Paul Joseph Steinhardt
December 25, 1952
|Thesis||Lattice theory of SU(N) flavor quantum electrodynamics in (1 + 1)-dimensions (1978)|
|Doctoral advisor||Sidney R. Coleman|
|Other academic advisors|
In addition, Steinhardt co-founded and is the current Director of the Princeton Center for Theoretical Science.
- 1 Education
- 2 Career and research
- 3 Inflationary cosmology
- 4 Cyclic/ekpyrotic theory of the universe
- 5 Dark energy and dark matter
- 6 Amorphous solids
- 7 Quasicrystals
- 8 Natural quasicrystals, meteoritics, and geoscience
- 9 Photonics
- 10 Honors and awards
- 11 References
Career and research
In cosmology, some of his important contributions include: He is one of the architects of inflationary cosmology which has become an essential part of the big bang theory. He also presented the first example of eternal inflation, which ultimately revealed the multiverse and caused him to doubt the inflation theory he had pioneered. He then became one of the developers of the leading competing alternatives to the big bang, the ekpyrotic and the cyclic theory of the universe. Steinhardt co-authored the first paper to compute the complete imprint of gravitational waves on the spectrum of temperature variations and on the B-mode polarization of the cosmic microwave background. He also co-authored the first paper to show that, based on observations at the time, most of the energy in the universe must be dark energy, enough to cause the universe to expand at an accelerating rate today. Several years later, the conclusion was confirmed by supernova observations. Working with various collaborators, he introduced the concept of quintessential dark energy to further explain the accelerating expansion of the universe. Steinhardt also co-authored the concept of strongly self-interacting dark matter to explain the missing mass of the universe, and to address anomalies previous theories of dark matter were unable to explain.
In condensed matter physics and geophysics: Steinhardt and his then-student Dov Levine invented the theoretical concept of quasicrystals, and even introduced the term "quasicrystal" to the lexicon. The two were also the first to correctly identify a puzzling aluminum-manganese alloy with icosahedral symmetry announced by Dan Shechtman and collaborators in 1984 as the first example of a quasicrystal to be made in a laboratory. The following decade, Steinhardt launched a worldwide search for a quasicrystal formed by a natural geophysical or astrophysical process; his team eventually discovered a candidate in a Florentine museum collection. Their discovery was recognized by the International Mineralogical Association as the first natural quasicrystal, and they accepted the team's proposed name for it, icosahedrite. Over the next year, disparate clues were pieced together about its origin, finally pointing to a remote mountain range in Chukotka, on Russia's Kamchatka peninsula. Two years later, Steinhardt initiated his first-ever field expedition, and led a team of geologists to the region. As a result of that expedition—and nearly three decades after first inventing the concept of this entirely new form of matter—Steinhardt managed to discover the first quasicrystal formed through a natural process, which his team later proved was part of a 4.5 billion year old meteorite. They also discovered a second quasicrystal named decagonite, along with a new crystalline mineral that was named steinhardtite, in his honor.
In photonics: Steinhardt has been an innovator in using quasicrystals and other novel patterns to design photonic materials, including a new class of material called hyperuniform disordered solids. More recently, Steinhardt helped design and fabricate a photonic quasicrystal (the quasicrystal analogue of a photonic crystal) for efficiently trapping and manipulating light in selected wavebands and is one of the leaders developing isotropic photonic materials known as hyperuniform disordered solids (HUDS). Steinhardt and his Princeton colleagues later helped form a company named Etaphase, Inc. to develop commercial applications for those materials.
I know of no other scientist, no other theoretical physicist alive who has a clearer focus on whether our theories and ideas are relevant to the real world. And that's always what he's after.
Development of theory
Beginning in the early 1980s, Steinhardt co-authored seminal papers that helped to lay the foundations of inflationary cosmology.
In 1982, Steinhardt and Andreas Albrecht (as well as Andrei Linde, working independently in Moscow) constructed the first inflationary models to produce sufficient accelerated expansion to smooth and flatten the universe and then "gracefully exit" (uniformly transform from inflationary expansion to a slowly expanding universe full of matter and radiation that evolves into the universe observed today). The inflation is caused by the energy stored in a scalar field (known today as the `inflaton’) that permeates space and varies with time from high energy to low energy. This work is especially significant, because it has been the prototype for most subsequent inflationary models.
The Albrecht-Steinhardt paper is also notable because it introduced the concept of Hubble friction in 1982, a gravitational effect that plays the key role in the "slow-roll" of the inflaton field (a very gradual change in its strength) enabling the generation of density fluctuations (see next item). Hubble friction and slow-roll is an essential element in most subsequent inflationary models.
James Bardeen, Steinhardt and Michael S. Turner then produced the first computation showing that density fluctuations can be generated from quantum fluctuations during inflation using a relativistically gauge method of calculation in 1983. They showed that it is possible to produce a nearly scale-invariant spectrum of fluctuations with a small tilt, properties later shown by observations of the cosmic microwave background to be features of our universe. Contemporaneous calculations by several other groups obtained similar conclusions using less rigorous methods.
Later that year, Steinhardt presented the first example of eternal inflation, which would eventually be shown to be a generic feature of inflationary models that leads to a multiverse of outcomes instead of the single smooth and flat universe, as originally hoped when first proposed. This was significant research, because Steinhardt would eventually argue (beginning in 2002) that if every outcome is allowed, inflation makes no predictions. Consequently, eternal inflation may be the Achilles heel of the inflationary picture.
Daile La and Steinhardt teamed up in 1989, and introduced extended inflation based on scalar-tensor theories of gravity that introduced a new way to solve the graceful exit problem through bubble nucleation and collisions rather than slow-roll.
Robert Crittenden, Rick Davis, J.R. Bond, G. Efstathiou and Steinhardt performed the first calculations of the complete imprint of gravitational waves on the cosmic microwave background temperature and E-mode and B-mode polarization power spectra in 1993.
Steinhardt broke ranks with Guth and Linde in 2002, when he began to voice his profound concern about the multiverse, which by this time had been established as an inevitable consequence of inflation and eternal inflation.
Steinhardt argued that the multiverse destroyed any predictive power of inflation because it produces an infinite variety of patches of space spanning every conceivable cosmological outcome, including an infinite number of patches that are not flat, not uniform, and nearly without scale-invariant perturbations. His opinion was unexpected and largely unwelcome in the scientific community, because Steinhardt had played an important role in developing both inflation and eternal inflation, which are the two important theoretical concepts which lead to the multiverse. His argument that the multiverse was a sign of the failure of the inflationary model was largely based in his belief in the long-standing scientific method, often referred to as falsifiability. He argued that because inflation produces a multiverse, where anything and everything is possible any number of times, the theory itself is so flexible that no observation or combination of observations can ever disprove it.
Steinhardt’s argument that inflation is eternal and produces anything and everything was something that leading proponents of the inflationary theory, including Alan Guth and Andrei Linde generally concede.
The inescapable conclusion of the theory’s lack of falsifiability, as Steinhardt often emphasized, was that none of the observations that had been made since the inflationary theory was first introduced could be viewed as supporting the theory, because any other result would have been equally possible in a multiverse. Inflation predicts nothing, Steinhardt argued, because it predicts everything.
Steinhardt also pointed out that inflation’s lack of falsifiability represented a significant break with 400 years of scientific tradition. The ability to disprove a theory through experiment has always been considered the key to distinguishing science from non-science, an argument frequently put forth by his mentor, Richard Feynman. “Our universe has a simple, natural structure,” Steinhardt said. “The multiverse idea is baroque, unnatural, untestable and, in the end, dangerous to science and society.”
As a result of his opposition to the multiverse, Steinhardt began to explore other cosmological theories which could describe the early universe just as well as inflation, such as the ekpyrotic and cyclic models.
The unlikeliness problem
Based on observations from the Planck satellite (reported in 2013) Anna Ijjas, Abraham Loeb and Steinhardt showed in a widely discussed pair of papers that the inflationary model was much less likely to explain our universe than previously thought. According to their analysis, the chances of obtaining a universe matching the observations after a period of inflation is less than one in a googolplex. Steinhardt and his team dubbed this result the “unlikeliness problem.”
The two papers also showed that the remaining inflationary models require more parameters, more fine-tuning of those parameters, and more unlikely initial conditions than the simplest models. As a result, the startling conclusion was that Steinhardt and his colleagues had essentially shown that, by the standards of normal scientific reasoning, the inflationary theory is unlikely to be correct based on the observations.
In 2015, the unlikeness problem was reaffirmed and strengthened by a subsequent round of measurements reported by the Planck satellite team. 
The multi-year analysis of observational data from the Planck Satellite by Steinhardt and his colleagues has made the unlikeliness problem one of the more significant challenges to the inflationary theory to date.
Cyclic/ekpyrotic theory of the universe
Working with Justin Khoury, Burt A. Ovrut and Neil Turok, Steinhardt introduced the ekpyrotic theory, which imagines a big bounce instead of a big bang. According to this model, the current expanding universe emerges from a bounce that occurred 13.7 billion years ago and that is a result of the preceding (contracting) universe. The smoothing and flattening of the universe and the generation of density variations occur during the phase of slow contraction before the bounce and remain after the bounce.
In the original example, the bounce corresponded to a collision and recoil of branes along an extra dimension in string theory (brane is derived from "membrane," a basic object in string theory). But more recent versions do not require extra dimensions or string theory; instead, quantum fields with potential energy evolving in space-time, similar to inflationary models, can be used.
Steinhardt and Turok then incorporated the ekpyrotic idea into a bolder proposal: the cyclic theory of the universe.
According to this model, the bounce represents the end of a cycle of evolution, the transition between a preceding period of contraction and the next period of expansion. This cycle repeats at regular intervals every trillion years or so. Dark energy is predicted by the model and plays an essential role stabilizing the cycles. Steinhardt and Turok proved that the cyclic process can repeat itself not just indefinitely, but also infinitely into the past and the future.
Earlier cyclic-type cosmological models had an entropy problem that the new cyclic model evades. According to the second law of thermodynamics, entropy always increases. The entropy produced in one cycle adds to the entropy produced in earlier cycles. If the universe contracts and the entropy produced in previous cycles lies within the observable universe at the beginning of the next cycle, the added entropy will cause the next cycle to be longer than the one before; or, projecting backwards in time, earlier cycles would have to be shorter and shorter, eventually reducing to zero duration. This reintroduces the problem that time must have a beginning. In the new cyclic model, the observable universe just after a bounce only occupies a tiny fraction of the space occupied by the observable universe a cycle earlier and so only contains a negligible fraction of the total entropy. The entropy building up over the cycles is almost entirely beyond what can be seen and cannot influence the next cycle.
Cyclic models have two big advantages over inflationary models. First, they do not produce a multiverse because the smoothing and flattening occur during contraction rather than inflation. This means that, unlike inflation, cyclic models make definite predictions and are falsifiable. Second, cyclic models explain why there must be dark energy. The accelerated expansion caused by dark energy starts the smoothing process and the decay of dark energy starts the contraction.
Because cyclic/ekpyrotic models do not produce a multiverse, they make definite predictions which, if shown to be wrong, disprove the theory. One prediction is that, unlike inflation, no detectable gravitational waves are generated during the smoothing and flattening process. A second prediction is that the current acceleration expansion must eventually stop and the vacuum must be eventually decay in order to initiate the next cycle. (Other predictions depend on the specific fields (or branes) that cause the contraction.)
The cyclic model may naturally explain why the cosmological constant is exponentially small and positive, compared to the enormous value expected by quantum gravity theories. The cosmological constant might begin large, as expected, but then slowly decay over the course of many cycles to the tiny value observed today.
The discovery of the Higgs field at the Large Hadron Collider (LHC) may provide added support for the cyclic model. Evidence from the LHC suggests that the current vacuum may decay in the future, according to calculations made by Steinhardt, Turok and Itzhak Bars. The decay of the current vacuum is required by the cyclic model in order to end the current phase of expansion, contract, bounce and a new era of expansion; the Higgs provides a possible mechanism of decay that can be tested. The Higgs field is a viable candidate for the field that drives the cycles of expansion and contraction.
Dark energy and dark matter
In 1995, Steinhardt and Jeremiah Ostriker used a concordance of cosmological observations to show there must be a non-zero dark energy component today, more than 65 percent of the total energy density, sufficient to cause the expansion of the universe to accelerate. This was verified three years later by supernova observations in 1998.
Working with colleagues, he subsequently introduced the concept of quintessence, a term used to describe a form of dark energy that varies with time. It was first posited by Steinhardt’s team as an alternative to the cosmological constant, which is (by definition) constant and static; quintessence is dynamic. Its energy density and pressure evolve over time.
The introduction of quintessence underscored the importance of measuring the equation-of-state of dark energy (the ratio of pressure divided by energy density, often symbolized by the letter "w"), in order to determine the cause of the acceleration of the known universe.
Steinhardt has also proposed various approaches for solving the "cosmological constant problem" using quintessence, including tracker and k-essence These forms of dark energy would have the property that they cannot dominate the expansion of the universe and initiate cosmic acceleration until later in the history of the universe, after galaxies form, in accord with astronomical observations.
The cosmological constant problem can be naturally resolved in the cyclic model, according to Steinhardt and Turok.
Most prior models of dark matter assumed it is composed of weakly interacting massive particles (also referred to as "WIMPs") that clump together through gravitational attraction only because the other forces between particles are too weak to affect their motion. The SIDM picture proposes that the other forces are strong enough for the particles to scatter off one another in regions where they would otherwise pile up (forming what cosmologists call a "cusp") in the center of galaxies. Also, the scattering would reduce the mass of dwarf galaxies in orbit around larger galaxies because the dark matter particles surrounding the larger galaxy would scatter and, thereby, strip off the dark matter surrounding the dwarf galaxy. This idea is particularly interesting because there is astronomical evidence suggesting that galaxy cores are not cuspy, and the most massive galaxies predicted by simulations based on WIMPs are not observed. If the SIDM picture is correct, searches for WIMP dark matter will find nothing, and different approaches will be needed to determine the identity of dark matter.
Steinhardt, Spergel and Jason Pollack have proposed that a small fraction of dark matter could have ultra-strong self-interactions, which would cause the particles to coalesce rapidly and collapse into seeds for early supermassive black holes.
He constructed the first computer generated continuous random network (CRN) model of glass and amorphous silicon in 1973, while still an undergraduate at Caltech. CRNs remain the leading model of amorphous silicon and other semiconductors today. Working with Richard Alben and D. Weaire, he used the computer model to predict structural and electronic properties.
Working with David Nelson and Marco Ronchetti, Steinhardt formulated mathematical expressions, known as "orientational order parameters", for computing the degree of alignment of interatomic bonds in liquids and solids in 1981. Applying them to computer simulations of monatomic supercooled liquids, they showed that the atoms form arrangements with finite-range icosahedral (soccer-ball like) bond orientational order as liquids cool. The order parameter continues to be used in diverse areas of chemistry and materials sciences; the discovery of icosahedral orientational order stimulated the discovery of quintessence.
1983: theoretical development
Motivated by the work on amorphous solids (described in the section above), Steinhardt and his then-student Dov Levine first introduced the theoretical concept of quasicrystals in a patent disclosure in 1983, a year before any other experimental evidence had been announced. They published their complete theory in December 1984, a few weeks after an intriguing new alloy had been reported by Dan Shechtman and his collaborators.
Their theory proposed the existence of a new phase of solid matter analogous to Penrose tilings with rotational symmetries previously thought to be impossible for solids. The paper, entitled "Quasicrystals: A New Class of Ordered Structures," coined the term "quasicrystal" to name the new phase. The word "quasicrystal" was meant to be a reference to the fact that the proposed atomic structure had quasiperiodic atomic ordering, rather than the periodic ordering characteristic of conventional crystals.
The paper showed that quasiperiodic structures could violate the rigorous mathematical symmetry restrictions for conventional periodic crystals that had been established for more than a century. They proved that all of the symmetries once thought to be forbidden for solids are actually possible for quasicrystals, including solids with axes of five-fold symmetry and three-dimensional icosahedral symmetry (the same as a soccer ball—the classic example of forbidden symmetry for solids that is described in contemporary textbooks on solid state physics).
Road to experimental realization
Working completely independently of Steinhardt and Dov Levine, a team composed of Dan Shechtman, Ilan Blech, Denis Gratias and John Cahn published their experimental discovery of an unusual alloy of aluminum and manganese. Its diffraction pattern consisted of sharp (though not perfectly point-like) spots arranged with icosahedral symmetry, and did not fit any known crystal structure. The team’s paper reported data about an alloy synthesized at the National Bureau of Standards (NBS) which Shechtman first viewed under a microscope in 1982, although the results were only first published in November 1984.
In December 1984, Steinhardt and Levine published their theory and proposed that the NBS alloy might be evidence for their theory about quasicrystals. The theorists demonstrated the close resemblance between the imperfect experimental pattern and the ideal pattern they had computed (and tried to patent) the previous year for an icosahedral quasicrystal.
The next two developments, a combination of experiment and theoretical explanation, proved that quasicrystals - as originally theorized by Steinhardt and Levine in 1983 - are physically plausible and realizable:
In 1987, An-Pang Tsai and his group synthesized the first stable icosahedral quasicrystal in their lab in Japan and showed that, compared to the NBS alloy, it had sharper diffraction spots arranged in much closer accordance with the 1983 quasicrystal theory. Whereas the imperfect diffraction pattern obtained from the NBS alloy allows room for alternative explanations (most notably one proposed by Linus Pauling), the results from the Tsai lab were unambiguously consistent with the quasicrystal predictions and inconsistent with the alternative ideas.
The following year, Steinhardt along with George Onoda, David DiVincenzo and Joshua Socolar showed that Penrose tiles could be randomly aggregated one-by-one to form a defect-free, two dimensional tiling using only local information about neighboring tiles to decide what tiles to attach next. This work proved there was no theoretical blockade to growing perfect quasicrystals, as had been previously thought.
Other contributions to the field
With collaborators, Steinhardt has also made significant contributions to understanding the quasicrystals’ unique mathematical and physical properties, including theories of how and why quasicrystals form and their elastic and hydrodynamics properties, laying the groundwork for many current applications.
Peter J. Lu and Steinhardt discovered a quasicrystalline Islamic tiling on the Darb-e Imam Shrine (1453 A.D.) in Isfahan, Iran constructed from girih tiles. In 2007, they revealed a conceptual breakthrough that enabled early artists to create increasingly complex periodic girih patterns over a period of centuries, culminating in a nearly perfect quasi-crystalline Penrose pattern five centuries before their discovery in the West.
Natural quasicrystals, meteoritics, and geoscience
In 1998, Steinhardt initiated an international search for a natural quasicrystal to demonstrate the stability of quasicrystal and the possibility of new forms of minerals that had not been noticed before.
After nearly a decade of failed searches and false positives, the team was joined by Italian scientist Luca Bindi, then curator of the mineral collection at the Universite’ di Firenze. In 2009, the first example of a natural quasicrystal was discovered in the basement of the museum. The tiny specimen, a few millimeters across, had been packed away in a box labeled "khatyrkite," and was thought to contain only an ordinary crystal composed of copper and aluminum. However, the specimen contained other minerals as well, including grains that Steinhardt and collaborators identified as a natural quasicrystal.
The International Mineralogical Association officially accepted the quasicrystal as a new mineral and designated its name, icosahedrite. The material had exactly the same atomic composition (Al63Cu24Fe13) as the first thermodynamically stable quasicrystal synthesized by An-Pang Tsai and his group in their laboratory in 1987.
Two years after identifying the museum sample, Steinhardt organized an international team of experts and led them on an expedition to its source, the remote Listventovyi stream in the Chukotka Autonomous Okrug in the northern half of the Kamchatka Peninsula in far eastern Russia. The team included Bindi and Valery Kryachko, the Russian ore geologist who had found the original samples of khatyrkite crystal while working at the Listventovyi stream in 1979; one of the samples Kryachko had recovered ended up at the Florence museum, where Bindi picked it out for investigation as part of Steinhardt's search for natural quasicrystals. Studies in Florence and Princeton eventually revealed that the sample contained grains of icosahedrite (in addition to the previously identified khatyrkite).
After digging and panning a ton and a half of clay along the banks of the Listvenitovyi stream in the Koryak Mountains, the team returned home with a few kilograms of separates. Eventually, after examining each grain of panned material one by one, a few were found to contain icosahedrite. Using these grains, Steinhardt's team was able to prove that the samples had originated from a meteorite formed 4.5 billion years ago (before there were planets), and had landed on the Earth about 15,000 years ago.
That discovery was considered both a scientific breakthrough and an enormous personal achievement. Nearly 30 years after first inventing the theoretical concept of an entirely new form of matter he originally named "quasicrystals," Steinhardt had discovered the first quasicrystal formed through a natural process. And he had managed to do so in one of the most remote regions of the world.
Steinhardt's Princeton lab continued to discover even more samples of icosahedrite, along with several additional new minerals. In 2014, one of those minerals was discovered to be a crystalline phase of aluminum, nickel and iron (Al38Ni33Fe30). It was later officially accepted by the International Mineralogical Association and named "steinhardtite." In 2015, yet another type of natural quasicrystal was discovered in a different grain of the meteorite. The second new natural quasicrystal was found to be a different mixture of aluminum, nickel and iron (Al71Ni24Fe5) and had a decagonal symmetry (a regularly stacking of atomic layers which each have 10-fold symmetry), and was named "decagonite."
A team of researchers including Steinhardt, Paul Chaikin, Weining Man and Mischa Megens designed and tested the first photonic quasicrystal with icosahedral symmetry in 2005. They were the first to demonstrate the existence of photonic band gaps ("PBGs"). These materials block light for a finite range of frequencies (or colors) and let pass light with frequencies outside that band, similar to the way in which a semiconductor blocks electrons for a finite range of energies.
Working with Salvatore Torquato and Marian Florescu, in 2009 Steinhardt discovered a new class of photonic materials called hyperuniform disordered solids (HUDS), and showed that solids consisting of a hyperuniform disordered arrangement of dielectric elements produce band gaps with perfect spherical symmetry. These materials, which act as isotropic semiconductors for light, can be used to control and manipulate light in a wide range of applications including optical communications, photonic computers, energy harvesting, non-linear optics and improved light sources.
The meta-material breakthroughs by Steinhardt and his Princeton colleagues were discovered to have valuable commercial applications. In 2012, the scientists helped create a start-up company named Etaphase, which will apply their discoveries to a wide range of high performance products. The inventions will be used in integrated circuits, structural materials, photonics, communications, chip-to-chip communications, intra-chip communications, sensors, datacomm, networking, and solar applications.
Honors and awards
- In 1986, Steinhardt was elected as a Fellow in the American Physical Society in recognition of his important contributions to cosmology and to the theoretical understanding of quasicrystals.
- In 1998, he was elected to the United States National Academy of Sciences for distinguished and continuing achievements in original research.
- In 2002, Steinhardt was honored for his pivotal role as one of the original architects of the inflationary model of the universe when he shared the prestigious P.A.M. Dirac Medal from the International Centre for Theoretical Physics with Alan Guth of MIT and Andrei Linde of Stanford.
- In 2010, Steinhardt received the Oliver E. Buckley Condensed Matter Prize of the American Physical Society for his pioneering contributions to the theory of quasicrystals, including the prediction of their diffraction pattern.
- In 2012, he received the John Scott Award for his work on quasicrystals, including the co-discovery of the first natural quasicrystal called icosahedrite during a successful geological expedition to Chukotka in Far Eastern Russia.
- In 2012, Steinhardt was named Simons Fellow in Theoretical Physics and Radcliffe Fellow at the Radcliffe Institute for Advanced Study at Harvard.
- In 2014, he received the Caltech Distinguished Alumni Award, which is the highest honor the Institute bestows upon its alumni.
- In 2014, the International Mineralogical Association accepted a new mineral from the Khatyrka meteorite into the official catalogue of natural minerals, and named it "steinhardtite" in his honor.
- Paul Steinhardt at the Mathematics Genealogy Project
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- Bardeen, J. M.; Steinhardt, P. J.; Turner, M. S. (1983). "Spontaneous Creation Of Almost Scale-Free Density Perturbations In An Inflationary Universe". Phys. Rev. D. 28 (4): 679–693. Bibcode:1983PhRvD..28..679B. doi:10.1103/PhysRevD.28.679.
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- Crittenden, R.; Bond, J.R.; Davis, R.L.; Efstathiou, G.E.; Steinhardt, P. J. (1993). "The Imprint of Gravitational Waves on the Cosmic Microwave Background". Phys. Rev. Lett. 71 (3): 324–327. arXiv:astro-ph/9303014. Bibcode:1993PhRvL..71..324C. doi:10.1103/PhysRevLett.71.324. PMID 10055242.
- Crittenden; Davis, R.L.; Steinhardt, P. J. (1993). "Polarization of the Microwave Background due to Primordial Gravitational Waves". Astrophysical Journal Letters. 417: L13–L16. arXiv:astro-ph/9306027. Bibcode:1993ApJ...417L..13C. doi:10.1086/187082.
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- Guth, Alan H.; Kaiser, David I.; Nomura, Yasunori (2014). "Inflationary paradigm after Planck 2013". Physics Letters. B733 (2014): 112–119. arXiv:1312.7619. Bibcode:2014PhLB..733..112G. doi:10.1016/j.physletb.2014.03.020.
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- Iijas, Anna; Steinhardt, Paul J.; Loeb, Abraham (2014). "Inflationary Schism". Phys. Lett. B. 7: 142–146. Bibcode:2014PhLB..736..142I. doi:10.1016/j.physletb.2014.07.012.
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- Steinhardt, P. J.; Turok, N. (2002-04-25). "A Cyclic Model of the Universe". Science. 296 (5572): 1436–1439. arXiv:hep-th/0111030. Bibcode:2002Sci...296.1436S. doi:10.1126/science.1070462. PMID 11976408.
- Steinhardt, P.J.; Turok, N.G. (2007). Endless Universe: beyond the Big Bang. Doubleday. ISBN 9780385509640.
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- Bars, I.; Steinhardt, P.J.; Turok, N (2013). "Cyclid cosmology, conformal symmetry and the metastability of the Higgs". Phys. Lett. B. 726 (1–3): 50–55. arXiv:1307.8106. Bibcode:2013PhLB..726...50B. doi:10.1016/j.physletb.2013.08.071.
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