Cosmology Day July 2018

Lecture 7: Hawking’s cosmology

Professor Thomas Hertog


Thomas Hertog is a Belgian cosmologist at KU Leuven University and a key collaborator of Professor Stephen Hawking.


Professor Hertog discussed the development of Hawking’s cosmology, starting from his big bang singularity theorems in the 1950s and ending with his efforts to get theoretical control over the multiverse which his work let him to. Professor Hertog attempted to provide some insight into the key overarching ideas that drove and bound together Hawking’s work on cosmology.

My notes from the lecture (if they don’t make sense then it is entirely my fault)


Stephen William Hawking CH CBE FRS FRSA (8 January 1942 – 14 March 2018) was an English theoretical physicist, cosmologist, and author, who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge at the time of his death. He was the Lucasian Professor of Mathematics at the University of Cambridge between 1979 and 2009.

Hawking could ask the right questions. He wanted the deepest level of understanding.

After graduating from Oxford University he began his PhD on cosmology at Trinity Hall, Cambridge, in October 1962.

Inspired by Roger Penrose’s theorem of a spacetime singularity in the centre of black holes, Hawking applied the same thinking to the entire universe; and, during 1965, he wrote his thesis on this topic. His thesis was approved in 1966 and he obtained his PhD degree in applied mathematics and theoretical physics, specialising in general relativity and cosmology, in March 1966.

On the Hoyle-Narlikar theory of gravitation Stephen William Hawking and Hermann Bondi

The ‘afterglow of creation’ – commonly known as the cosmic background radiation – is the left-over heat from the fireball of the big bang in which the Universe was born 13.7 billion years ago.

It provides a unique insight into our Universe’s infancy – a ‘baby photo’ described somewhat dramatically by Stephen Hawking as ‘the discovery of the century, if not of all time’ and by experimentalist and Nobel Prize winner George Smoot as ‘like seeing the face of God’.

Did the Universe have a beginning?


Georges Henri Joseph Édouard Lemaître, RAS Associate (17 July 1894 – 20 June 1966) was a Belgian Roman Catholic priest, astronomer, and professor of physics at the Catholic University of Leuven. He proposed on theoretical grounds that the universe is expanding, which was observationally confirmed soon afterwards by Edwin Hubble. He was the first to derive what is now known as Hubble’s law, or the Hubble-Lemaître law, and made the first estimation of what is now called the Hubble constant, which he published in 1927, two years before Hubble’s article. Lemaître also proposed what became known as the “Big Bang theory” of the creation of the universe, originally calling it the “hypothesis of the primeval atom” or the “Cosmic Egg”.


Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

In 1917, Einstein applied the general theory of relativity to the structure of the universe as a whole. He discovered that the general field equations predicted a universe that was dynamic, either contracting or expanding. As observational evidence for a dynamic universe was not known at the time, Einstein introduced a new term, the cosmological constant, to the field equations, in order to allow the theory to predict a static universe. The modified field equations predicted a static universe of closed curvature, in accordance with Einstein’s understanding of Mach’s principle in these years. This model became known as the Einstein World or Einstein’s static universe.

In theoretical physics, particularly in discussions of gravitation theories, Mach’s principle (or Mach’s conjecture) is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach.

Einstein didn’t like the idea of an expanding universe as his relativity theory broke down at the singularity which is believed to be the origin of the “Big Bang”.

The initial singularity was a singularity of seemingly infinite density thought to have contained all of the mass and space-time of the Universe before quantum fluctuations caused it to rapidly expand in the Big Bang and subsequent inflation, creating the present-day Universe. The initial singularity is part of the Planck epoch, the earliest period of time in the history of the universe.

Hawking believed that a singularity was inevitable.

The Big Bang singularity was a moment in time.

In theoretical physics, the Hartle–Hawking state, named after James Hartle and Stephen Hawking, is a proposal concerning the state of the Universe prior to the Planck epoch.

Hartle and Hawking suggest that if we could travel backwards in time towards the beginning of the Universe, we would note that quite near what might otherwise have been the beginning, time gives way to space such that at first there is only space and no time. Beginnings are entities that have to do with time; because time did not exist before the Big Bang, the concept of a beginning of the Universe is meaningless. According to the Hartle–Hawking proposal, the Universe has no origin as we would understand it: the Universe was a singularity in both space and time, pre-Big Bang. Thus, the Hartle–Hawking state Universe has no beginning, but it is not the steady state Universe of Hoyle; it simply has no initial boundaries in time or space.

If the Universe has no beginning Hawking asked “Do we need a God?”.

There needs to be something special about the boundary conditions at the time of the “Big Bang”.

Long before the emergence of planets, stars, or galaxies, the universe consisted of an exploding quantum soup of elementary” particles. Encoded in this formless, shapeless soup were seeds of cosmic structure, which over billions of years grew into the beautiful and complex universe we observe today.

At present gravitation and spacetime cannot be married to quantum theory.

Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics, and where quantum effects cannot be ignored, such as near compact astrophysical objects where the effects of gravity are strong.

The current understanding of gravity is based on Albert Einstein’s general theory of relativity, which is formulated within the framework of classical physics. On the other hand, the other three fundamental forces of physics are described within the framework of quantum mechanics and quantum field theory, radically different formalisms for describing physical phenomena. It is sometimes argued that a quantum mechanical description of gravity is necessary on the grounds that one cannot consistently couple a classical system to a quantum one.

While a quantum theory of gravity may be needed to reconcile general relativity with the principles of quantum mechanics, difficulties arise when applying the usual prescriptions of quantum field theory to the force of gravity via graviton bosons. The problem is that the theory one gets in this way is not renormalizable and therefore cannot be used to make meaningful physical predictions. As a result, theorists have taken up more radical approaches to the problem of quantum gravity, the most popular approaches being string theory and loop quantum gravity. Although some quantum gravity theories, such as string theory, try to unify gravity with the other fundamental forces, others, such as loop quantum gravity, make no such attempt; instead, they make an effort to quantize the gravitational field while it is kept separate from the other forces.

Strictly speaking, the aim of quantum gravity is only to describe the quantum behaviour of the gravitational field and should not be confused with the objective of unifying all fundamental interactions into a single mathematical framework. A theory of quantum gravity that is also a grand unification of all known interactions is sometimes referred to as The Theory of Everything (TOE). While any substantial improvement into the present understanding of gravity would aid further work towards unification, the study of quantum gravity is a field in its own right with various branches having different approaches to unification.

One of the difficulties of formulating a quantum gravity theory is that quantum gravitational effects only appear at length scales near the Planck scale, around 10−35 metre, a scale far smaller, and equivalently far larger in energy, than those currently accessible by high energy particle accelerators. Therefore physicists lack experimental data which could distinguish between the competing theories which have been proposed.

In the 1970s Hawking became interested in Black Holes.

A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary of the region from which no escape is possible is called the event horizon. Although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it essentially impossible to observe.

Work by James Bardeen, Jacob Bekenstein, Carter, and Hawking in the early 1970s led to the formulation of black hole thermodynamics. These laws describe the behaviour of a black hole in close analogy to the laws of thermodynamics by relating mass to energy, area to entropy, and surface gravity to temperature. The analogy was completed when Hawking, in 1974, showed that quantum field theory predicts that black holes should radiate like a black body with a temperature proportional to the surface gravity of the black hole.

Particle Creation by Black Holes S. W. Hawking

In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies.

So quantum theory allows for the spontaneous creation and annihilation of particles.

A quantum fluctuation is the temporary appearance of energetic particles out of empty space, as allowed by the uncertainty principle.

Hawking came up with an ingenious thought experiment

Hawking radiation is blackbody radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974.

In the 1970s, Hawking described the mother of all thought experiments, one that still keeps physicists awake at night.

It starts with a black hole. According to Einstein’s relativity, a black hole is a region of space-time where gravity has become so strong that nothing can escape. But Hawking asked himself how quantum particles behave in the vicinity of such an object. After all, quantum mechanics is a theory of probabilities; maybe what’s impossible according to Einstein is possible in the quantum realm.

And indeed it is. Hawking’s calculations revealed, as he put it in his characteristically mischievous way that “black holes ain’t so black.” They actually emit a steady, faint stream of particles, now known as Hawking radiation. These particles carry away bits of the mass of the hole, so that it will eventually disappear entirely, a phenomenon known as Hawking evaporation.

So here’s the thought experiment: Throw a book into the black hole. The book carries information. Perhaps that information is about physics, perhaps that information is the plot of a romance novel — it could be any kind of information. But as far as anyone knows, the outgoing Hawking radiation is the same no matter what went into the black hole. The information is apparently lost — where did it go?

Thus we have the “black hole information loss puzzle,” perhaps Hawking’s most profound gift to physics. At issue is the fate of the principle of the conservation of information. Without general relativity, quantum mechanics predicts that information is conserved; likewise, without quantum mechanics, general relativity predicts that information is conserved, even if some of it is hidden inside a black hole. It is therefore bothersome that putting the two theories together seems to lead to information just disappearing.

For a long time Hawking argued, against the intuition of most other leading physicists, that information was simply erased from the universe, and we would have to learn to deal with it. But eventually he changed his mind (something he always was admirably willing to do), conceding in 2004 that information was probably somehow retained in the outgoing radiation. The matter, however, is very far from settled.

Hawking’s information-loss thought experiment is the single biggest clue we have to how quantum gravity might operate. Even if we don’t have the full theory yet, we know a lot about quantum mechanics and a lot about gravity. Together those are enough to convince us that Hawking radiation is real, even if it has never been directly observed. This means that any eventual theory of quantum gravity will have to explain either how information somehow escapes from black holes or how it is destroyed.

The Bekenstein-Hawking entropy or black hole entropy is the amount of entropy that must be assigned to a black hole in order for it to comply with the laws of thermodynamics as they are interpreted by observers external to that black hole. This is particularly true for the first and second laws. Black hole entropy is a concept with geometric root but with many physical consequences. It ties together notions from gravitation, thermodynamics and quantum theory, and is thus regarded as a window into the as yet mostly hidden world of quantum gravity.

Formula for black hole entropy

If A stands for the surface area of a black hole (area of the event horizon), then the black hole entropy, in dimensionless form, is given by

SBH = kc3A/4ħG

A stands for the surface area of a black hole (area of the event horizon), G, ħ and c denote, respectively, Newton’s gravity constant, the Planck-Dirac constant (h/(2π)) and the speed of light.

Euclidean Quantum Gravity Stephen W. Hawking

In the book is the first hint of holography

The theory of inflation, which was first proposed by Alan Guth in 1981, soon became a “need” of modern cosmology and various modified models of inflationary Universe were proposed.

In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from 10−36 seconds after the conjectured Big Bang singularity to sometime between 10−33 and 10−32 seconds after the singularity.

Guth (1981) The point of cosmic inflation is to solve cosmological tuning problems (and the monopole problem)

Starobinsky (1981) The point of cosmic inflation is to “help find” a unique highly symmetric (highly tuned!) initial state with no singularity


Alan Harvey Guth (born February 27, 1947) is an American theoretical physicist and cosmologist. Guth has researched elementary particle theory (and how particle theory is applicable to the early universe).


Alexei Alexandrovich Starobinsky (born 19 April 1948) is a Soviet and Russian astrophysicist and cosmologist. He received the Kavli Prize in Astrophysics “for pioneering the theory of cosmic inflation”, together with Alan Guth and Andrei Linde in 2014.


Andrei Dmitriyevich Linde (born March 2, 1948) is a Russian-American theoretical physicist and the Harald Trap Friis Professor of Physics at Stanford University.


To understand how the current universe looks the same in all directions, we need the very early universe to undergo a first-order phase transition at the age of about 10−36 to 10−33 seconds. In this interval the radius of the universe would have increased from sub-quantum size to roughly 1 cm. This inflation would be due to an inflation field (quanta: inflatons) with a huge, constant vacuum energy. At the end of the transition that vacuum energy would have to be zero. The huge expansion would drastically cool the universe, but the required decay of the inflatons into particles with enormous kinetic energy would reheat things. The scale factor during expansion would be like eHt, and the key fact is that the quantum fluctuations existing during the inflationary era wold be magnified to macroscopic size at the end of the era, and should still be seen in the Big Flash temperature and density variations. In fact the level of variation seen by WMAP and Planck on the last scattering surface is around 10−4 to 10−5, precisely as expected from the inflationary scenario. These density variations were the seeds of all the structure currently seen in our universe, augmented by the driven standing waves existing in the later soup of fermions and photons. This era of intense particle creation in a compact space-time would also create distinctive gravitational radiation, and would produce a unique quadrupole signature in the Big Flash, which it is very, very important to be able to observe.

Guth (1981) first proposed the cosmic inflation which is now called old inflation. He pictured the inflation as the first-order phase transitioning from metastable state (or false vacuum) to the true stable one (or true vacuum).

New inflation (by Linde (1982a), Linde (1982b), Linde (1982d), Linde (1982c), and Albrecht and Steinhardt (1982)) was proposed not so long after the problems of the old inflation had been revealed. New inflation happened through the second-order phase transition, instead of the first-order phase transition proposed in the old inflation. Also, the new inflation introduced inflaton as the scalar field governing the evolution of the early universe. Three big problems in cosmology

Inflation turned virtual particles into real particles.



What happened before inflation?


As you travel further back in time, more interesting things happen at higher and higher energies. When the big bang was first conceived, the extrapolation went all the way back to a singularity, when all the matter and energy in the Universe was concentrated at a single point, where the expansion was arbitrarily fast, and the temperature was practically infinite.

But when inflation came along, all of that changed. No longer can you extrapolate all the way back to a singularity. If we wound the clock of the Universe backwards, we would discover something remarkable. At some point, about 10-30 seconds before we would anticipate running into that singularity, the Universe instead would undergo inflation (in reverse, if we’re looking backwards), and we have no evidence for anything that came before it.

The Big Bang, instead of being a singularity, is the set of initial conditions of an extremely hot, dense, expanding Universe that exists immediately after the end of inflation.


Was there a singularity before inflation? Possibly, but at this point, we have no way of knowing. Inflation is the first thing we can say anything definitive about, but it definitely comes before what we traditionally call “The Big Bang”. So maybe you should admit that “Starts with a Bang” isn’t really the starting point of everything, after all, just the start of where our observable Universe comes from.

Yes, inflation happens before the Big Bang, and ever since its acceptance, has removed the necessity of a singularity at the start.

Predicting a Prior for Planck “Landscape of potentials”

Cosmic structure formation in Hybrid Inflation models

Relative weighting of models for inflation enables sharp predictions to be made

The boundary conditions of the universe Hawking, S.W.…proc..563H State, Pontificia Academia Scientiarum, 1982

The questions of what the boundary conditions of thy universe are and where they should be imposed are considered.

The no-boundary proposal


One of the biggest outstanding problems with current theories of the origin of the universe is how to deal with the fact that it apparently began with a singularity − a point of infinite density and zero size at the beginning of time. To dodge the mathematical problems that arise from this, Prof. Hawking and James Hartle conceived the no-boundary proposal, in which the singularity is eliminated by a mathematical manoeuvre akin to rounding off a sharp corner. To make it work, the theorists proposed that time as we experience it blends with space to become “imaginary time” near the beginning of the universe. The concept of time vanishes at the beginning just as the concept of south vanishes at the south pole. The proposal has been debated and challenged, most recently by Neil Turok and collaborators, but it remains a conceptual milestone on the way to a deep understanding of the origins of the universe.

Spontaneous creation of a pair of universes from “nothing”

The No-Boundary Measure of the Universe

The no-boundary wave function peaked around inflationary universes. It explained why inflation started in the first place.

The multiverse is a hypothetical group of multiple universes including the universe in which we live. Together, these universes comprise everything that exists: the entirety of space, time, matter, energy, the physical laws and the constants that describe them. The different universes within the multiverse are called “parallel universes”, “other universes”, or “alternative universes”.

Populating the Landscape: A Top Down Approach S.W. Hawking and Thomas Hertog

Stephen Hawking, along with Thomas Hertog of CERN, proposed that the universe’s initial conditions consisted of a superposition of many possible initial conditions, only a small fraction of which contributed to the conditions we see today. According to their theory, it is inevitable that we find our universe’s “fine-tuned” physical constants, as the current universe “selects” only those past histories that led to the present conditions. In this way, top-down cosmology provides an anthropic explanation for why we find ourselves in a universe that allows matter and life, without invoking the ontic existence of the Multiverse.

The top-down approach is based on the principle that radiation smoothed out the matter density fluctuations to produce large pancakes. These pancakes accrete matter after recombination and grow until they collapse and fragment into galaxies.


Large pancakes of matter form first then fragment into galaxy-sized lumps

This scenario has the advantage of predicting that there should be large sheets of galaxies with low density voids between the sheets. Clusters of galaxies form where the sheets intersect.

Eternal inflation is a hypothetical inflationary universe model, which is itself an outgrowth or extension of the Big Bang theory.

According to eternal inflation, the inflationary phase of the universe’s expansion lasts forever throughout most of the universe. Because the regions expand exponentially rapidly, most of the volume of the universe at any given time is inflating. Eternal inflation, therefore, produces a hypothetically infinite multiverse, in which only an insignificant fractal volume ends inflation.

Inflation Theory Part 1 – Eternal Inflation: What Caused The Big Bang

Eternal Inflation Alan H. Guth


Above left is a schematic diagram to illustrate the fractal structure of the universe created by eternal inflation. The four horizontal bars represent a patch of the universe at four evenly spaced successive times. The expansion of the universe is not shown, but each horizontal bar is actually a factor of three larger than the preceding bar, so each region of repulsive-gravity material is actually the same size as the others. During the time interval between bars, 1/3 of each region of repulsive-gravity material decays to form a pocket universe. The process repeats ad infinitum, producing an infinite number of pocket universes.

The top bar of the above right diagram indicates a region of false vacuum. The evolution of this region is shown by the successive bars moving downward, except that the expansion could not be shown while still fitting all the bars on the page. So the region is shown as having a fixed size in commoving coordinates, while the scale factor, which is not shown, increases from each bar to the next. As a concrete example, suppose that the scale factor for each bar is three times larger than for the previous bar. If we follow the region of false vacuum as it evolves from the situation shown in the top bar to the situation shown in the second bar, in about one third of the region the scalar field rolls down the hill of the potential energy diagram, precipitating a local big bang that will evolve into something that will eventually appear to its inhabitants as a universe. This local big bang region is shown in grey and labelled “Universe.” Meanwhile, however, the space has expanded so much that each of the two remaining regions of false vacuum is the same size as the starting region. Thus, if we follow the region for another time interval of the same duration, each of these regions of false vacuum will break up, with about one third of each evolving into a local universe, as shown on the third bar from the top. Now there are four remaining regions of false vacuum, and again each is as large as the starting region. This process will repeat itself literally forever, producing a kind of a fractal structure to the universe, resulting in an infinite number of the local universes shown in grey. There is no standard name for these local universes, but they are often called bubble universes. I prefer, however, to call them pocket universes, to avoid the suggestion that they are round. While bubbles formed in first-order phase transitions are round , the local universes formed in eternal new inflation are generally very irregular, as can be seen for example in the two-dimensional simulation by Vanchurin, Vilenkin, and Winitzki.

The diagram is of course an idealization. The real universe is three dimensional, while the diagram illustrates a schematic one-dimensional universe.

Eternal inflation and its implications Alan H. Guth


Inflation occurs as the scalar field rolls down a hill of the potential energy diagram, as shown above, starting high on the hill. As the field rolls down the hill, quantum fluctuations will be superimposed on top of the classical motion.

Quantum creation leads to a multiverse

The holographic principle is a principle of string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region—preferably a light-like boundary like a gravitational horizon. First proposed by Gerard ‘t Hooft, it was given a precise string-theory interpretation by Leonard Susskind who combined his ideas with previous ones of ‘t Hooft and Charles Thorn. As pointed out by Raphael Bousso, Thorn observed in 1978 that string theory admits a lower-dimensional description in which gravity emerges from it in what would now be called a holographic way. The prime example of holography is the AdS/CFT correspondence.

The holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected. In the case of a black hole, the insight was that the informational content of all the objects that have fallen into the hole might be entirely contained in surface fluctuations of the event horizon. The holographic principle resolves the black hole information paradox within the framework of string theory. However, there exist classical solutions to the Einstein equations that allow values of the entropy larger than those allowed by an area law, hence in principle larger than those of a black hole. These are the so-called “Wheeler’s bags of gold”. The existence of such solutions conflicts with the holographic interpretation, and their effects in a quantum theory of gravity including the holographic principle are not yet fully understood.

In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories.

The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997. Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Polyakov, and by Edward Witten. By 2015, Maldacena’s article had over 10,000 citations, becoming the most highly cited article in the field of high energy physics.

The Large N Limit of Superconformal field theories and supergravity Juan Maldacena


Juan Martín Maldacena (September 10, 1968 in Buenos Aires, Argentina) is a theoretical physicist.

In 1997, theoretical physicist Juan Maldacena proposed that an audacious model of the Universe in which gravity arises from infinitesimally thin, vibrating strings could be reinterpreted in terms of well-established physics. The mathematically intricate world of strings, which exist in nine dimensions of space plus one of time, would be merely a hologram: the real action would play out in a simpler, flatter cosmos where there is no gravity.

Maldacena’s idea thrilled physicists because it offered a way to put the popular but still unproven theory of strings on solid footing — and because it solved apparent inconsistencies between quantum physics and Einstein’s theory of gravity. It provided physicists with a mathematical Rosetta stone, a ‘duality’, that allowed them to translate back and forth between the two languages, and solve problems in one model that seemed intractable in the other and vice versa. But although the validity of Maldacena’s ideas has pretty much been taken for granted ever since, a rigorous proof has been elusive.


The AdS/CFT duality relates a theory on the boundary of a region to a theory with gravity in the interior. The AdS/CFT duality is a concrete example of holography, which suggests that all the information about the interior of some region is actually contained on the boundary.

In the AdS/CFT correspondence, one considers, in addition to a theory of quantum gravity, a certain kind of quantum field theory called a conformal field theory. This is a particularly symmetric and mathematically well behaved type of quantum field theory. Such theories are often studied in the context of string theory, where they are associated with the surface swept out by a string propagating through spacetime, and in statistical mechanics, where they model systems at a thermodynamic critical point.


In mathematics and physics, a de Sitter space is the analogue in Minkowski space, or spacetime, of a sphere in ordinary Euclidean space.

Holographic Quantum Cosmology | Thomas Hertog


A sketch of the timeline of the holographic Universe. Time runs from left to right. The far left denotes the holographic phase and the image is blurry because space and time are not yet well defined. At the end of this phase (denoted by the black fluctuating ellipse) the Universe enters a geometric phase, which can now be described by Einstein’s equations (a manageable Universe occurs). The cosmic microwave background was emitted about 375,000 years later. Patterns imprinted in it carry information about the very early Universe and seed the development of structures of stars and galaxies in the late time Universe (far right). A smooth exit from eternal inflation. Credit: Paul McFadden.

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