spatially closed and hence finite despite having no boundary. It was homogeneously filled with dilute matter and likewise could be ascribed a finite mass. On the other hand, Einstein assumed the universe to be temporally infinite, meaning that its radius of curvature did not vary in time.

To secure a static universe, which he and others thought was justified by current astronomical data, Einstein introduced in his equations a term that counteracted the gravitational attraction and was expressed by the average density of matter. The new term was given by an exceedingly small but nonzero “cosmological constant,” a new constant of nature similar to the attenuation factor that Seeliger had introduced in his Newtonian cosmology of the late 1890s. The cosmological constant is usually designated by the Greek letter Λ ‎ (lambda). Einstein was not much concerned about the agreement of his model with astronomical observations. “Whether, from the standpoint of present astronomical knowledge, it is tenable, will not here be discussed,” he wrote at the end of his paper ( Einstein, 1952 , p. 188).

Later the same year, after having studied Einstein’s theory, the Dutch astronomer Willem de Sitter found another solution to the field equations that represented a very different world model. De Sitter’s universe was empty—that is, containing no matter at all—and yet it was spatially closed like Einstein’s. Moreover, de Sitter and his contemporaries conceived it as a model of a static universe and not, as later cosmologists would do, as representing an exponentially expanding universe. Also like Einstein’s model, the one proposed by de Sitter made use of the cosmological constant but in this case related in a different way to the radius of curvature. Of course, in de Sitter’s model, the cosmological constant was unrelated to the density of matter (which he took to be zero).

Although Einstein admitted that de Sitter’s alternative world model was mathematically correct, he insisted that it was wrong on physical grounds and that it disagreed with the spirit of general relativity theory. All the same, de Sitter’s alternative attracted much attention among the few physicists, astronomers, and mathematicians specializing in relativistic cosmology ( Kerzberg, 1989 ). One reason why astronomers found the model to be attractive was that it, contrary to the Einstein world, predicted a kind of galactic redshifts that were possibly related to the redshifts observed by Slipher. In his 1929 paper on the linear redshift–distance law, Hubble suggested that the redshifts might in part be explained on the basis of de Sitter’s theory.

During the 1920s, physicists and astronomers discussed in detail which of the two rival relativistic theories of the universe was the most satisfactory. Among those active in the debate were the British astronomer Arthur Eddington, the German mathematician Hermann Weyl, and, in the United States, Howard Robertson and Richard Tolman. However, no agreement or clarification was reached. Only at the end of the decade did they vaguely recognize that there might be other models that incorporated features of both the Einstein world and the de Sitter world. With a few exceptions, they did not seriously contemplate that the universe as a whole might be dynamic rather than static. As late as 1929 , Einstein wrote in an article for the Encyclopaedia Britannica ,

Nothing certain is known of what the properties of the space-time continuum may be as a whole. Through the general theory of relativity, however, the view that the continuum is infinite in its time-like extent but finite in its space-like extent has gained in probability. ( Kerzberg, 1989 , p. 335)

This view would not last for long.

The collapse of the static universe paradigm was foreshadowed by Alexander Friedmann, a Russian theoretical physicist, in a paper published in the widely read Zeitschrift für Physik in 1922 . Contrary to other workers in the field, Friedmann concluded from a systematic investigation of Einstein’s field equations that there were other solutions than the two static models associated with the names of Einstein and de Sitter. The general solution, he proved, included a variety of dynamical world models, that is, models in which the radius of curvature depended on the time parameter. For example, he found a class of solutions that formally included expanding world models with an absolute beginning in time. Friedmann even referred to “the time since the creation of the world . . . to the present state” ( Tropp et al., 1993 , p. 158). Moreover, he derived from the field equations that there were solutions corresponding to a “cyclic universe” in which the size of the closed universe oscillated or pulsated between zero and a maximum value. And this was not all, for in a follow-up paper 2 years later, he discussed as the first one the possibility of hyperbolic and spatially infinite world models with a constant negative curvature.

Remarkably, although Einstein knew about Friedmann’s work and responded critically to it, it made no impact at all on mainstream cosmology. When the Belgian physicist and expert in general relativity Georges Lemaître in 1927 arrived at essentially the same equations as Friedmann, he was unaware of the work of his Russian colleague, who died prematurely in 1925 . From a mathematical point of view, Lemaître’s paper, written in French and published in a not widely circulated journal ( Annales Scientifique Bruxelles ), was similar to Friedmann’s, but otherwise, it was strikingly different as it was of a physical rather than mathematical nature ( Farrell, 2005 ; Lambert, 2011 ). Lemaître explicitly argued that the one and only real universe was expanding, and he supported the claim by explaining the galactic redshifts as a result of the expansion. As he emphasized, the redshifts were caused not by galaxies moving through space but by galaxies being carried with the expanding space. On this basis, he even derived the redshift–distance law announced by Hubble 2 years later and estimated for the “Hubble constant” a value of 625 km/s/Mpc.

Like Einstein’s static universe, Lemaître’s expanding version of it was positively curved and hence of finite size. By the late 1920s, Einstein had abandoned the idea of a cosmological constant, which he now considered a mistake, but in Lemaître’s thinking about the universe, it was a crucial and indispensable quantity that he kept to throughout his life. His work of 1927 is often said to be the beginning of big-bang cosmology, but this is a mistake. It pictured the universe as gradually evolving from the static Einstein state and thus without a definite beginning in time.

For about 3 years, Lemaître’s brilliant paper shared the fate of Friedmann’s work: It was ignored and most likely simply unknown to the majority of contemporary cosmologists. Remarkably, not a single scientific paper referred to it until the spring of 1930 . Einstein was one of the very few who did know about the paper, but at the time, he was convinced that the universe was static and consequently rejected Lemaître’s model. Many years later, Lemaître recalled a conversation he had with Einstein in the autumn of 1927 :

After some favorable technical remarks, he concluded by saying that from the physical point of view that appeared completely abominable to him. . . . I had the impression that Einstein was hardly aware of the astronomical facts. ( Lemaître, 1958 , p. 131)

4. From Expanding Universe to Primeval Atom

Attitudes concerning the expanding universe changed dramatically in the early part of 1930 after Eddington belatedly studied or restudied Lemaître’s paper. He now understood that the expanding model answered most of the problems that had plagued cosmology and that it supplied Hubble’s empirical redshift–distance law with the necessary theoretical foundation. De Sitter reached the same conclusion, writing in a letter to Shapley in April 1930 that Lemaître’s theory of the expanding universe was “the true solution, or at least a possible solution, which must be somewhere near the truth” ( Smith, 1982 , p. 187). With the enthusiastic support of Eddington and de Sitter, the expanding universe became quickly accepted by most specialists, including Einstein—although in his case with some delay. The conversion from a static to an expanding world picture amounted to nothing less than a paradigm shift in cosmological thought.

This is not to say that everyone converted to the new paradigm, for some leading astronomers either adopted an agnostic attitude or suggested alternative models that explained the redshifts on the basis of a static universe. To the first group belonged Hubble, who never endorsed the expanding universe as uncontrovertibly true. The second group was represented by the innovative Swiss American astrophysicist Fritz Zwicky, who pioneered what became known as the “tired light” alternative widely discussed in the 1930s ( Kragh, 2017 ). Nonetheless, by the early 1930s the expanding universe was broadly accepted and had even entered the public arena in the form of popular books such as de Sitter’s Kosmos from 1932 and Eddington’s best-selling The Expanding Universe from 1933 .

The new revelation also gave rise to expanding universe models that differed from Lemaître’s original 1927 model or what, because of certain refinements added by Eddington, was often referred to as the “Eddington–Lemaître model.” Thus, in the spring of 1931 , Einstein suggested a cyclic model similar to the one discussed by Friedmann 9 years earlier. The new Einstein universe belonged in a formal sense to the big-bang category insofar that it postulated a sudden beginning in time. The same was the case with the more important flat-space model with no cosmological constant that Einstein and de Sitter jointly proposed in 1932 . According to this model, the universe expanded forever but at a continually slower rate. A simple calculation showed that its age t * was given by 2/3 times the inverse Hubble constant also known as the Hubble time ( T = 1/ H ).

The influential Einstein–de Sitter model was of the big-bang type only in a formal mathematical sense and not in a physical–realist sense. In this respect, it differed markedly from Lemaître’s daring proposal in Nature of May 9, 1931 , that the birth of this universe was due to a cataclysmic explosive event in the far past. The Belgian physicist imagined that there once existed what he metaphorically called a highly radioactive “primeval atom” in which the mass of the entire universe was concentrated. Only with the violent decay of this “atom” would space and time come into existence. The original super-atom was not an abstract pointlike “singularity” but a real body of finite size and density. Lemaître assumed, if only as an illustration or analogy, that this original body had a density of the same order as an atomic nucleus, namely, 10 15 g / cm 3 . Otherwise, it had no physical properties and was therefore inaccessible to scientific inquiry ( Kragh & Lambert, 2007 ).

Lemaître’s note in Nature was no more than an enigmatic and spirited speculation, but later in 1931 , he developed it into a quantitative theory that deservedly has been called the first example ever of a relativistic big-bang universe. First of all, according to this model, the universe had only existed in a finite period of time. As a result of the original creation event, the universe had expanded rapidly, and due to the cosmological constant, it was presently in a state of accelerated expansion that would continue indefinitely. Why believe in this apparently fantastic scenario? Would there be any traces left of the original explosion that could be examined today? Lemaître thought so, arguing—incorrectly, it turned out—that the poorly understood cosmic rays were electrically charged fossils of the disintegration of the primeval atom several billion years ago. The cosmic rays, he said poetically, were “one of the most curious of the hieroglyphs of our astronomical library” ( Kragh, 1996 , p. 53).

Lemaître’s primeval atom theory received positive attention in the popular press but was met with reservation and sometimes hostility in the scientific community. A universe with literally a beginning in time—not only as a mathematical abstraction but as a physical reality—was widely considered unacceptable and especially so if the hypothesis was not supported by observational evidence. Although Eddington was one of the pioneers of the expanding universe, he flatly dismissed Lemaître’s new theory on both scientific and philosophical grounds. As he stated on one occasion, he found the notion of a created universe to be “repugnant.” Other astronomers and physicists likewise rejected the primeval atom theory or simply chose to ignore it. The Canadian astronomer John Plaskett said about Lemaître’s hypothesis that it was “the wildest speculation of all . . . an example of speculation run mad without a shred of evidence to support it” ( Plaskett, 1932 ).

To make a long story short, through the 1930s, big-bang solutions to the cosmological field equations were rarely taken seriously or assigned physical reality. In the few cases where scientists entertained ideas somewhat similar to Lemaître’s, they did not refer to the Belgian physicist’s primeval atom hypothesis. To mention but one example, in 1938 , the German physicist Carl Friedrich von Weizsäcker speculated that the early universe was extremely hot (about 10 11 K) and of nuclear density. He thought that nuclear reactions in the primeval state of the universe might have produced not only some of the chemical elements but also the energy necessary for the subsequent expansion ( Drischner, 2014 ). Although Weizsäcker’s cosmological scenario had elements in common with the earlier primeval atom hypothesis, he did not refer to Lemaître’s work. The idea of an original nuclear explosion or radioactive decay was only further developed after World War II and then in a version that owed little to Lemaître’s primeval atom theory of the early 1930s.

5. Two Rival Theories of the Universe

In the late 1940s, two completely different theories of the expanding universe saw the light of day. One of them, primarily due to the Russian American nuclear physicist George Gamow, was a typical big-bang theory insofar that it assumed that the universe was born in an extremely hot and dense state a finite time ago and since then had continued to evolve. The other theory of the universe, known as the “steady-state theory,” was proposed by three young Cambridge physicists, Hermann Bondi, Thomas Gold, and Fred Hoyle. It claimed that the universe was eternal in the past as well as in the future and that on a large scale, it had never changed significantly. Whereas Hoyle’s more mathematical version of the steady-state theory was based on equations similar to those proposed by Einstein, the Bondi–Gold version denied that the theory of general relativity was valid for the universe as a whole. Although the two versions differed in some respects, they led to the same picture of an everlasting and eternally expanding universe. The rivalry between the big-bang theory and the steady-state theory colored much of the cosmological debate from 1948 to about the mid-1960s ( Kragh, 1996 ).

In a letter to Einstein of September 1946 , Gamow stated the essence of his thoughts about the explosive beginning of the universe:

It is important to remember that in order to explain the present relative abundance of the chemical elements one must agree that in “the Days of Creation” the mean density and temp. of the Universe was 10 7 gm / cm 3 and 10 10 K. ( Kragh, 1996 , p. 110)

He first assumed that the mechanism of element formation was proton–neutron reactions, with the protons formed by the radioactive decay of a primeval hot soup of neutrons. To develop his theory quantitatively, Gamow entered a close and fruitful collaboration with two assistants, Ralph Alpher and Robert Herman. The three physicists realized that the high temperature required the very early universe to be composed mainly by electromagnetic radiation and not by matter. As the universe expanded and grew colder, the ratio between radiation density and matter density would decrease and eventually lead to the present matter-dominated universe.

Importantly, in a brief paper of 1948 , Alpher and Herman argued that the original radiation had by now shifted to the microwave area and with a very low intensity corresponding to a temperature of around 5 K. In other words, they predicted the existence of the weak cosmic microwave background that many years later would be detected and then be interpreted as convincing evidence for the hot big-bang theory. However, at the time, the prediction failed to attract attention among physicists and astronomers outside Gamow’s small research group ( Alpher & Herman, 2001 ).

The calculations of Gamow and his collaborators resulted in promising results for the cosmological formation of helium but not for the heavier elements. All attempts to build up elements heavier than helium by means of realistic thermonuclear reactions failed, which was widely seen as a grave difficulty for Gamow’s theory of the early universe. This theory reached its climax in 1953 , when Alpher, Herman, and James Follin presented a detailed and much improved version of it. The complex calculations of the three authors followed the evolution of the early universe from t = 10 − 4 seconds after the initial explosion at t = 0 to around 10 minutes later. In this brief period of time, they found that helium would be formed in an amount of 32% of the total mass of the universe. This would turn out to be an approximately correct figure, but in the early 1950s, there were no observational results with which the theoretical figure could be compared. For this reason, it was ineffective as an argument in favor of the big-bang theory.

Strangely, from a modern perspective, with the work of Alpher, Herman, and Follin, big-bang cosmology came to an almost complete halt and was only resurrected a decade later by other physicists. From 1954 to 1963 , only a single scientific paper was devoted to the theory of Gamow and his collaborators.

There were additional reasons why the majority of physicists and astronomers tended to disbelieve finite-age relativistic cosmologies in general and the big-bang theory in particular. Most of the evolutionary models belonging to this class predicted an age of the universe t * smaller than the Hubble time and therefore smaller than the age of the stars and even of the Earth! Based on Hubble’s observations, the accepted Hubble time was 1.8 billion years, whereas the Earth was known to have an age of 4 billion years or possibly more ( Kragh, 1996 , pp. 73–79, 271–273). The “age paradox” only became less paradoxical when astronomical measurements in the mid-1950s showed that the value of the Hubble time was too low by a factor of 2 or more. By the end of the decade, a value of 10–15 billion years was broadly accepted (the present values are T = 14.4 billion years and t * = 13.7 billion years).

Of course, since there was no beginning of the universe posited by the steady-state theory, there was also no age paradox. In its early version, this theory built conceptually on the “perfect cosmological principle,” according to which the universe in its large-scale features was homogeneous in both space and time. It followed from the principle that the average density of matter must remain constant and therefore that new matter, perhaps in the form of hydrogen atoms, must continually be created as the universe expands. Although the necessary creation rate was miniscule, no more than 10 − 43 g / s / cm 3 , it seemed to violate the fundamental law of energy conservation and was for this reason a most controversial element in the new cosmology.

Contrary to the wide class of relativistic evolution theories, the theory of Hoyle and his allies led to several definite predictions that could be tested by means of experiment and observation not only in principle but also in practice. In philosophical terminology, the theory was highly falsifiable. For example, the steady-state universe was infinite and geometrically flat, and it expanded exponentially at a constant rate. The predicted matter density happened to be the same as that of the Einstein–de Sitter model—namely, around 5 × 10 − 28 g / cm 3 . Moreover, as shown by the Irish physicist William McCrea in 1950 , it followed from the steady-state theory that there was a definite age distribution of the galaxies and that their average age was equal to one third times the Hubble time. McCrea was during the 1950s an active and valuable supporter of the steady-state theory, which he preferred for both scientific and philosophical reasons.

There was no hot and dense state in the steady-state theory, which consequently had to account for the abundance and distribution of the elements solely in terms of nuclear reactions taking place in stellar bodies of various kinds. In an important collaborative work with William Fowler, Margaret Burbidge, and Geoffrey Burbidge, Hoyle produced in 1957 a comprehensive theory of stellar nucleosynthesis that explained the formation of practically all elements and isotopes from the lightest to the heaviest. This pioneering theory is generally known under the acronym B2FH, a reference to Burbidge, Burbidge, Fowler, and Hoyle. The successful B2FH theory was not explicitly derived from steady-state assumptions, but because it made no use of the hypothesis of a big bang, it was sometimes taken as indirect support of steady-state cosmology.

6. A Cosmological Controversy

The more than decade-long discussion concerning the foundation of cosmology and the evolution of the universe was in part of a philosophical nature and attracted no less attention among philosophers than among physicists and astronomers ( Kragh, 2022 ). By the mid-1950s, it was still uncertain if the universe as a whole could be the subject of scientific inquiry in the same way as parts of it such as stars and galaxies. If cosmology were accepted as physical science, what would the criteria for theory choice be?

Characteristic for the climate at the time, in 1954 , Bondi discussed with Gerald Whitrow, an esteemed physicist and astronomer, the question “Is Physically Cosmology a Science?” in the pages of the British Journal for the Philosophy of Science . The question was seriously meant, not rhetorical. Inspired by the influential Austrian British philosopher Karl Popper, Bondi emphasized falsifiability as the hallmark of science, and from this perspective, he saw no reason why cosmology should not be a genuine science. Whitrow was not so sure. Bondi also suggested that since the steady-state theory was undoubtedly more easily falsifiable than the evolution theories based on general relativity, from a methodological point of view, the first theory was a better choice than the latter.

Despite the role played by methodological and other philosophical considerations in the cosmological controversy, it was first and foremost about observations and how they related to the predictions of the two theories. It was generally agreed that ultimately, the controversy would be settled by new observational and experimental data.

If the curvature of space and the manner in which the rate of expansion increased or decreased could be determined, it would be possible to discriminate between the two rival cosmological models. This approach was followed by the leading American astronomer Allan Sandage, who in 1956 reported observations that apparently disagreed with the steady-state theory. However, by closer inspection, his data turned out to be inconclusive, and neither did later results from optical astronomy unambiguously refute the theory defended by Hoyle, Gold, and Bondi. The new science of radio astronomy promised to do better. Radio astronomy turned into radio cosmology in 1955 , when Martin Ryle and his group at Cambridge University concluded that the distribution of radio sources or what at the time was called “radio stars” disagreed with the steady-state theory ( Sullivan, 1990 ).

However, the announcement that the theory had now been disproved was premature. The Cambridge results were contradicted by measurements made by Bernard Mills and his team of radio astronomers in Sydney who obtained quite different results from the Southern Hemisphere. For a while, it seemed that radio astronomy was as unable as optical astronomy to distinguish between the two competing models of the universe. This was not the case, though, for in the early 1960s, new and more reliable data from both Cambridge and Sydney showed otherwise—namely, that observations of radio sources provided strong evidence against the steady-state theory. The consensus among the radio astronomers did not conclusively refute the theory, but it left it in a seriously weakened state. Hoyle and what remained of the steady-state camp admitted as much, and yet they thought that the discrepancy between observations and theory was not inevitable. Perhaps it would disappear in a suitably revised version of the steady-state theory.

Although the half-forgotten Alpher–Herman prediction of a cosmic background radiation played almost no role in the cosmological controversy, in the early 1960s, a few physicists came to the same conclusion. In about 1963 , the Princeton physicist Robert Dicke hypothesized a cosmic black-body radiation originating from the collapse of a pre–big-bang universe and its subsequent rebirth. Dicke’s speculation about a cyclic universe was unfruitful except that it revived the idea of a cosmic background made up of low-intensity electromagnetic waves. In 1964 , James Peebles, also at Princeton, calculated the properties of the assumed radiation, which he estimated to have a present temperature of around 10 K. Neither Dicke nor Peebles were at the time aware of the old prediction of Alpher and Herman. Nor were they aware of the results obtained by two researchers at Bell Laboratories while examining an antenna for use in radio astronomy ( Peebles et al., 2009 ).

Arno Penzias and Robert Wilson found to their surprise an excess temperature in their antenna of 3.5 K, which somehow was of cosmic origin but without associating their finding with cosmological models of the early universe. When Dicke and his group in Princeton were informed about the anomalous excess temperature, they immediately understood that Penzias and Wilson serendipitously had discovered a fossil radiation from the very early universe. The discovery was made public in the New York Times of May 21, 1965 (“Signals Imply a ‘Big Bang’ Universe”), and the details were given in the July 1965 issue of Physical Review , which included the observational results of Penzias and Wilson alongside with the big-bang interpretation due to Dicke and his collaborators.

When further work confirmed the background radiation at other wavelengths than the one used by Penzias and Wilson (7.3 cm), there was little doubt: It was the result of a very early and hot phase in the history of the universe and therefore crucial evidence that the steady-state theory was wrong. Hoyle and his collaborator Jayant Narlikar disagreed, but their sustained attempts to explain the data in terms of starlight and in agreement with a modified steady-state theory won no acceptance.

The discovery of the cosmic microwave background had important implications for the nuclear processes leading to the formation of helium in the early universe. Calculations of this kind had been performed by Gamow’s group in the early 1950s, and they now reappeared in a much-improved version. In 1966 , Peebles derived from big-bang calculations a cosmic abundance of helium of 26%–28% by mass depending on the value of the present density of matter. The figure agreed nicely with earlier observations by Donald Osterbrock and John Rogerson, who found around 25%. The amount of helium in the universe thus provided one more confirmation of the hot big-bang model.

Yet another confirmation, should one be needed, was provided by the strange quasars discovered in 1963 . Recognized to be extragalactic objects, the quasars were cosmologically important in the same way that the spiral galaxies were. In a study of 1966 , Dennis Sciama and Martin Rees investigated the distribution of quasars and their redshifts in relation to the prediction of the steady-state theory. They concluded that observational data and theory could not possibly be brought into agreement. Sciama had for years defended the steady-state theory, but now he lost his faith.

7. A New Cosmos

The presently accepted theory of the universe and its evolution in time is conveniently labeled the “big-bang theory,” but this is a slightly anachronistic phrase if used for the early development of the theory. The term big bang was originally coined by Hoyle, an archenemy of this kind of cosmological theory, in a BBC broadcast of March 20, 1945 . “I cannot see any good reason for preferring the big bang idea,” he said. “Indeed it seems to me to be in the philosophical sense a distinctly unsatisfactory notion, since it puts the basic assumption out of sight where it can never be challenged by a direct appeal to observation” ( Kragh, 2014 ). One might expect that the catchy phrase quickly caught on and was frequently used in the cosmological controversy, but this was far from the case. Gamow disliked the term, and although Hoyle referred to it in his popular book The Nature of the Universe from 1950 , he only used it again in 1965 . In their seminal paper in the Astrophysical Journal of July 1965 , Dicke and his coauthors did not employ the term but instead wrote about the “primeval fireball” as the state of the early universe giving birth to the microwave background. In fact, the later so popular phrase only appeared insignificantly in the scientific literature until the early 1970s, when it finally gained momentum.

Nonetheless, the year 1965 can reasonably be seen as the beginning of a new chapter in the annals of cosmology, the year in which the modern big-bang theory was born. Of course, the new theory was not accepted instantly or by all physicists and astronomers (not to mention the philosophers). For a decade or more, it was resisted by a substantial minority of scientists who suggested alternative cosmologies without an explosive beginning of the universe at large. By the late 1960s, former steady-state supporters such as Bondi, Gold, Sciama, and McCrea had either abandoned the theory of an eternal universe or withdrawn from research in cosmology. However, Hoyle continued until his death in 2001 to fight what he thought was the semireligious and unacceptable big-bang orthodoxy. Together with Narlikar and a few other collaborators, he developed a series of alternative theories that had in common with the original theory, only that they preserved the idea of an eternal universe with continual creation of matter.

Another kind of alternative cosmological theory was promoted by the prominent Swedish physicist Hannes Alfvén, who in the 1960s introduced a “plasma cosmology” based on the assumption of equal amounts of matter and antimatter in the universe. Alfvén’s favored cosmology shared with Hoyle’s that it denied a beginning of the universe as a whole, but otherwise it was quite different. None of the two alternatives succeeded in attracting wide interest, and by the 1980s, they had become marginalized. The same was the case with versions of the old “tired-light” hypothesis, which not only denied the big bang but also that the universe was expanding. By the late 1980s, respected journals of astrophysics and cosmology no longer accepted papers belonging to the tired-light category.

Less than a decade after the discovery of the cosmic microwave background, the hot big-bang theory of the universe had acquired an almost paradigmatic status. At the same time, cosmology experienced a strong quantitative growth such as indicated by bibliometric data ( Ryan & Shepley, 1976 ). The annual number of research articles on cosmology had on average been around 30 in the period from 1950 to 1962 , a very low number. Between 1962 and 1972 , it increased from 50 to 250. Another indication of cosmology’s growing maturity as a scientific discipline is the emergence of textbooks on the subject and the content and style of those books. Before 1970 , there were only a couple of textbooks, and the most used of these, Bondi’s Cosmology from 1952 with a revised edition of 1956 , covered very different and incompatible theories of the universe. Peebles’s Physical Cosmology from 1971 was entirely different as it essentially identified cosmology with the new theory of an explosive beginning of the universe and its subsequent evolution governed by the laws of general relativity. Peebles’s book was quickly followed by others in the same genre, of which Steven Weinberg’s Gravitation and Cosmology from 1972 was arguably the most important.

For the first time ever, cosmology became a full-time professional occupation for trained scientists with a shared view of what their field of research was all about. With the professionalization and increased academic status followed a de facto exclusion of the philosophers and amateur scientists who had previously been significant actors in the development of cosmological thoughts. The designation “cosmologist” had traditionally been reserved for philosophers speculating about the meaning of the universe and not widely used, if used at all, by the scientists engaged in cosmological research. Now some astronomers and physicists slowly began to perceive themselves as cosmologists in a different and strictly scientific sense.

It is tempting to speak of the conceptual change that occurred in cosmology in the 1960s as a “revolution” and the victorious big-bang theory as a new “paradigm.” However, the terminology is hardly appropriate if understood in the strong sense of the terms associated with Thomas Kuhn’s philosophy of science ( Marx & Bornmann, 2010 ). Something new and very important did happen in the period, and yet the new picture of the finite-age universe was not incommensurable with earlier views such as held by Lemaître and Gamow. After all, as far as fundamental physics is concerned, the new consensus theory rested safely on old ground, namely, general relativity and nuclear physics as described by quantum mechanics. Whereas the change was not truly revolutionary in Kuhn’s sense, in a sociological perspective, it led to a kind of cosmological paradigm that, despite all later developments, has lasted until the present.

When James Peebles received the Nobel Prize in physics in 2019 , the universe looked significantly different from the one he had co-created half a century earlier ( Peebles, 2020 ). The modern developments are not parts of the present article, but three of the most important discoveries deserve brief mention:

According to the inflation theory introduced in the early 1980s by Alan Guth and others, the universe underwent a kind of phase transition very shortly after the magical “creation” moment at t = 0, and as a result, it inflated by a gigantic factor ( Guth, 1997 ). The brief inflation era produced a universe filled with hot radiation energy, which subsequently expanded at the much slower rate of the standard cosmological theory. In a sense, inflation explained the big bang. The inflation theory, which exists in several versions, was highly successful and is today broadly accepted because of its explanatory power and agreement with observations. However, it has also been severely criticized on methodological and other grounds.

As Zwicky argued as early as 1933 , there must be a large amount of unseen or dark matter in the universe. Zwicky’s hypothesis was confirmed more than 40 years later when observations indicated that dark matter is much more abundant than ordinary matter. Moreover, most of the dark matter seems to be “exotic,” that is, made up of particles not described by the standard model of elementary particles. What these particles are and if dark matter is real or not still belong to the unsolved problems that challenge modern cosmologists ( Sanders, 2010 ). According to most experts, around 27% of the total energy of the universe consists of dark matter.

Not only is the universe filled with dark matter, but it is also filled with dark energy , of which there is even more ( Kragh & Overduin, 2014 ). Observations of supernovae redshifts in the late 1990s proved that the universe is in a state of acceleration not unlike what Lemaître had suggested much earlier. To explain the acceleration, some repulsive cosmic force is needed, which to most cosmologists means Einstein’s cosmological lambda constant. That this constant can be interpreted as a vacuum energy density was first pointed out by Lemaître in 1934 but only provided with an explanation based on quantum field theory much later—namely, by the Russian physicist Yakov Zeldovich in the late 1960s. However, there is no agreement between the measured value of the cosmological constant and calculations of the vacuum energy density. Although there is still some dispute concerning the nature of this constant and its role in cosmology, the present consensus model known as the Λ ‎CDM model (Lambda Cold Dark Matter) identifies the dark energy as a manifestation of the cosmological constant. Observations and theory lead to a universe where 68% of the energy is dark.

8. Conclusion

Although cosmology as a physical science only emerged in the 20th century , the quest of understanding the universe has always been a challenge for philosophically minded people. The kinds of “cosmologies” discussed by Anaximander and other pre-Socratic philosophers—or those discussed in the Middle Ages by Thomas Aquinas and other scholastics—were of course very different from the cosmological models based on modern physical theory. And yet it would be a mistake to believe that people’s present ideas of the universe have nothing at all in common with the speculations of the past. Across a time span of more than two millennia, there has been a remarkable measure of permanence in the kind of cosmic questions people have wanted to know the answers to.

While some of the old questions have now been deemed scientifically meaningless and excluded from the scientific discourse about the universe, others are still part of what cosmologists are concerned with either directly or indirectly. To the first category belongs the question of the purpose of the universe and God’s role in the cosmic history. But there are other and equally majestic questions that may well be answered by accepted scientific methods and therefore belong, at least in principle, to the domain of modern cosmology. Is the world at large finite or infinite? Has the world always existed, or did it once come into existence? If there were an absolute origin, how can it possibly be explained without recourse to a divine creator? To mention but one more of the old but not obsolete questions: Does advanced life exist elsewhere in the universe?

The latter question is not only of astronomical relevance because of the discovery since the 1990s of numerous exoplanets revolving around stars different from the Sun. It also relates to cosmological thinking in the form of the so-called anthropic principle introduced by the astrophysicist Brandon Carter in 1974 . According to the simplest version of this principle or argument, carbon-based life could not have originated in a universe evolving just slightly different from the one observed . In another version, it states that important aspects of the universe can only be explained by the existence of human life. Whatever the precise meaning of the controversial anthropic principle, it has attracted massive attention among cosmologists as well as philosophers ( Carr, 2007 ; Vidal, 2014 ). While some see it as a new foundation of cosmological thinking, others deny its scientific legitimacy. Interestingly, the present debate concerning the anthropic principle and its consequences is in some ways strikingly similar to the cosmological controversy of the past between the steady-state theory and relativistic evolution theories.

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Physics 20B. Cosmology. Lec 24: Review of Modern Cosmology (English)

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Lecture 24: Review of Modern Cosmology

Course Description:  "Cook's Tour" of the universe. Ancient world models. Evidence for universal expansion; the size and age of the universe and how it all began. The long-range future and how to decide the right model. Anthropic principle.

Required attribution: Bullock, James. Physics 20B (UCI Open: University of California, Irvine), ../courses/physics_20b_cosmology.html. [Access date]. License: Creative Commons Attribution-ShareAlike 3.0 United States License.

Modern Cosmology

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Title: modern cosmology, an amuse-gueule.

Abstract: This essay is a nontechnical primer for a broader audience, in which I paint a broad-brush picture of modern cosmology. I begin by reviewing the evidence for the big bang, including the expansion of our Universe, the cosmic microwave background, and the primordial abundances of the light elements. Next, I discuss how these and other cosmological observations can be well explained by means of the concordance model of cosmology, putting a particular emphasis on the composition of the cosmic energy budget in terms of visible matter, dark matter, and dark energy. This sets the stage for a short overview of the history of the Universe from the earliest moments of its existence all the way to the accelerated expansion at late times and beyond. Finally, I summarize the current status of the field, including the challenges it is currently facing such as the Hubble tension, and conclude with an outlook onto the bright future that awaits us in the coming years and decades. The text is complemented by an extensive bibliography serving as a guide for readers who wish to delve deeper.

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Understanding Modern Physics I: Relativity and Cosmology

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Introduction to special relativity, general relativity and cosmology.

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There are 5 modules in this course

Course Overview: https://youtu.be/xyF-MmGNxd0

The 20th century was known as the century of physics. In the past 120 years, concepts such as space, time, energy, entropy and particles were understood to much deeper levels. New paradigms of thinking such as relativity and quantum mechanics emerged. This course is the first course in the Understanding Modern Physics series, which covers an introduction to special relativity, general relativity and cosmology. We will find: (i) How space and time are relative to observers, and unified into a more fundamental construction of spacetime; (ii) Why the spacetime is not absolute, but rather curves in response to matter, and how gravity emerge as a result of such spacetime curvature; and (iii) What is the framework to understand the evolution of the whole universe, and how that is related to problems such as the origin of space and matter, and the fate of our universe. Note: the videos with a (*) are optional. They provide complementary information but not in the learning objective or assignment questions. Feel free to choose to watch them or not.

Special Relativity: Relativity of Space and Time

What's included.

14 videos 1 quiz

14 videos • Total 104 minutes

• Introduction • 7 minutes • Preview module
• Principles of Relativity • 7 minutes
• Speed of Light • 6 minutes
• Postulates • 11 minutes
• Light Clock • 4 minutes
• Time Dilation • 13 minutes
• 4-Step Reasoning • 4 minutes
• Twin Paradox • 5 minutes
• Physical Picture • 12 minutes
• Interferometers (* optional video) • 6 minutes
• Ether (* optional video) • 10 minutes
• Speed of Light from Maxwell (* optional video) • 6 minutes
• Units (* optional video) • 3 minutes
• Weekly Summary • 4 minutes

1 quiz • Total 30 minutes

• I - 1 • 30 minutes

Special Relativity: Length, Simultaneity and Lorentz Transformation

17 videos 1 quiz

17 videos • Total 119 minutes

• Introduction • 2 minutes • Preview module
• Length Contraction • 6 minutes
• Length in 4-Step Reasoning (* optional video) • 16 minutes
• Velocity Addition: First Look (* optional video) • 10 minutes
• Terrell Rotation (* optional video) • 5 minutes
• Device for "Same Time" • 7 minutes
• Spacetime Diagram • 5 minutes
• Relativity of Same Time • 7 minutes
• Twin Paradox Revisited (* optional video) • 4 minutes
• Causality • 11 minutes
• No Perfect Rigid Body • 3 minutes
• Ladder Paradox • 9 minutes
• Lorentz Transform, Space Part • 6 minutes
• Lorentz Transform, Time Part • 6 minutes
• Effects From Lorentz Transform • 6 minutes
• Velocity Addition • 5 minutes
• Summary • 4 minutes

1 quiz • Total 60 minutes

• I - 2 • 60 minutes

Special Relativity: Geometry, Energy and Momentum

15 videos 1 quiz

15 videos • Total 109 minutes

• Geometry • 7 minutes
• Lorentzian and Euclidean • 9 minutes
• Invariant Interval • 7 minutes
• Conserved Quantities • 6 minutes
• Newtonian Momentum • 5 minutes
• Momentum • 11 minutes
• Kinetic Energy • 9 minutes
• Mass Conservation? • 5 minutes
• Energy • 7 minutes
• 4D Vectors • 11 minutes
• Doppler Effect (* optional video) • 7 minutes
• Spacetime Metric (* optional video) • 2 minutes
• Natural Units (* optional video) • 7 minutes
• Summary • 6 minutes
• I - 3 • 60 minutes

General Relativity

12 videos 1 quiz

12 videos • Total 100 minutes

• Why gravity is related to inertia? • 5 minutes
• Equivalence Principle • 8 minutes
• Time Dilation • 8 minutes
• Schwarzschild Metric • 6 minutes
• Light Bending • 9 minutes
• Gravitational Lensing • 9 minutes
• Black Holes • 8 minutes
• Journey of Falling in • 7 minutes
• Are Black Holes Black? • 12 minutes
• Gravitational Waves • 11 minutes
• Summary • 5 minutes
• I - 4 • 30 minutes

8 videos 1 quiz

8 videos • Total 86 minutes

• Olber's Paradox • 10 minutes • Preview module
• The Observable Universe • 10 minutes
• Cosmological Principle • 7 minutes
• Friedmann Equation • 13 minutes
• Hubble Law • 10 minutes
• Continuity Equation • 10 minutes
• Matter Content • 12 minutes
• Origins and Summary • 10 minutes
• I - 5 • 30 minutes

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Recommended if you're interested in Physics and Astronomy

The Hong Kong University of Science and Technology

Understanding Modern Physics III: Simplicity and Complexity

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129 reviews

Reviewed on Apr 16, 2022

It was a great experience,it was my first online course its help me in understandin physics principle in more details and help in my degree course

Reviewed on Aug 18, 2023

everything you need to know about modern physics. start here

Reviewed on Jan 13, 2022

Yi Wang does an excellent job of capturing the most important elements of the subject and demonstrating how they can be best understood. Well done!

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The Oxford Handbook of the History of Modern Cosmology

Emeritus professor, Niels Bohr Institute, University of Copenhagen

Malcolm Longair is Director of Development and Outreach, Jacksonian Professor Emeritus of Natural Philosophy at the Cavendish Laboratory, University of Cambridge, UK. He also holds the position of Editor-in-Chief of the Biographical Memoirs of Fellows of the Royal Society (2016-2020). Longair has previously held positions as a university lecturer in the Department of Physics, University of Cambridge (1970--80); Astronomer Royal for Scotland, Regius Professor of Astronomy, University of Edinburgh, Director of the Royal Observatory, Edinburgh (1980-90); Jacksonian Professor of Natural Philosophy, University of Cambridge (1991-2008); and Head of Cavendish Laboratory (1997-2005).

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Although some of the observational and conceptual roots of modern cosmology can be traced back to the nineteenth century, it was only in the twentieth century that the study of the universe as a whole emerged as a genuine physical science. The development through the twentieth and now well into the twenty-first century has been far from smooth, but in spite of a number of false trails it has been tremendously fruitful and surprisingly successful scientifically. The volume presents a comprehensive overview of the development of cosmology from about 1860 to the most recent discoveries. It describes and explains the historical background to what we know about the universe today and what people in the past thought they knew about the universe, starting with the first observations of spiral nebulae and ending with the discovery of gravitational waves. The book is organized into thirteen roughly chronologically ordered chapters, some focusing on theory and others more on observations and technological advances. A few of the chapters are of a more general nature, relating to larger contexts such as politics, philosophy and religious world views. The chapters are written by eight different authors, some of whom are astrophysicists or cosmologists while others have backgrounds in the history and philosophy of science. Each chapter can be read separately but also has a symbiotic relation with the other chapters. As a result, the book describes the history of modern cosmology coherently, comprehensively and with ample references to the relevant sources.

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March 1, 1954

Modern Cosmology

Its theories utilize evidence from such things as the decay of atoms, the metabolism of stars and the flight of galaxies to describe the structure of the Universe in space and time

By George Gamow

Modern cosmology: Facts and ideas

• Published: 22 November 2008
• Volume 63 , pages 299–314, ( 2008 )

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• A. M. Cherepashchuk 1 &
• A. D. Chernin 1  

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A brief overview of the main observational facts and key ideas of modern cosmology is given. It is emphasized that present-day cosmology is a fundamental science about the evolution of the Universe based on verified empirical data and a reliable physical theory.

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Modern Cosmology, an Amuse-Gueule

Cosmogenesis

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A. M. Cherepashchuk and A. D. Chernin, Universe, Life, Black Holes (Fryazino, 2003) [in Russian].

A. M. Cherepashchuk and A. D. Chernin, Horizons of the Universe (Novosibirsk, 2005) [in Russian].

A. D. Chernin, Usp. Fiz. Nauk 178 , 267 (2008).

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Ya. B. Zeldovich, Usp. Fiz. Nauk 95 , 209 (1968).

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Sternberg Astronomical Institute, Universitetskii pr. 13, Moscow, 119992, Russia

A. M. Cherepashchuk & A. D. Chernin

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Original Russian Text © A.M. Cherepashchuk, A.D. Chernin, 2008, published in Vestnik Moskovskogo Universiteta. Fizika, 2008, No. 5, pp. 3–19.

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Cherepashchuk, A.M., Chernin, A.D. Modern cosmology: Facts and ideas. Moscow Univ. Phys. 63 , 299–314 (2008). https://doi.org/10.3103/S0027134908050019

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Received : 20 February 2008

Published : 22 November 2008

Issue Date : October 2008

DOI : https://doi.org/10.3103/S0027134908050019

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The Gallagher brothers are reviving Oasis. Here's a look at their decades-long feud

Rachel Treisman

Liam (right) and Noel Gallagher, pictured in London in February 1999, were members of the beloved band Oasis from 1991 until it broke up in 2009. PA Images via Getty Images hide caption

Fifteen years after Oasis imploded, the dueling brothers at the heart of the beloved British band have confirmed fans' far-fetched dreams of a reunion tour.

On Monday, Noel and Liam Gallagher — the band’s main songwriter and singer, respectively — each tweeted out an 11-second video with the date and time of Aug. 27 at 8 a.m., flickering in the Oasis font. The Oasis Instagram account shared it, too.

The posts came a day after Britain’s Sunday Times reported that anonymous industry insiders were “adamant” that the two would reunite for a series of high-profile concerts next summer, including a headlining slot at the Glastonbury Festival and what would be a record 10-night run at Wembley Stadium.

R.E.M. reunites for the first time in 15 years, performs 'Losing My Religion'

“See you down the front,” Liam then replied in a thread about the article on the social platform X.

He also posted a cryptic tweet of his own: “I never did like that word FORMER.”

Oasis, which started as a five-person band out of Manchester in 1991, is credited with reviving the genre of Britpop music, fueled in part by its rivalry with the London-based band Blur .

The group rose to fame with its 1994 album, Definitely Maybe , and cemented its legacy with such hits as “Wonderwall,” “Don’t Look Back in Anger” and “Champagne Supernova.” As of three decades later, Oasis had sold 75 million records worldwide.

The Gallaghers, the only constant members of the band's ever-changing lineup, became known for their splashy antics onstage and across the tabloid pages — as well as their own deepening rivalry.

Liam Gallagher Has A Lot To Be Happy About

Years of insults, incidents and infighting culminated on Aug. 28, 2009, when the band broke up minutes before they were set to take the stage at the Rock en Seine festival in Paris.

“People will write and say what they like, but I simply could not go on working with Liam a day longer,” Noel wrote in a statement at the time. He would later recount a backstage argument in which his younger brother grabbed his guitar and started “wielding it like an axe.”

The Gallaghers went their separate ways, but continued their careers in music: Liam and other Oasis members kept the band going under the name Beady Eye, while Noel formed his own band, High Flying Birds.

The two have performed their old hits separately over the years, though never with each other.

And they’ve publicly floated the prospect of a reunion more than once, including in 2015 and 2018. But such a reconciliation had yet to materialize — and the two continued taking very public swipes at each other on social media.

Eagle-eyed hopefuls, however, had spotted hints that detente could be on the horizon.

While performing at the Reading Festival on Sunday, Liam dedicated the Oasis song “Half The World Away” to his brother and “Cigarettes & Alcohol” to haters of the band. As his set ended, the onstage screens flashed that same Aug. 27 announcement video.

The Gallagher brothers are reuniting for an Oasis tour 15 years after breaking up

On Tuesday morning, almost exactly 15 years to the day after Oasis broke up, the brothers confirmed they will indeed be getting back together .

They shared identical videos advertising " Oasis live '25 ," a series of concerts across the United Kingdom and Ireland next July and August, with tickets going on sale this weekend. The year 2025, notably, will mark 30 years since the release of their second album, (What’s The Story) Morning Glory.

"The guns have fallen silent. The stars have aligned," the brothers also tweeted. "The great wait is over. Come see. It will not be televised."

The hints and ensuing announcement have caused quite a stir on social media, where scores of the group’s fans have been posting about the cultural significance — and personal finance implications — of an Oasis reunion tour.

And many have joked that those looking to buy tickets should aim for the first night only , lest the famously fight-prone brothers break up the band once again. It wouldn't be the first time.

A brief timeline of the Gallagher feud, from fisticuffs to legal battles

Singer Liam Gallagher (left) and his brother Noel Gallagher, of the band Oasis, perform on stage in San Francisco in 1997. After the band split, they pursued separate musical careers and continued to trade public insults. Dave Hogan/Getty Images hide caption

Noel and Liam’s tumultuous relationship goes back decades, blazing a trail of barbs and bad blood that is nearly impossible to detail in full (though outlets like Pitchfork , Rolling Stone and Radio X have bravely attempted).

Here are some of the highlights, or low points:

• April 1994: Shortly before their commercial breakthrough, the brothers sit for an interview with NME’s John Harris and spend the entire time bickering — over the meaning of rock and roll, a drunken incident on a ferry several months earlier and how much they hated each other, among other things. A year later, they release the audio as a single called “ Wibbling Rivalry ” (subtitle: “fourteen minutes of verbal mayhem”).
• September 1994: During a concert in Los Angeles, Liam hits Noel over the head with a tambourine and slings insults at both the band and the American audience. Noel storms off and briefly quits the band, but rejoins later that tour and uses the incident as inspiration for the track “Talk Tonight.”
• Spring 1995: While recording (What’s The Story) Morning Glory , Liam brings a group of visitors to the studio while Noel is trying to work. Noel, trying to get them to leave, ends up hitting Liam on the head with a cricket bat. Years later, the rescued cricket bat was sold at auction with a certificate of authenticity.
• August 1996: Liam pulls out of their MTV Unplugged taping at the last minute, blaming a sore throat, then proceeds to spend the entire performance heckling his brother from the crowd and drinking champagne. Later that month, Liam also pulls out of a U.S. tour just before it starts — he ends up joining a few days in, but the tour is canceled after two weeks.
• May 2000: During a night of drinking in Barcelona, Liam questions the paternity of Noel’s daughter, prompting Noel to leave the band once again. He doesn't return for the rest of the European tour.
• December 2002: After experiencing issues with his voice and walking offstage early at several shows throughout the fall, Liam wraps up the year by kicking a German police officer in the chest, sparking a brawl at a Munich bar and losing several teeth in the process. “I haven’t spoken to him directly,” Noel said at the time. “Why should we talk after what’s gone on?”
• October 2005: Noel says in an interview that his brother is “frightened to death” of him: “I can read him and I can f****** play him like a slightly disused arcade game. I can make him make decisions that he thinks are his but really they’re mine.”
• 2009:  Noel tells  Q  magazine at one point that Liam is “the angriest man you'll ever meet. He's like a man with a fork in a world of soup.” (Dishing it back 10 years later, Liam posted a video of himself in 2019 eating soup with a fork .)
• August 2009: Oasis abruptly cancels its set at the V Festival in England, with Liam blaming a case of laryngitis. Years later, Noel claims his brother was actually hungover, which prompted Liam to sue him for libel (he eventually dropped the suit). That month, Noel quits the band for real. He told NPR about it in 2012 : “We were backstage waiting to go onstage to 30,000 people in Paris. The tour manager came in and said, 'Five minutes!' We broke up within that five minutes. I'm not proud of that, but all things come to an end."

The brothers kept slinging insults years after the breakup

The Gallaghers kept fighting long after they were no longer bandmates, insulting each other in awards speeches, interviews and social media posts (including the infamous 2016 “potato” tweet , which became a yearslong bit of its own).

There were occasional public calls for reconciliation interspersed throughout the drama, giving fans brief glimmers of hope.

Liam hinted at a truce in 2017 but called it off a month later. In 2018, he tweeted at his brother suggesting a reunion, but after no reply wrote, “I’ll take that as a NO then.” During the peak of the COVID-19 pandemic, Noel released a previously hidden song recorded in 1995, to Liam’s apparent displeasure.

But by 2023, both brothers seemed to be taking the idea of a reunion more seriously.

Notably, that same year, Blur — their Britpop archrival, which had been on hiatus since 2015 — reunited to release an album and perform at a number of summer festivals. The Ballad of Darren was critically acclaimed, with outlets from the BBC to Mojo to NPR including it on their best-of-the-year lists. The group closed out its triumphant 2023 by announcing another hiatus .

Music Interviews

Noel gallagher: flying high after oasis.

“If Oasis hadn’t had reached their potential, and there was something left to do, it would be different, but I just don’t see what the point would be,” Noel said in a January 2023 interview . “It would be make a load of money, I’ve got a load of money. To do some monumental [venue] I’ve already done them."

But he left the door open to the possibility, adding: “Now, that’s not saying, in 10 years' time, it won’t appeal to me..."

Two months later, Liam responded to a fan’s tweet asking if there was any chance Oasis might get back together.

“It’s happening,” he wrote .

And now it's official.

Correction Aug. 27, 2024

This story has been updated to reflect that it was Noel who called Liam "a man with a fork in a world of soup."

• Liam Gallagher
• Noel Gallagher

Explore the dark with Quarter Moon tour, Observe the Moon Night | Sky Guy

The James Webb Space Telescope continues making historic findings. This time it is how galaxies in the early universe formed. Some galaxies are so massive that they do not fit into the standard model of cosmology. A new study led by the University of Texas at Austin suggests black holes are making these galaxies appear more massive than they really are.

“We are still seeing more galaxies than predicted, although none of them are so massive that they ‘break’ the universe,” said University of Texas at Austin graduate student Katherine Chworowsky,

Morning sky: Mars rises around 2:30 a.m. early on and by 2 a.m. late in the month. Jupiter rises around 2 a.m. in early September and rises around midnight later in the month. Mercury is too low in the east for viewing without optical aid. Watch the Moon pass a couple of bright stars and planets, see below for dates.

Evening sky: Saturn rises around 9 pm in early September and by 7:30 pm late in the month. Venus is low in the west. Watch the Moon pass a couple of bright stars and planets, see below for dates.

2nd : New Moon.

5th: Moon close to Venus very low in the west just after sunset. Optical aid will help.

6th: Moon near bright star Spica in Virgo low in the west at sunset. Optical aid will help.

7th – 8th : Saturn reaches opposition meaning it rises as the Sun sets.

9th: Mercury near the bright star Regulus in Leo in the east before sunrise. Optical aid will help.

10th: Moon near bright star Antares (“Rival of Mars”), a supergiant star, in the evening sky.

11th : First quarter Moon.

13th: Come view the Quarter Moon and tour the night sky with the Tallahassee Astronomical Society. This FREE event is held monthly at Cascades Park in downtown Tallahassee. See the moon, any planets that are up, and such other heavenly wonders as may be in the sky. Quarter Moon viewing is the perfect time to bring your family, home school group, or other small group to see the sky, ask questions about astronomy, and see some telescopes in action! 8 p.m. through 9:30 p.m.

14th: Tallahassee Astronomical Society’s free planetarium show, “September Skies over Tallahassee,” at the planetarium at the Challenger Learning Center (not recommended for children under 5). Doors close at 6 p.m.

14th: Join TAS at the Challenger Learning Center on Kleman Plaza, downtown Tallahassee, from 6-9 p.m. for the International Observe the Moon Night! We will have a planetarium shown, moon-gazing through telescopes, create moon-inspired art, and listen to moon-themed music!

17th: Full Moon close to Saturn in the evening into the morning sky.

22nd: Moon passes through the bright star cluster Pleiades (“Subaru”) in the morning sky.

22nd: Autumnal equinox (“equal night”) for the Northern Hemisphere.

24th: Last quarter Moon.

25th : Crescent Moon near Mars in the early morning sky.

26th: Moon near bright star Pollux in Gemini before sunrise.

29th: Crescent Moon near bright star Regulus (“Little Prince”) in Leo in early morning sky.

Check out TAS’s events calendar at tallystargazers.org

Ken Kopczynski is a former president of the Tallahassee Astronomical Society, a local group of amateur astronomers.

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Moscow, the captivating capital of Russia, can’t help you without being charmed, as everyone gets!

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Domodedovo International Airport, the largest airport in Russia and other one is Sheremetyevo International Airport.

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Moscow Metro is just fascinating, a sort of museum in itself. Fine architecture, interiors and absolute majesty, the metro stations of Moscow City Tour are undoubtedly the most exhilarating experiences.

Another pleasant and wonderful way to explore the city is by taking a Moskva River cruise with city guide. You can have great sights that include Moscow Kremlin, Saint Basil’s Cathedral and Cathedral of Christ the Saviour.

Some must visit places in Moscow are:

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Modern Cosmology PDF

Download Modern Cosmology PDF

Description

Modern Cosmology, Second Edition, provides a detailed introduction to the field of cosmology. Beginning with the smooth, homogeneous universe described by a Friedmann-Lemaître-Robertson-Walker metric, this trusted resource includes careful treatments of dark energy, big bang nucleosynthesis, recombination, and dark matter. The reader is then introduced to perturbations about an FLRW universe: their evolution with the Einstein-Boltzmann equations, their primordial generation by inflation, and their observational consequences: the acoustic peaks in the CMB; the E/B decomposition in polarization; gravitational lensing of the CMB and large-scale structure; and the BAO standard ruler and redshift-space distortions in galaxy clustering. The Second Edition now also covers nonlinear structure formation including perturbation theory and simulations. The book concludes with a substantially updated chapter on data analysis. Modern Cosmology, Second Edition, shows how modern observations are rapidly revolutionizing our picture of the universe, and supplies readers with all the tools needed to work in cosmology....

Table of Contents

Contents About the authors Preface 1 The concordance model of cosmology 1.1 A nutshell history of the universe 1.2 The Hubble diagram 1.3 Big Bang nucleosynthesis 1.4 The cosmic microwave background 1.5 Structure in the universe 1.6 ΛCDM: the concordance model of cosmology 1.7 Summary and outlook Exercises 2 The expanding universe 2.1 Expanding space 2.1.1 The metric 2.1.2 The geodesic equation 2.2 Distances 2.3 Evolution of energy 2.4 Cosmic inventory 2.4.1 Photons 2.4.2 Baryons 2.4.3 Dark matter 2.4.4 Neutrinos 2.4.5 Epoch of matter-radiation equality 2.4.6 Dark energy 2.5 Summary Exercises 3 The fundamental equations of cosmology 3.1 Einstein equations 3.2 Boltzmann equation 3.2.1 Boltzmann equation for particles in a harmonic potential 3.2.2 Boltzmann equation in an expanding universe 3.2.3 Collision terms 3.3 Beyond the homogeneous universe 3.3.1 Perturbed spacetime 3.3.2 The geodesic equation 3.3.3 The collisionless Boltzmann equation for radiation 3.3.4 The collisionless Boltzmann equation for massive particles 3.4 Summary Exercises 4 The origin of species 4.1 The homogeneous Boltzmann equation revisited 4.2 Big Bang nucleosynthesis 4.2.1 Neutron abundance 4.2.2 Light element abundances 4.3 Recombination 4.4 Dark matter 4.5 Summary Exercises 5 The inhomogeneous universe: matter & radiation 5.1 The collisionless Boltzmann equation for photons 5.2 Collision terms: Compton scattering 5.3 The Boltzmann equation for photons 5.4 The Boltzmann equation for cold dark matter 5.5 The Boltzmann equation for baryons 5.6 The Boltzmann equation for neutrinos 5.7 Summary Exercises 6 The inhomogeneous universe: gravity 6.1 Scalar-vector-tensor decomposition 6.2 From gauge to gauge 6.3 The Einstein equations for scalar perturbations 6.3.1 Ricci tensor 6.3.2 Two components of the Einstein equations 6.4 Tensor perturbations 6.4.1 Christoffel symbol for tensor perturbations 6.4.2 Ricci tensor for tensor perturbations 6.4.3 Einstein equations for tensor perturbations 6.4.4 Verifying the decomposition theorem 6.5 Summary Exercises 7 Initial conditions 7.1 The horizon problem and a solution 7.2 Inflation 7.3 Gravitational wave production 7.3.1 Quantizing the harmonic oscillator 7.3.2 Tensor perturbations 7.4 Scalar perturbations 7.4.1 Scalar field perturbations around an unperturbed background 7.4.2 Super-horizon perturbations 7.4.3 Spatially flat slicing 7.5 The Einstein-Boltzmann equations at early times 7.6 Summary Exercises 8 Growth of structure: linear theory 8.1 Prelude 8.1.1 Three stages of evolution 8.1.2 Closing the Boltzmann hierarchy 8.2 Large scales 8.2.1 Super-horizon solution 8.2.2 Through horizon crossing 8.3 Small scales 8.3.1 Horizon crossing 8.3.2 Sub-horizon evolution 8.4 The transfer function 8.5 The growth factor 8.6 Beyond cold dark matter and radiation 8.6.1 Baryons 8.6.2 Massive neutrinos 8.6.3 Dark energy 8.7 Summary Exercises 9 The cosmic microwave background 9.1 Overview 9.2 Large-scale anisotropies 9.3 Acoustic oscillations 9.3.1 Tightly-coupled limit of the Boltzmann equations 9.3.2 Tightly-coupled solutions 9.4 Diffusion damping 9.5 Inhomogeneities to anisotropies 9.5.1 Free streaming 9.5.2 The angular power spectrum 9.6 The CMB power spectrum 9.6.1 Large angular scales 9.6.2 Acoustic peaks 9.7 Cosmological parameters 9.7.1 Curvature and Λ 9.7.2 Amplitude, spectral index, and optical depth 9.7.3 Baryon and CDM densities 9.8 Summary Exercises 10 The polarized CMB 10.1 Polarization 10.2 Generating polarization from Compton scattering 10.3 Polarization from a single plane wave 10.4 Boltzmann solution 10.5 Polarization power spectra 10.6 Detecting gravitational waves 10.7 Summary Exercises 11 Probes of structure: tracers 11.1 Galaxy clustering 11.1.1 Galaxy statistics 11.1.2 Redshift-space distortions 11.1.3 BAO and Alcock-Paczyński 11.2 Angular correlations 11.3 The Sunyaev-Zel'dovich effect 11.4 Summary Exercises 12 Growth of structure: beyond linear theory 12.1 Prelude 12.2 Perturbation theory 12.3 Simulations 12.4 Dark matter halos 12.4.1 Halo masses and profiles 12.4.2 The halo mass function 12.5 Galaxy clusters 12.6 Galaxy clustering and bias 12.7 The halo model 12.8 Summary Exercises 13 Probes of structure: lensing 13.1 Overview 13.2 Photon geodesics 13.3 CMB lensing 13.4 Galaxy shapes 13.5 Weak-lensing statistics 13.5.1 Shear power spectrum 13.5.2 Shear correlation function 13.5.3 Shear cross-correlations 13.6 Summary Exercises 14 Analysis and inference 14.1 The likelihood function 14.2 Overview: from raw data to parameter constraints 14.3 Mapmaking 14.4 Two-point functions 14.4.1 CMB power spectrum 14.4.2 Galaxy power spectrum 14.5 The Fisher matrix 14.6 Sampling the likelihood function 14.7 Summary Exercises A Solutions to selected exercises Chapter 1 Exercise 1.1 Exercise 1.2 Exercise 1.4 Chapter 2 Exercise 2.1 Exercise 2.4 Exercise 2.5 Exercise 2.7 Exercise 2.8 Exercise 2.11 Chapter 3 Exercise 3.2 Exercise 3.6 Exercise 3.7 Exercise 3.8 Exercise 3.12 Chapter 4 Exercise 4.1 Exercise 4.6 Exercise 4.9 Chapter 5 Exercise 5.3 Exercise 5.4 Chapter 6 Exercise 6.1 Exercise 6.3 Exercise 6.8 Chapter 7 Exercise 7.2 Exercise 7.12 Exercise 7.13 Chapter 8 Exercise 8.4 Exercise 8.8 Exercise 8.13 Chapter 9 Exercise 9.2 Exercise 9.9 Exercise 9.16 Exercise 9.17 Chapter 10 Exercise 10.1 Exercise 10.6 Chapter 11 Exercise 11.1 Exercise 11.4 Exercise 11.8 Chapter 12 Exercise 12.4 Exercise 12.10 Exercise 12.13 Chapter 13 Exercise 13.1 Exercise 13.4 Chapter 14 Exercise 14.4 Exercise 14.10 B Numbers B.1 Physical constants B.2 Astrophysical constants B.3 Fiducial cosmology C Special functions C.1 Legendre polynomials C.2 Spherical harmonics C.3 Spherical Bessel functions C.4 Fourier transforms C.5 Miscellaneous D Symbols D.1 Mathematical and geometrical definitions D.2 Frequently used relations D.3 Symbol definitions Bibliography Index

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