Paul Steinhardt

Paul J. Steinhardt
Born February 25, 1952
Washington, D.C.
Residence United States
Nationality American
Fields Theoretical physics
Cosmology
Institutions
Alma mater
Doctoral advisor Sidney R. Coleman
Other academic advisors
Known for
Notable awards Guggenheim Fellowship (1994)
Dirac Medal, ICTP (2002)
Oliver E. Buckley Condensed Matter Prize (2010)
John Scott Award (2012)
Simons Fellow (2012)
Radcliffe Fellow, Harvard (2012)
Caltech Distinguished Alumni Award (2014)
Website
http://wwwphy.princeton.edu/~steinh/

Paul Joseph Steinhardt (born December 25, 1952) is an American theoretical physicist and cosmologist who is currently the Albert Einstein Professor in Science at Princeton University.[1]

Steinhardt's research has spanned problems in particle physics, astrophysics, cosmology, condensed matter physics, geoscience and photonics. He is well known as one of the architects of inflationary cosmology and the leading competing alternative known as the cyclic theory of the universe. He is also one of the originators of the concept of quasicrystals, a state of matter with rotational symmetries once thought to be impossible for solids. He initiated and led a successful effort to discover the first natural quasicrystal, icosahedrite, and has been an innovator in using quasicrystals and other novel patterns to design photonic materials.

In addition, Steinhardt co-founded and is the current Director of the Princeton Center for Theoretical Science.[2]

Academic work

Steinhardt is best known for his work in theoretical physical cosmology, where he helped develop the theory of cosmic inflation, which attempts to explain the homogeneity and geometry of the universe and the origin of the fluctuations that seeded the formation of galaxies and large-scale structure. He introduced the concept of quintessence, a time-varying form of dark energy to explain the current accelerating expansion of the universe. His recent work has been on the ekpyrotic and cyclic theories of the Universe. The cyclic/ekpyrotic theory is a radical alternative to big bang/inflationary cosmology in which the big bang is replaced by a big bounce that recurs periodically and the key events shaping the large-scale structure of the universe occur before the bounce.

Steinhardt also works in condensed matter physics. Working with Dov Levine, he introduced the concept of quasicrystals (and coined the term) in 1983 and has made numerous contributions to understanding their mathematical and physical properties. 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).

Inflationary cosmology

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[3] (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).[4] 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,[3] 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 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.[5] 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.[6] 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.[7]

Robert Crittenden, Rick Davis, J.R. Bond, G. Efstathiou and Steinhardt performed the first calculations of the imprint of gravitational waves on the cosmic microwave background temperature and E-mode and B-mode polarization power spectra in 1993.[8][9]

Steinhardt’s major contributions to the inflationary theory were recognized in 2002, when he shared the Dirac Prize with Alan Guth of M.I.T. and Andrei Linde of Stanford.[10]

Criticism

Around the same time, Steinhardt publicly broke ranks with Guth and Linde when he began to voice his profound concern about the multiverse producing 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 without nearly scale-invariant perturbations. Steinhardt’s warnings were unexpected, because he had played an important role in developing inflation and revealing the multiverse when he introduced the first example of eternal inflation in 1983.

He has emphasized that due to the multiverse, the inflationary model is so flexible that no observation or combination of observations can disprove it. Also, none of the observations that have been made since the theory was first introduced in 1981 can be interpreted as actually supporting the theory, because any other result would fit the theory just as well.[11] This is significant, because the ability to disprove a theory through experiment has always been considered the key to distinguishing science from non-science.[12]

In a widely discussed pair of papers published in 2013 and 2014, Anna Ijjas, Abraham Loeb and Steinhardt showed that observations from the Planck satellite reported in 2013 disfavor the simplest textbook inflationary models.[13][14] These papers showed that the remaining models require more parameters, more fine-tuning of those parameters, and more unlikely initial conditions than the simplest models. This predicament for inflationary cosmology is often referred to as "the unlikeliness problem." These conclusions were reaffirmed and strengthened by measurements subsequently reported by the Planck satellite team in 2015.

Cyclic/ekpyrotic theory of the universe

Early development

Steinhardt has been a leading developer of cyclic and ekpyrotic cosmology.

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.[15] 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.[11][16]

Steinhardt and Turok then incorporated the ekpyrotic idea into a bolder proposal: the cyclic theory of the universe.[17]

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.

Notable features

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.[11] 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.

Predictions

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.[18] 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.)

Two predictions that apply to all cyclic/ekpyrotic models: (1) unlike inflation, no detectable gravitational waves are generated during the smoothing and flattening process; and, (2) the current acceleration expansion must eventually stop and the vacuum must be eventually decay in order to initiate the next cycle.

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.[19] 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.[20] 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

Steinhardt has made significant contributions researching the "dark side" of the universe: dark energy, the cosmological constant problem 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.[21] This was verified three years later by supernova observations in 1998.[22][23][24]

Working with colleagues, he subsequently introduced the concept of quintessence, a term used to describe a form of dark energy that varies with time.[25] 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[26] and k-essence[27] 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.[19]

In 2000, David Spergel and Steinhardt first introduced the concept of strongly self-interacting dark matter (SIDM) to explain various anomalies in standard cold dark models.[28]

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.[29]

Amorphous solids

Steinhardt's research in condensed matter has centered on the properties of non-periodic solids: glasses, amorphous semiconductors, metals and quasicrystals.

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.[30][31]

Working with David Nelson and Marco Ronchetti, Steinhard 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.[32]

Quasicrystals

1983: theoretical development

Motivated by the work on amorphous solids (described in the section above), Steinhardt and Dov Levine first introduced the theoretical concept of quasicrystals in a patent disclosure in 1983,[33] a year before any other experimental evidence had been announced. They published their complete theory in December 1984,[34] 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-dimensonal 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.[35] 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.[34] 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.

Experimental realization

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.[36] 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.[37]

Other contributions to the field

Girih tile quasicrystal pattern on right half of spandrel at Darb-e Imam Shrine

With collaborators, Steinhardt has also made significant contributions to understanding the quasicrystals’ unique mathematical and physical properties,[38] including theories of how and why quasicrystals form[39] and their elastic and hydrodynamics properties,[40] laying the groundwork for many current applications.

Peter Lu and Steinhardt discovered a quasicrystalline Islamic tiling on the Darb-e Imam Shrine (1453 A.D.) in Isfahan, Iran constructed from girih tiles. [41] 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.[42]

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.[43] 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 100 microns sized 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.[43] 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.[36]

Two years after identifying the museum sample, Steinhardt organized an international team of experts and led them on an expedition to its source, a remote stream in the Chukotka Autonomous Okrug in the northern half of the Kamchatka peninsula in far eastern Russia. There the team discovered more samples of icosahedrite and proved that it originated from a meteorite formed 4.5 billion years ago (before there were planets) and landed on the Earth about 15,000 years ago. After returning home with those samples, more samples of icosahedrite were discovered in the laboratory, 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."[44] 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).[45]

Photonics

Steinhardt’s research on quasicrystals and other non-crystalline solids expanded into work on designer materials with novel photonic and phononic properties.

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").[46] 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, Steinhardt discovered a new class of photonic materials called hyperuniform disordered solids (HUDS) in 2009, and showed that solids consisting of a hyperuniform disordered arrangement of dielectric elements produce band gaps with perfect spherical symmetry.[47][48] 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.

As a result of these discoveries, a start up company was founded in 2012 to develop the scientific discoveries into a range of high performance applications, including novel photonic integrated circuits for use in computer data centers, sensors, improved photovoltaics and displays.[49]

Honors and awards

References

  1. ↑ http://www.physics.princeton.edu/~steinh/webbrief/
  2. ↑ http://pcts.princeton.edu/pcts/faculty_fellows.html
  3. ↑ 3.0 3.1 Albrecht, A.; Steinhardt, P. J. (1982). "Cosmology For Grand Unified Theories With Radiatively Induced Symmetry Breaking". Phys. Rev. Lett. 48 (17): 1220–1223. Bibcode:1982PhRvL..48.1220A. doi:10.1103/PhysRevLett.48.1220.
  4. ↑ Albrecht, A.; Steinhardt, P.J. (1982). "Reheating an Inflationary Universe". Phys.Rev.Lett 48 (20): 1437–1440. Bibcode:1982PhRvL..48.1437. doi:10.1103/PhysRevLett.48.1437.
  5. ↑ 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. doi:10.1103/PhysRevD.28.679.
  6. ↑ Gibbons, Gary W.; Hawking, Stephen W.; Siklos, S.T.C., eds. (1983). The Very early universe: proceedings of the Nuffield workshop, Cambridge, 21 June to 9 July, 1982. Cambridge University Press. ISBN 0-521-31677-4.
  7. ↑ La, D.; Steinhardt, P. J. (1989). "Extended Inflationary Cosmology". Phys. Rev. Lett. 62 (4): 376–378. Bibcode:1989PhRvL..62.376. doi:10.1103/PhysRevLett.62.376.
  8. ↑ Crittenden, R.; Bond, J.R.; Davis, R.L.; Efstathiou, G.E.; Steinhardt, P. J. (1989). "The Imprint of Gravitational Waves on the Cosmic Microwave Background". Phys. Rev. Lett. 71 (3): 324–327. Bibcode:1993PhRvL..71.324. doi:10.1103/PhysRevLett.71.324.
  9. ↑ Crittenden; Davis, R.L.; Steinhardt, P. J. (1989). "Polarization of the Microwave Background due to Primordial Gravitational Waves" (PDF). Astrophysical Journal Letters 417: L13. Bibcode:1993ApJ…417L..13C. doi:10.1086/187082.
  10. ↑ 10.0 10.1 https://www.ictp.it/about-ictp/prizes-awards/the-dirac-medal/the-medallists-%281%29.aspx
  11. ↑ 11.0 11.1 11.2 Steinhardt, Paul J. (April 2011). "Inflation Debate: Is the theory at the heart of modern cosmology deeply flawed?" (PDF). Scientific American.
  12. ↑ "Feynman on Scientific Method". YouTube. Retrieved July 28, 2012.
  13. ↑ Iijas, Anna; Loeb, Abraham; Steinhardt, Paul (2013). "Inflationary Paradigm in trouble after Planck 2013". Phys. Lett.B. 723 (4-5): 261–266. doi:10.1016/j.physletb.2013.05.023.
  14. ↑ Iijas, Anna; Steinhardt, Paul J.; Loeb, Abraham (2014). "Inflationary Schism after Planck 2013". Phys. Lett.B. 7: 142–146. doi:10.1016/j.physletb.2014.07.012.
  15. ↑ Khoury, J.; Ovrut, B.; Steinhardt, P. J. (2001). "The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang". Phys. Rev.D 64 (12): 123522. doi:10.1103/PhysRevD.64.123522.
  16. ↑ http://www.physics.princeton.edu/~steinh/vaasrev.pdf
  17. ↑ Steinhardt, P. J.; Turok, N. (2002-04-25). "A Cyclic Model of the Universe". Science 296 (5572): 1436–1439. arXiv:hep-th/0111030v2. Bibcode:2002Sci...296.1436S. doi:10.1126/science.1070462. PMID 11976408.
  18. ↑ Steinhardt, P.J.; Turok, N.G. (2007). Endless Universe: beyond the Big Bang. Doubleday. ISBN 9780385509640.
  19. ↑ 19.0 19.1 Steinhardt, P. J.; Turok, N. (2006). "Why the cosmological constant is small and positive". Science 312 (5777): 1180–1182. arXiv:astro-ph/0605173v1. doi:10.1126/science.1126231.
  20. ↑ Bars, I.; Steinhardt, P.J.; Turok, N (2013). "Cyclid cosmology, conformal symmetry and the metastability of the Higgs". Phys. Lett.B.2013.08.071 726 (1-3): 50–55. doi:10.1016/j.physletb.2013.08.071.
  21. ↑ Ostriker, J. P.; Steinhardt, P.J. (1995). "The observational case for a low-density Universe with a non-zero cosmological constant". Nature 377: 600–602. doi:10.1038/377600a0.
  22. ↑ Riess, A.; al., et (1989). "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant". The Astronomical Journal 116 (3). Bibcode:1998AJ….116.1009R. doi:10.1086/300499.
  23. ↑ Perlmutter, S.; al., et (1999). "Measurements of Omega and Lamba from 42 High-Redshift Supernovae". The Astrophysical Journal 517 (2). doi:10.1086/307221.
  24. ↑ Bahcall, N.A.; Ostriker, J.P.; Perlmutter, S.; Steinhardt, P.J. (1999). "The Cosmic Triangle: Revealing the State of the Universe". Science 284: 1481–1488. doi:10.1126/science.284.5419.1481.
  25. ↑ Caldwell, R.R.; Dave, R.; Steinhardt, P.J. (1998). "Cosmological Imprint of an Energy Component with General Equation-of-State". Phys.Rev.Lett. 80 (8): 1582–1585. doi:10.1103/PhysRevLett.80.1582.
  26. ↑ Steinhardt, P.J.; Wang, L.; Zlatev, I (1999). "Cosmological Tracking Solutions". Phys.Rev.D 59 (12). doi:10.1103/PhysRevD.59.123504.
  27. ↑ Amendariz-Picon, C.; Mukhanov, V.; Steinhardt, P.J. (2000). "A Dynamical Solution to the Problem of a Small Cosmological Constant and Late-tome Cosmic Acceleration". Phys.Rev.Lett. 85 (21): 4438–4441. doi:10.1103/PhysRevLett.85.4438.
  28. ↑ Spergel, D.N.; Steinhardt, P.J. (2000). "Observational Evidence for Self-Interacting Cold Dark Matter". Phys.Rev.Lett. 84 (17): 3760–3763. doi:10.1103/PhysRevLett.84.3760.
  29. ↑ Pollack, Jason; Spergel, David N.; Steinhardt, Paul J. (2014). "Supermassive Black Holes from Ultra-Strongly Self-Interacting Dark Matter".
  30. ↑ Steinhardt, P. J.; Alben, R.; Duffy, M. G.; Polk, D. E. (1973). "Relaxed Continuous Random Network Models". Phys.Rev.B. 8 (12): 6021–6023.
  31. ↑ Alben, R.; Weaire, D.; Steinhardt, P.J. (1973). "One Band Density of States for the Polk Model". Journal of Physics 6 (20). doi:10.1088/0022-3719/6/20/003.
  32. ↑ Steinhardt, P. J.; Nelson, D.; Ronchetti, M. (1981). "Icosahedral Bond Orientational Order in Supercooled Liquids". Phys.Rev.Lett. 47 (18): 1297–1300.
  33. ↑ Steinhardt, P. J. (2013). "Quasicrystals: a brief history of the impossible". doi:10.1007/s12210-012-0203-3.
  34. ↑ 34.0 34.1 Levine, D.; Steinhardt, P.J. (1984). "Quasicrystals: A New Class of Ordered Structures" (PDF). Phys.Rev.Lett 53 (26): 2477–2480. doi:10.1103/PhysRevLett.53.2477.
  35. ↑ Shechtman, D.; Blech, D.; Gratias; Cahn, J.W. (1984). "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry". Phys.Rev.Lett 53 (20): 1951–1953. doi:10.1103/PhysRevLett.53.1951.
  36. ↑ 36.0 36.1 Tsai, An-Pang; Inoue, Tsuyoshi; Masumoto (1987). "A Stable Quasicrystal in Al-Cu-Fe System". Japanese Journal of Applied Physics 26 (Part 2, Number 9). doi:10.1143/JJAP.26.L1505.
  37. ↑ Onoda, G.; Steinhardt, P.J.; DiVincenzo, D.; Socolar, J. (1988). "Growing Perfect Quasicrystals". Phys.Rev.Lett. 60 (25): 2653–2656. doi:10.1103/PhysRevLett.60.2653.
  38. ↑ Socolar, J.; Steinhardt, P.J. (1986). "Quasicrystals II: Unit Cell Configurations" (PDF). Phys.Rev.B. 34 (2): 617–647. doi:10.1103/PhysRevB.34.617.
  39. ↑ Jeong, H.C.; Steinhardt, P.J. (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature 382: 431–433. doi:10.1038/382431a0.
  40. ↑ Levine, D.; Lubensky, T.; Ostlund, S.; Ramaswamy, S.; Steinhardt, P.J.; Toner, J. (1985). "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals". Phys.Rev.Lett. 54 (14): 1520–1523. doi:10.1103/PhysRevLett.54.1520.
  41. ↑ Lu, P.; Steinhardt, P.J. (2007). "Decagonal and Quasicrystalline Tilings in Medieval Islamic Architecture". Science 315 (5815): 1106–1110. doi:10.1126/science.1135491.
  42. ↑ Lu, P.; Deffreyes, K.; Steinhardt, P.J.; Yao (2001). "Identifying and Indexing Icosahedral Quasicrystals from Powder Diffraction Patterns". Phys.Rev.Lett. 87 (27): 275507. doi:10.1103/PhysRevLett.87.275507.
  43. ↑ 43.0 43.1 Bindi, L.; Steinhardt, P.J.; Yao, N.; Lu, P. (2009). "Natural Quasicrystals". Science 324 (5932): 1306–1309. doi:10.1126/science.1170827.
  44. ↑ 44.0 44.1 http://wwwphy.princeton.edu/~steinh/STEINHARDTITE%20LETTER.pdf
  45. ↑ Bindi, Luca; Yao, Nan; Lin, Chaney; Hollister, Lincoln S.; Andronicos, Christopher L.; Distler, Vadim V.; Eddy, Michael P.; Kostin, Alexander; Kryachko, Valery; MacPherson, Glenn J.; Steinhardt, William M.; Yudovskaya, Marina; Steinhardt, Paul J. (2015). "Natural quasicrystal with decagonal symmetry". Scientific Reports 5. doi:10.1038/srep09111.
  46. ↑ Man, W.; Megens, M.; Steinhardt, P.J.; Chaikin, P. (2005). "Experimental Measurement of the Photonic Properties of Icosahedral Quasicrystals". Nature 436: 993–996. doi:10.1038/nature03977.
  47. ↑ Florescu, M.; Torquato, S.; Steinhardt, Paul J. (2009). "Designer disordered materials with large, complete photonic band gaps". Proceedings of the National Academy of Sciences 106 (49). doi:10.1073/pnas.0907744106.
  48. ↑ Man, W.; al., et (2013). "Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids". Proceedings of the National Academy of Sciences 110 (40): 15886–15891. doi:10.1073/pnas.1307879110.
  49. ↑ http://cherrypit.princeton.edu/photonics-story-April-2013.pdf
  50. ↑ http://www.aps.org/programs/honors/fellowships/archive-all.cfm?initial=S&year=&unit_id=&institution=Princeton+University
  51. ↑ http://www.nasonline.org/member-directory/members/3005837.html
  52. ↑ http://www.aps.org/programs/honors/prizes/prizerecipient.cfm?first_nm=Paul&last_nm=Steinhardt&year=2010
  53. ↑ http://www.garfield.library.upenn.edu/johnscottaward/js2011-2020.html
  54. ↑ https://www.simonsfoundation.org/funding/funding-opportunities/mathematics-physical-sciences/simons-fellow-program/simons-fellows-awardees-theoretical-physics/2012-simons-fellows-awardees-theoretical-physics/
  55. ↑ https://www.radcliffe.harvard.edu/people/paul-steinhardt
  56. ↑ http://www.alumni.caltech.edu/distinguished-alumni-awards-2014/