Cosmological principle

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Unsolved problem in physics:

Is the universe homogeneous and isotropic at large enough scales, as claimed by the cosmological principle and assumed by all models that use the Friedmann–Lemaître–Robertson–Walker metric, including the current version of the ΛCDM model, or is the universe inhomogeneous or anisotropic?[1][2][3]

In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is uniformly isotropic and homogeneous when viewed on a large enough scale, since the forces are expected to act equally throughout the universe on a large scale, and should, therefore, produce no observable inequalities in the large-scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.


Astronomer William Keel explains:

The cosmological principle is usually stated formally as 'Viewed on a sufficiently large scale, the properties of the universe are the same for all observers.' This amounts to the strongly philosophical statement that the part of the universe which we can see is a fair sample, and that the same physical laws apply throughout. In essence, this in a sense says that the universe is knowable and is playing fair with scientists.[4]

The cosmological principle depends on a definition of "observer", and contains an implicit qualification and two testable consequences.

"Observers" means any observer at any location in the universe, not simply any human observer at any location on Earth: as Andrew Liddle puts it, "the cosmological principle [means that] the universe looks the same whoever and wherever you are."[5]

The qualification is that variation in physical structures can be overlooked, provided this does not imperil the uniformity of conclusions drawn from observation: the Sun is different from the Earth, our galaxy is different from a black hole, some galaxies advance toward rather than recede from us, and the universe has a "foamy" texture of galaxy clusters and voids, but none of these different structures appears to violate the basic laws of physics.

The two testable structural consequences of the cosmological principle are homogeneity and isotropy. Homogeneity means that the same observational evidence is available to observers at different locations in the universe ("the part of the universe which we can see is a fair sample"). Isotropy means that the same observational evidence is available by looking in any direction in the universe ("the same physical laws apply throughout").[dubious ] The principles are distinct but closely related, because a universe that appears isotropic from any two (for a spherical geometry, three) locations must also be homogeneous.


The cosmological principle is first clearly asserted in the Philosophiæ Naturalis Principia Mathematica (1687) of Isaac Newton.[dubious ] In contrast to some earlier classical or medieval cosmologies, in which Earth rested at the center of universe, Newton conceptualized the Earth as a sphere in orbital motion around the Sun within an empty space that extended uniformly in all directions to immeasurably large distances. He then showed, through a series of mathematical proofs on detailed observational data of the motions of planets and comets, that their motions could be explained by a single principle of "universal gravitation" that applied as well to the orbits of the Galilean moons around Jupiter, the Moon around the Earth, the Earth around the Sun, and to falling bodies on Earth. That is, he asserted the equivalent material nature of all bodies within the Solar System, the identical nature of the Sun and distant stars and thus the uniform extension of the physical laws of motion to a great distance beyond the observational location of Earth itself.


Since the 1990s, observations assuming the cosmological principle have concluded that around 68% of the mass–energy density of the universe can be attributed to dark energy, which led to the development of the ΛCDM model.[6][7][8]

Observations show that more distant galaxies are closer together and have lower content of chemical elements heavier than lithium.[citation needed] Applying the cosmological principle, this suggests that heavier elements were not created in the Big Bang but were produced by nucleosynthesis in giant stars and expelled across a series of supernovae explosions and new star formation from the supernovae remnants, which means heavier elements would accumulate over time. Another observation is that the furthest galaxies (earlier time) are often more fragmentary, interacting and unusually shaped than local galaxies (recent time), suggesting evolution in galaxy structure as well.

A related implication of the cosmological principle is that the largest discrete structures in the universe are in mechanical equilibrium. Homogeneity and isotropy of matter at the largest scales would suggest that the largest discrete structures are parts of a single indiscrete form, like the crumbs which make up the interior of a cake. At extreme cosmological distances, the property of mechanical equilibrium in surfaces lateral to the line of sight can be empirically tested; however, under the assumption of the cosmological principle, it cannot be detected parallel to the line of sight (see timeline of the universe).

Cosmologists agree that in accordance with observations of distant galaxies, a universe must be non-static if it follows the cosmological principle. In 1923, Alexander Friedmann set out a variant of Albert Einstein's equations of general relativity that describe the dynamics of a homogeneous isotropic universe.[9][10] Independently, Georges Lemaître derived in 1927 the equations of an expanding universe from the General Relativity equations.[11] Thus, a non-static universe is also implied, independent of observations of distant galaxies, as the result of applying the cosmological principle to general relativity.


Karl Popper criticized the cosmological principle on the grounds that it makes "our lack of knowledge a principle of knowing something". He summarized his position as:

the "cosmological principles" were, I fear, dogmas that should not have been proposed.[12]


Although the universe is inhomogeneous at smaller scales, according to the ΛCDM model it ought to be isotropic and statistically homogeneous on scales larger than 250 million light years. However, recent findings (the Axis of Evil for example) have suggested that violations of the cosmological principle exist in the universe and thus have called the ΛCDM model into question, with some authors suggesting that the cosmological principle is now obsolete and the Friedmann–Lemaître–Robertson–Walker metric breaks down in the late universe.[1]

Violations of isotropy[edit]

The cosmic microwave background (CMB) is predicted by the ΛCDM model to be isotropic, that is to say that its intensity is about the same whichever direction we look at.[13] Data from the Planck Mission shows hemispheric bias in 2 respects: one with respect to average temperature (i.e. temperature fluctuations), the second with respect to larger variations in the degree of perturbations (i.e. densities),[14][15] the collaboration noted that these features are not strongly statistically inconsistent with isotropy.[16] Some authors say that the universe around Earth is isotropic at high significance by studies of the cosmic microwave background temperature maps.[17] There are however claims of isotropy violations from galaxy clusters,[2][3] quasars,[18] and type Ia supernovae.[19]

Violations of homogeneity[edit]

The cosmological principle implies that at a sufficiently large scale, the universe is homogeneous. Based on N-body simulations in a ΛCDM universe, Yadav and his colleagues showed that the spatial distribution of galaxies is statistically homogeneous if averaged over scales of 260/h Mpc or more.[20]

A number of observations have been reported to be in conflict with predictions of maximal structure sizes:

  • The Clowes–Campusano LQG, discovered in 1991, has a length of 580 Mpc, and is marginally larger than the consistent scale.
  • The Sloan Great Wall, discovered in 2003, has a length of 423 Mpc,[21] which is only just consistent with the cosmological principle.
  • U1.11, a large quasar group discovered in 2011, has a length of 780 Mpc, and is two times larger than the upper limit of the homogeneity scale.
  • The Huge-LQG, discovered in 2012, is three times longer than, and twice as wide as is predicted possible according to these current models, and so challenges our understanding of the universe on large scales.
  • In November 2013, a new structure 10 billion light years away measuring 2000–3000 Mpc (more than seven times that of the Sloan Great Wall) was discovered, the Hercules–Corona Borealis Great Wall, putting further doubt on the validity of the cosmological principle.[22]
  • In September 2020, a 4.9σ conflict was found between the kinematic explanation of the CMB dipole and the measurement of the dipole in the angular distribution of a flux-limited, all-sky sample of 1.36 million quasars.[23]
  • In June 2021, the Giant Arc was discovered, a structure spanning approximately 1000 Mpc.[24] It is located 2820 Mpc away and consists of galaxies, galactic clusters, gas, and dust.
  • In January 2024, the Big Ring was discovered. It is located 9.2 billion light years away from Earth has a diameter of 1.3 billion light years or around the size of 15 full Moons as seen from Earth.[25]

However, as pointed out by Seshadri Nadathur in 2013 using statistical properties,[26] the existence of structures larger than the homogeneous scale (260/h Mpc by Yadav's estimation)[20] does not necessarily violate the cosmological principle in the ΛCDM model (see Huge-LQG § Dispute).[27]

CMB dipole[edit]

Unsolved problem in physics:

Is the CMB dipole purely kinematic, or does it signal anisotropy of the universe, resulting in the breakdown of the FLRW metric and the cosmological principle?[1]

The cosmic microwave background (CMB) provides a snapshot of a largely isotropic and homogeneous universe. The largest scale feature of the CMB is the dipole anisotropy; it is typically subtracted from maps due to its large amplitude. The standard interpretation of the dipole is that it is due to the Doppler effect caused by the motion of the solar system with respect to the CMB rest-frame.

Several studies have reported dipoles in the large scale distribution of galaxies that align with the CMB dipole direction, but indicate a larger amplitude than would be caused by the CMB dipole velocity.[28] A similar dipole is seen in data of radio galaxies, however the amplitude of the dipole depends on the observing frequency showing that these anomalous features cannot be purely kinematic.[29] Other authors have found radio dipoles consistent with the CMB expectation.[30] Further claims of anisotropy along the CMB dipole axis have been made with respect to the Hubble diagram of type Ia supernovae[31] and quasars.[32] Separately, the CMB dipole direction has emerged as a preferred direction in some studies of alignments in quasar polarizations,[33] strong lensing time delay,[34] type Ia supernovae,[35] and standard candles.[36] Some authors have argued that the correlation of distant effects with the dipole direction may indicate that its origin is not kinematic.

Alternatively, Planck data has been used to estimate the velocity with respect to the CMB independently of the dipole, by measuring the subtle aberrations and distortions of fluctuations caused by relativistic beaming[37] and separately using the Sunyaev-Zeldovich effect.[38] These studies found a velocity consistent with the value obtained from the dipole, indicating it is consistent with being entirely kinematic. Measurements of the velocity field of galaxies in the local universe show that on short scales galaxies are moving with the local group, and that the average mean velocity decreases with increasing distance.[39] This follows the expectation if the CMB dipole were due to the local peculiar velocity field, it becomes more homogeneous on large scales. Surveys of the local volume have been used to reveal a low density region in the opposite direction to the CMB dipole,[40] potentially explaining the origin of the local bulk flow.

Perfect cosmological principle[edit]

The perfect cosmological principle is an extension of the cosmological principle, and states that the universe is homogeneous and isotropic in space and time. In this view the universe looks the same everywhere (on the large scale), the same as it always has and always will. The perfect cosmological principle underpins Steady State theory and emerges[clarification needed] from chaotic inflation theory.[41][42][43]

See also[edit]


  1. ^ a b c Abdalla, Elcio; Abellán, Guillermo Franco; Aboubrahim, Armin (11 Mar 2022), "Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies", Journal of High Energy Astrophysics, 34: 49, arXiv:2203.06142v1, Bibcode:2022JHEAp..34...49A, doi:10.1016/j.jheap.2022.04.002, S2CID 247411131
  2. ^ a b Billings, Lee (April 15, 2020). "Do We Live in a Lopsided Universe?". Scientific American. Retrieved March 24, 2022.
  3. ^ a b Migkas, K.; Schellenberger, G.; Reiprich, T. H.; Pacaud, F.; Ramos-Ceja, M. E.; Lovisari, L. (8 April 2020). "Probing cosmic isotropy with a new X-ray galaxy cluster sample through the LX-T scaling relation". Astronomy & Astrophysics. 636 (April 2020): 42. arXiv:2004.03305. Bibcode:2020A&A...636A..15M. doi:10.1051/0004-6361/201936602. S2CID 215238834. Retrieved 24 March 2022.
  4. ^ Keel, William C. (2007). The Road to Galaxy Formation (2nd ed.). Springer-Praxis. p. 2. ISBN 978-3-540-72534-3.
  5. ^ Liddle, Andrew (2003). An Introduction to Modern Cosmology (2nd ed.). John Wiley & Sons. p. 2. ISBN 978-0-470-84835-7.
  6. ^ Ellis, G. F. R. (2009). "Dark energy and inhomogeneity". Journal of Physics: Conference Series. 189 (1): 012011. Bibcode:2009JPhCS.189a2011E. doi:10.1088/1742-6596/189/1/012011. S2CID 250670331.
  7. ^ Colin, Jacques; Mohayaee, Roya; Rameez, Mohamed; Sarkar, Subir (20 November 2019). "Evidence for anisotropy of cosmic acceleration". Astronomy and Astrophysics. 631: L13. arXiv:1808.04597. Bibcode:2019A&A...631L..13C. doi:10.1051/0004-6361/201936373. S2CID 208175643. Retrieved 25 March 2022.
  8. ^ Redd, N. T. (2013). "What is Dark Energy?". Archived from the original on 19 May 2016. Retrieved 28 October 2018.
  9. ^ Alexander Friedmann (1923). Die Welt als Raum und Zeit (The World as Space and Time). Ostwalds Klassiker der exakten Wissenschaften. ISBN 978-3-8171-3287-4. OCLC 248202523..
  10. ^ Tropp, Ėduard Abramovich; Frenkel, Viktor Ya.; Chernin, Artur Davidovich (1993). Alexander A. Friedmann: The Man who Made the Universe Expand. Cambridge University Press. p. 219. ISBN 978-0-521-38470-4.
  11. ^ Lemaître, Georges (1927). "Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques". Annales de la Société Scientifique de Bruxelles. A47 (5): 49–56. Bibcode:1927ASSB...47...49L. translated by A. S. Eddington: Lemaître, Georges (1931). "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ". Monthly Notices of the Royal Astronomical Society. 91 (5): 483–490. Bibcode:1931MNRAS..91..483L. doi:10.1093/mnras/91.5.483.
  12. ^ Helge Kragh: "The most philosophically of all the sciences": Karl Popper and physical cosmology Archived 2013-07-20 at the Wayback Machine (2012)
  13. ^ "Australian study backs major assumption of cosmology". 17 September 2012.
  14. ^ "Simple but challenging: the Universe according to Planck". ESA Science & Technology. October 5, 2016 [March 21, 2013]. Retrieved October 29, 2016.
  15. ^ Planck Collaboration; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B.; Bartolo, N.; Basak, S.; Benabed, K.; Bersanelli, M.; Bielewicz, P.; Bock, J. J.; Bond, J. R. (2020-09-01). "Planck 2018 results. VII. Isotropy and statistics of the CMB". Astronomy and Astrophysics. 641: A7. arXiv:1906.02552. Bibcode:2020A&A...641A...7P. doi:10.1051/0004-6361/201935201. hdl:10138/320318. ISSN 0004-6361.
  16. ^ Planck Collaboration; Aghanim, N.; Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B.; Bartolo, N.; Basak, S.; Battye, R.; Benabed, K.; Bernard, J. -P. (2020-09-01). "Planck 2018 results. I. Overview and the cosmological legacy of Planck". Astronomy and Astrophysics. 641: A1. arXiv:1807.06205. Bibcode:2020A&A...641A...1P. doi:10.1051/0004-6361/201833880. hdl:10138/320876. ISSN 0004-6361. S2CID 119185252.
  17. ^ Saadeh D, Feeney SM, Pontzen A, Peiris HV, McEwen, JD (2016). "How Isotropic is the Universe?". Physical Review Letters. 117 (13): 131302. arXiv:1605.07178. Bibcode:2016PhRvL.117m1302S. doi:10.1103/PhysRevLett.117.131302. PMID 27715088. S2CID 453412.
  18. ^ Secrest, Nathan J.; von Hausegger, Sebastian; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (February 25, 2021). "A Test of the Cosmological Principle with Quasars". The Astrophysical Journal Letters. 908 (2): L51. arXiv:2009.14826. Bibcode:2021ApJ...908L..51S. doi:10.3847/2041-8213/abdd40. S2CID 222066749.
  19. ^ Javanmardi, B.; Porciani, C.; Kroupa, P.; Pflamm-Altenburg, J. (August 27, 2015). "Probing the Isotropy of Cosmic Acceleration Traced By Type Ia Supernovae". The Astrophysical Journal Letters. 810 (1): 47. arXiv:1507.07560. Bibcode:2015ApJ...810...47J. doi:10.1088/0004-637X/810/1/47. S2CID 54958680. Retrieved March 24, 2022.
  20. ^ a b Yadav, Jaswant; Bagla, J. S.; Khandai, Nishikanta (25 February 2010). "Fractal dimension as a measure of the scale of homogeneity". Monthly Notices of the Royal Astronomical Society. 405 (3): 2009–2015. arXiv:1001.0617. Bibcode:2010MNRAS.405.2009Y. doi:10.1111/j.1365-2966.2010.16612.x. S2CID 118603499.
  21. ^ Gott, J. Richard III; et al. (May 2005). "A Map of the Universe". The Astrophysical Journal. 624 (2): 463–484. arXiv:astro-ph/0310571. Bibcode:2005ApJ...624..463G. doi:10.1086/428890. S2CID 9654355.
  22. ^ Horvath, I.; Hakkila, J.; Bagoly, Z. (2013). "The largest structure of the Universe, defined by Gamma-Ray Bursts". arXiv:1311.1104. {{cite journal}}: Cite journal requires |journal= (help)
  23. ^ Secrest, Nathan; von Hausegger, Sebastian; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (2021-02-01). "A Test of the Cosmological Principle with Quasars". The Astrophysical Journal Letters. 908 (2): L51. arXiv:2009.14826. Bibcode:2021ApJ...908L..51S. doi:10.3847/2041-8213/abdd40. ISSN 2041-8205. S2CID 222066749.
  24. ^ "Line of galaxies is so big it breaks our understanding of the universe".
  25. ^ "A Big Cosmological Mystery". University of Central Lancashire. Retrieved 2024-01-15.
  26. ^ Nadathur, Seshadri (2013). "Seeing patterns in noise: gigaparsec-scale 'structures' that do not violate homogeneity". Monthly Notices of the Royal Astronomical Society. 434 (1): 398–406. arXiv:1306.1700. Bibcode:2013MNRAS.434..398N. doi:10.1093/mnras/stt1028. S2CID 119220579.
  27. ^ Sylos-Labini F, Tekhanovich D, Baryshev Y (2014). "Spatial density fluctuations and selection effects in galaxy redshift surveys". Journal of Cosmology and Astroparticle Physics. 7 (13): 35. arXiv:1406.5899. Bibcode:2014JCAP...07..035S. doi:10.1088/1475-7516/2014/07/035. S2CID 118393719.
  28. ^ Secrest, Nathan; von Hausegger, Sebastian; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (25 February 2021). "A Test of the Cosmological Principle with Quasars". The Astrophysical Journal. 908 (2): L51. arXiv:2009.14826. Bibcode:2021ApJ...908L..51S. doi:10.3847/2041-8213/abdd40. ISSN 2041-8213. S2CID 222066749.
  29. ^ Siewert, Thilo M.; Schmidt-Rubart, Matthias; Schwarz, Dominik J. (2021). "Cosmic radio dipole: Estimators and frequency dependence". Astronomy & Astrophysics. 653: A9. arXiv:2010.08366. Bibcode:2021A&A...653A...9S. doi:10.1051/0004-6361/202039840. S2CID 223953708.
  30. ^ Darling, Jeremy (2022-06-01). "The Universe is Brighter in the Direction of Our Motion: Galaxy Counts and Fluxes are Consistent with the CMB Dipole". The Astrophysical Journal. 931 (2): L14. arXiv:2205.06880. Bibcode:2022ApJ...931L..14D. doi:10.3847/2041-8213/ac6f08. ISSN 0004-637X.
  31. ^ Singal, Ashok K. (2022). "Peculiar motion of Solar system from the Hubble diagram of supernovae Ia and its implications for cosmology". Monthly Notices of the Royal Astronomical Society. 515 (4): 5969–5980. arXiv:2106.11968. doi:10.1093/mnras/stac1986.
  32. ^ Singal, Ashok K. (2022). "Solar system peculiar motion from the Hubble diagram of quasars and testing the cosmological principle". Monthly Notices of the Royal Astronomical Society. 511 (2): 1819–1829. arXiv:2107.09390. doi:10.1093/mnras/stac144.
  33. ^ Hutsemekers, D.; Cabanac, R.; Lamy, H.; Sluse, D. (October 2005). "Mapping extreme-scale alignments of quasar polarization vectors". Astronomy & Astrophysics. 441 (3): 915–930. arXiv:astro-ph/0507274. Bibcode:2005A&A...441..915H. doi:10.1051/0004-6361:20053337. ISSN 0004-6361. S2CID 14626666.
  34. ^ Krishnan, Chethan; Mohayaee, Roya; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (16 September 2021). "Does Hubble Tension Signal a Breakdown in FLRW Cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv:2105.09790. Bibcode:2021CQGra..38r4001K. doi:10.1088/1361-6382/ac1a81. ISSN 0264-9381. S2CID 234790314.
  35. ^ Krishnan, Chethan; Mohayaee, Roya; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (2022). "Hints of FLRW breakdown from supernovae". Physical Review D. 105 (6): 063514. arXiv:2106.02532. Bibcode:2022PhRvD.105f3514K. doi:10.1103/PhysRevD.105.063514. S2CID 235352881.
  36. ^ Luongo, Orlando; Muccino, Marco; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (2022). "Larger H0 values in the CMB dipole direction". Physical Review D. 105 (10): 103510. arXiv:2108.13228. Bibcode:2022PhRvD.105j3510L. doi:10.1103/PhysRevD.105.103510. S2CID 248713777.
  37. ^ Planck Collaboration; Aghanim, N.; Armitage-Caplan, C.; Arnaud, M.; Ashdown, M.; Atrio-Barandela, F.; Aumont, J.; Baccigalupi, C.; Banday, A. J.; Barreiro, R. B.; Bartlett, J. G.; Benabed, K.; Benoit-Lévy, A.; Bernard, J. -P.; Bersanelli, M. (2014-11-01). "Planck 2013 results. XXVII. Doppler boosting of the CMB: Eppur si muove". Astronomy and Astrophysics. 571: A27. arXiv:1303.5087. Bibcode:2014A&A...571A..27P. doi:10.1051/0004-6361/201321556. hdl:10138/233688. ISSN 0004-6361. S2CID 5398329.
  38. ^ Planck Collaboration; Akrami, Y.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A. J.; Barreiro, R. B.; Bartolo, N.; Basak, S.; Benabed, K.; Bernard, J. -P.; Bersanelli, M.; Bielewicz, P.; Bond, J. R. (2020-12-01). "Planck intermediate results. LVI. Detection of the CMB dipole through modulation of the thermal Sunyaev-Zeldovich effect: Eppur si muove II". Astronomy and Astrophysics. 644: A100. arXiv:2003.12646. Bibcode:2020A&A...644A.100P. doi:10.1051/0004-6361/202038053. hdl:10138/324269. ISSN 0004-6361. S2CID 214713774.
  39. ^ Avila, Felipe; Oliveira, Jezebel; Dias, Mariana L. S.; Bernui, Armando (2023-02-01). "The bulk flow motion and the Hubble-Lemaître law in the Local Universe with the ALFALFA survey". Brazilian Journal of Physics. 53 (2): 49. arXiv:2302.04978. Bibcode:2023BrJPh..53...49A. doi:10.1007/s13538-023-01259-z. ISSN 0103-9733. S2CID 256631872.
  40. ^ Hoffman, Yehuda; Pomarède, Daniel; Tully, R. Brent; Courtois, Hélène M. (2017-01-01). "The dipole repeller". Nature Astronomy. 1 (2): 0036. arXiv:1702.02483. Bibcode:2017NatAs...1E..36H. doi:10.1038/s41550-016-0036. ISSN 2397-3366. S2CID 7537393.
  41. ^ Aguirre, Anthony & Gratton, Steven (2003). "Inflation without a beginning: A null boundary proposal". Physical Review D. 67 (8): 083515. arXiv:gr-qc/0301042. Bibcode:2003PhRvD..67h3515A. doi:10.1103/PhysRevD.67.083515. S2CID 37260723.
  42. ^ Aguirre, Anthony & Gratton, Steven (2002). "Steady-State Eternal Inflation". Physical Review D. 65 (8): 083507. arXiv:astro-ph/0111191. Bibcode:2002PhRvD..65h3507A. doi:10.1103/PhysRevD.65.083507. S2CID 118974302.
  43. ^ Gribbin, John. "Inflation for Beginners". Archived from the original on 2010-03-26. Retrieved 2017-02-01.