Nonperturbative Approaches in Field TheoryView this Special Issue
Research Article | Open Access
Steffen Hahn, Ralf Hofmann, "Cosmic Microwave Background as a Thermal Gas of SU(2) Photons: Implications for the High- Cosmological Model and the Value of ", Advances in High Energy Physics, vol. 2017, Article ID 7525121, 9 pages, 2017. https://doi.org/10.1155/2017/7525121
Cosmic Microwave Background as a Thermal Gas of SU(2) Photons: Implications for the High- Cosmological Model and the Value of
Presently, we are facing a tension in the most basic cosmological parameter, the Hubble constant . This tension arises when fitting the Lambda-cold-dark-matter model (CDM) to the high-precision temperature-temperature (TT) power spectrum of the Cosmic Microwave Background (CMB) and to local cosmological observations. We propose a resolution of this problem by postulating that the thermal photon gas of the CMB obeys an SU() rather than U() gauge principle, suggesting a high- cosmological model which is void of dark-matter. Observationally, we rely on precise low-frequency intensity measurements in the CMB spectrum and on a recent model independent (low-) extraction of the relation between the comoving sound horizon at the end of the baryon drag epoch and (). We point out that the commonly employed condition for baryon-velocity freeze-out is imprecise, judged by a careful inspection of the formal solution to the associated Euler equation. As a consequence, the above-mentioned tension actually transforms into a discrepancy. To make contact with successful low- CDM cosmology we propose an interpolation based on percolated/depercolated vortices of a Planck-scale axion condensate. For a first consistency test of such an all- model we compute the angular scale of the sound horizon at photon decoupling.
Since the pioneering work by Yang and Mills  on the definition of a local four-dimensional, classical, and minimal field theory, which is based on the nonabelian gauge group SU(), much progress has been made in elucidating the role of topologically stabilized and (anti)-self-dual field configurations in building the nonperturbative ground state and influencing the properties of its excitations [2–8]. In particular, the deconfining phase is subject to a highly accurate thermal ground state estimate [9, 10], being composed of so-called Harrington-Shepard (anti)calorons . This (cosmologically relevant) ground state invokes both an adjoint Higgs mechanism [12–15], rendering two out of three directions of the SU() algebra massive (free thermal quasiparticles), and a chiral anomaly [2, 3, 5, 6], giving mass to the Goldstone mode induced by the associated dynamical breaking of this global symmetry. Radiative corrections to thermodynamical quantities, evaluated on the level of free thermal (quasi)particles, are minute and well under control [9, 10]. Note that this is in contrast to the large effects of radiative corrections attributed to the effective QCD action at zero temperature in [16, 17] which are exploited as potential inducers of vacuum energy in the cosmological context in [18–22]. However, it was argued in [23, 24] that QCD condensates, which contribute to the trace anomaly of the energy-momentum tensor (as implied by the effective action), do not act cosmologically.
Postulating that thermal photon gases obey an SU() rather than a U() gauge principle, the SU() Yang-Mills scale can be inferred from low-(radio)frequency spectral intensity measurements, for example , of the Cosmic Microwave Background (CMB) , prompting the name . Below we will use the name synonymously for the implied cosmological model. To investigate the consequences of this postulate towards the equation of state radiative corrections are entirely negligible . When subjecting local energy conservation in a Friedmann-Lemaître-Robertson-Walker (FLRW) universe to this equation of state the numerical temperature ()-redshift () relation () of the CMB follows; see Figure 1 [27, 28], where a comparison with the conventional U() photon gas is shown. The curvature of ( K denoting today’s CMB temperature) at low is due to the influence of the SU() Yang-Mills mass scale on the equation of state. In  an argument is given why recent observational “extractions” of , which claim no deviations from the conventional behavior , are circular. One has at high and therefore a lower slope compared to the conventional case. In an approximation, where recombination at is subjected to thermodynamics, the decoupling condition is where denotes the Thomson photon-electron scattering rate at the decoupling temperature K. We have where and denote the respective ratios of today’s energy densities in baryons and cold dark matter to the critical density. Since this roughly matches . If a strong matter domination can be assumed during recombination then should be equal to but, due to matter-radiation equality occurring at in , this assumption is not quite met, explaining the mild discrepancy between and . Still, we take this rough argument and the desired minimality of the cosmological model as motivations to omit cold dark matter in the high- cosmological model which operates down to recombination and well beyond it.
Concerning the number of massless neutrinos , a conservative input is used: . This high- model, composed of , baryonic matter, and massless neutrinos (), is sufficient to predict the sound horizon at the end of the baryon drag epoch which, in turn, can be confronted with the - relation, recently extracted from local cosmological observations , to determine the value of . The value of , as computed in a high- model, rather sensitively depends on the definition of redshift for baryon-velocity () freeze-out. Usually, is identified with the maximum position of the so-called drag visibility function [31, 32]. However, inspecting the solution of the corresponding Euler equation, given as a functional of , one concludes that this definition applies only in the limit of zero peak width. Realistic results for the ionization fraction , obtained by numerical integration of the according Boltzmann hierarchy (recfast ), imply that the width of this peak extends over several hundred units of redshift in both cases CDM and . As a consequence, a more precise definition of is in order which associates with the left flank of . Therefore, we will in the following refer to this corrected redshift for the freeze-out of as . Our value , after intersection with the - relation of , determines the value of in good agreement with the value obtained in . Also, we would like to point out that, as a consequence of the corrected baryon-velocity freeze-out condition, the value of in CDM, obtained by this method, is now at a discrepancy with the value published in .
To be able to compute the CMB power spectra, our consistent high- cosmological model of (3) needs to be connected to the observationally well cross-checked CDM low- parametrization of the universe’s composition. To facilitate such an interpolation, a candidate real scalar field representing the dark sector is the so-called Planck-scale axion (PSA) condensate [35–37] which rests on chiral symmetry breaking within the Planckian epoch and the axial anomaly invoked by deconfining thermal ground states of Yang-Mills theories. Notice that the only Yang-Mills theory exhibiting the deconfining phase from today to well beyond recombination is . A model, where undergoes coherent and damped oscillations at late times such as to effectively represent CDM, is falsified by the redshift , where the universe’s expansion starts to accelerate, being too high. This prompts the idea that interpolation between at high and CDM at low is achieved by the U() topologically stabilized solitonic configurations (vortices) of the PSA condensate occurring in percolated form (due to a Berezinskii-Kosterlitz-Thouless phase transition following their very creation during a nonthermal phase transition at very high ) down to intermediate where a depercolation transition partially liberates them to effectively represent a pressureless vortex gas. Whether or not the cores of depercolated PSA vortices properly serve as dark-matter halos in spiral galaxies to explain the observed flattening of rotation curves and the lensing signatures of bullet galaxies is an open question. Likewise, it is not yet guaranteed that this new cosmological model, which exhibits radiation domination and baryon freeze-out prior to photon decoupling, explains the observed angular power spectra of the CMB.
This work is organized as follows. In Section 2 we explain our high- cosmological model , introduced in , and compare it with the conventional CDM cosmology. The modification of decoupling conditions due to finite widths visibility functions is discussed in Section 3. Based on this, we perform the computation of and confront it with the - relation of  to determine the value of . Subsequently, in Section 5 we investigate whether coherent and damped oscillations of the PSA field can realistically represent CDM at low , with a negative result. According to  we are thus led to propose an interpolation between high- and low- CDM in terms of percolated PSA vortices which, at some intermediate redshift , partially undergo a depercolation transition. Such a model is demonstrated to be consistent with the extremely well observed angular scale of the sound horizon at photon decoupling . Finally, we summarize our results and provide an outlook on how the new model can be tested further by confrontation with the power spectra of various CMB angular correlation functions.
2. Definition of Cosmological Model
In a flat FLRW universe, a cosmological model is given in terms of the -dependence of the Hubble parameter where is today’s cosmological expansion rate and . Here is the fraction of the energy density of fluid to the critical density today. The function is determined by energy conservation subject to fluid ’s equation of state. From now on we work in supernatural units () where Newton’s constant has units of inverse mass squared. Table 1 lists the parameter values used subsequently.
2.1. The Conventional CDM Model
In the conventional high- CDM model is given as Here nonrelativistic matter decomposes into baryonic () and cold dark matter (CDM). The radiation component contains photons with two polarizations, two relativistic vector modes with three polarizations each, and flavours of massless neutrinos with two polarizations each. is today’s fraction of photonic energy density to critical energy density (for details see ).
2.2. Modifications of CDM towards
In high- the Hubble parameter is given as In this case, only baryonic matter is present. We reiterate that both models, (2) and (3), need to be supplemented by a dark sector to yield successful low- CDM cosmology; see (32). The radiation sector is modified due to a different number of relativistic degrees of freedom and due to the high- temperature-redshift relation ; for details see [27, 28].
3. The End of Recombination
The comoving sound horizon at redshift is defined as whereby denotes the sound velocity in the primordial baryon-electron-photon plasma, given as The function is determined by 3/4 of the ratio of energy densities in baryons and photons. In CDM we have whereas in one obtains The values of can be read off Table 1.
3.1. Conventional Freeze-Out
The final stages of recombination can be characterized in a twofold way. One considers either (i) photon temperature freeze-out, which is relevant for the peak structure in the temperature-temperature (TT) angular power spectrum of the CMB or (ii) baryon-velocity freeze-out, which is detectable in the matter correlation function (galaxy counts). Concerning case (i), the conventional criterion, which fixes the redshift , reads where denotes the total cross section for Thomson scattering, is the ionization fraction (calculated with recfast), and refers to the density of free electrons just before hydrogen recombination, given as Here denotes the helium mass fraction in baryons (see Table 1). Concerning case (ii), the conventional criterion is defined as
3.2. Corrected Freeze-Out
We now show that conditions (8) and (10) are imprecise due to the finite widths of the respective visibility functions. To see this, we have to analyze the formal solution of the Boltzmann hierarchy for the temperature perturbation and of the Euler equation for [31, 32, 40]. Since the argument is similar for both cases we focus on the latter only. The Euler equation reads where is the comoving wave number (omitted as a subscript in the following), denotes the (relative) dipole of the temperature anisotropy , and represents the Newtonian gravitational potential. The overdot demands differentiation with respect to conformal time. Transforming the conformal time to a redshift dependence, the solution of (11) is Here is defined as and the visibility function is represented by In order to study freeze-out the function in (12) is considered slowly varying. Therefore, the variability of the integral solely depends on within its peak region. In both cases CDM and function exhibits a broad peak in dependence of whose shape and maxima do not depend on ; see Figure 2. Note that (10) describes the maxima of . However, due to the finite width the integral in (12) is not saturated at but rather ceases to vary for where lf denotes the maxima of the derivative of . Therefore, defines the freeze-out point more realistically than . According to Figure 2’s caption the values of deviate substantially. Namely, An analogous discussion applies to photon temperature freeze-out with the following results (see ):
4. The Value of
Subjecting the freeze-out redshifts of (15) to (4) under consideration of (2) and (3) yields In Figure 3, these ( independent) values of the sound horizon are confronted with the - relation of . Note the good agreement between the values of implied by in and the extraction performed in . On the other hand, reproduces the value of published in  which exhibits a tension compared to . However, according to Figure 3, the more realistic freeze-out value in CDM entails Thus, in CDM there actually is a 5 discrepancy between the value of quoted in  and obtained by confrontation of with the - relation of .
5. Planck-Scale-Axion Field and Interpolation of High- with Low- Cosmology
Here we would like to analyze cosmological models which link low- CDM with high- . We assume a dark sector which originates from a real, minimally coupled scalar field, a pseudo Nambu-Goldstone mode of dynamical chiral symmetry occurring at the Planck scale [35, 36], whose potential is due to the chiral U anomaly invoked by (anti)calorons of the deconfining, thermal ground state of Yang-Mills theories [1–3, 6, 42–44]. This prompts the name Planck-scale axion (PSA). The only Yang-Mills theory, which is deconfining well above recombination, is because otherwise there would not be just one species of photons.
The radiatively protected potential for the axion condensate , arising due to the thermal ground state of [43, 44], reads as follows: where eV, is a dimensionless factor of order unity, and the reduced Planck mass reads With a canonical kinetic term for the according equation of motion is where an overdot signals the derivative with respect to cosmological time.
In a first attempt at a CDM - interpolation we assume spatially homogeneous -field dynamics subject to CDM constraints at low . It turns out, however, that such a model predicts a value of , defined as the zero of the deacceleration parameter of about which is much higher than the realistic value ~0.7 obtained in CDM. Therefore, as a second proposal we abolish the energy density arising from spatially homogeneous configurations of the field . Rather, we conceive the dark-matter sector in CDM as a piece of energy density due to depercolated topological solitons (vortices) of the field which percolate instantaneously into a dark-energy like piece at some redshift such that . The origin of such a vortex percolate, with hierarchically ordered core sizes, could be due to Hagedorn transitions of Yang-Mills theories in the early universe which are accompanied by Berezinskii-Kosterlitz-Thouless transitions in the axionic sector. Today’s value of would then be interpreted in terms of not-yet depercolated vortices. Indeed, in such an interpolation between CDM and a value of can be fitted to the angular size of the sound horizon at photon decoupling. At the extra contribution to dark-energy amounts to ~0.65% of the baryonic energy density which is consistent with .
5.1. Spatially Homogeneous, Coherent Oscillations
Here we discuss a cosmological model where the interpolation between CDM and is attempted by a spatially homogeneous PSA field undergoing damped and coherent oscillations at late times. This models a pressureless component (cold dark matter) and component with negative pressure (dark-energy). Notice that these two components represent fluids that are not separately conserved.
The Hubble equation reads Here denotes radiation-like energy density including (for radiation energy density is severely suppressed in the cosmological model; for the thermal ground state and the masses of the vector modes of can be neglected) and three flavours of massless neutrinos; is the energy density of baryons, in addition to the energy density associated with the spatially homogeneous PSA field which evolves temporally. Eqs. (21) and (23) can be cast into fully dimensionless equations by rescaling with powers of in the following way: In general, dimensionless quantities (after rescaling with the appropriate power of ) are indicated by the hat-symbol. After rescaling and in dependence of (21) and (23) transmute into where a prime demands -differentiation. In (26) we approximate as With the initial conditions for sufficiently large (no roll; in practice one safely can choose ) the solution to (25) subject to (26) is unique. To fix the values of in (19) and in (28) we demand and that coincides with typical fit value obtained in CDM cosmology : Figure 4 shows the deacceleration parameter for the model defined by (25), (26), (29), and (30). Obviously, this model is falsified by a much too high value of the zero of .
5.2. Percolated and Unpercolated Vortices
Here the basic idea invokes the fact that a PSA field , due to nonthermal phase transitions of the Hagedorn type (e.g., there should be an Yang-Mills theory of scale going confining at ) is subject to U winding and in this way creation of a density of percolated topological solitons (vortex percolate) with a hierarchical ordering of core sizes. Percolation could be understood as a Berezinskii-Kosterlitz-Thouless phase transition [45, 46]. Effectively, this percolate represents homogeneous, constant energy density. As the universe expands the vortex percolate is increasingly stretched, and at around some critical redshift it releases a part of its solitons characterized by some specific core size. The ensuing vortex gas acts cosmologically like pressureless matter. Vortices of larger core sizes remain trapped in the percolate. For this scenario to be a consistent interpolation of and CDM we need to assure that [47, 48].
With the definition of (27) the cosmological model to be considered thus reads where is the dark sector energy density, defined as where and are today’s values of the dark-energy and cold-dark-matter densities associated with (30) and the value quoted in Table 1, respectively.
In order to fix the value of we confront the model of (31) and (32) with the observed angular scale of the sound horizon at CMB photon decoupling, occurring at . Theoretically, is given as To match , as extracted in  from the TT power spectrum, we require ; see Figure 5. This yields a percentage of vacuum energy at CMB photon decoupling of about The omission of vacuum energy in our high- cosmological model of (3) thus is justified for the interpolating model defined in (31) and (32).
6. Summary and Outlook
In the present work we have analyzed, based on a modified temperature-redshift relation for the CMB which, in turn, derives from the postulate that thermal photon gases are subject to an SU() rather than a U() gauge principle, a high- cosmological model which is void of dark-matter and considers three species of massless neutrinos. Such a model predicts (after a reconsideration of baryon-velocity freeze-out) a value of the sound horizon which, together with a model independent extraction of the - relation from cosmologically local observations in , yields good agreement with the value of determined by low- observations in . The same - relation predicts a low value of in standard CDM cosmology which is at a discrepancy with the value given in .
Motivated by the above results, an interpolation between CDM at low and our new high- model is called for. In a first attempt, we have investigated whether coherent and damped oscillations of a Planck-scale axion condensate can realistically accomplish this, with a negative result. With  we were thus led to propose an interpolation in terms of percolated PSA vortices which, at some intermediate , partially undergo a depercolation transition. We have demonstrated this model to be consistent with the angular scale of the sound horizon at photon decoupling.
The new model needs to be tested against the various CMB angular spectra. Our hope is that radiative corrections in SU() Yang-Mills thermodynamics, which play out at low , are capable of explaining the large-angle anomalies of the CMB .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
- C. N. Yang and R. L. Mills, “Conservation of isotopic spin and isotopic gauge invariance,” Physical Review, vol. 96, no. 1, pp. 191–195, 1954.
- S. L. Adler and W. A. Bardeen, “Absence of higher-order corrections in the anomalous axial-vector divergence equation,” Physical Review, vol. 182, no. 5, pp. 1517–1536, 1969.
- J. S. Bell and R. Jackiw, “A PCAC puzzle: π0 → γγ in the σ-model,” Il Nuovo Cimento A, vol. 60, no. 1, pp. 47–61, 1969.
- T. Banks and A. Casher, “Chiral symmetry breaking in confining theories,” Nuclear Physics, Section B, vol. 169, no. 1-2, pp. 103–125, 1980.
- G. 't Hooft, “How instantons solve the U(1) problem,” Physics Reports, vol. 142, no. 6, pp. 357–387, 1986.
- K. Fujikawa, “Path integral for gauge theories with fermions,” Physical Review D, vol. 21, no. 10, pp. 2848–2858, 1980, Erratum to: Physical Review D, vol. 22, p. 1499, 1980.
- D. Dyakonov and V. Yu. Petrov, “Instanton-based vacuum from the Feynman variational principle,” Nuclear Physics B, vol. 245, pp. 259–292, 1984.
- T. Schäfer and E. V. Shuryak, “Instantons in QCD,” Reviews of Modern Physics, vol. 70, no. 2, pp. 323–425, 1998.
- R. Hofmann, The Thermodynamics of Quantum Yang-Mills Theory: Theory and Applications, World Scientific, 2nd edition, 2016.
- I. Bischer, T. Grandou, and R. Hofmann, “Massive loops in thermal SU(2) Yang-Mills theory: radiative corrections to the pressure beyond two loops,” International Journal of Modern Physics A, vol. 32, no. 19-20, Article ID 1750118, 2017.
- B. J. Harrington and H. K. Shepard, “Periodic Euclidean solutions and the finite-temperature Yang-Mills gas,” Physical Review D, vol. 17, no. 8, pp. 2122–2125, 1978.
- P. W. Anderson, “Plasmons, gauge invariance, and mass,” Physical Review, vol. 130, no. 1, pp. 439–442, 1963.
- P. W. Higgs, “Broken symmetries and the masses of gauge bosons,” Physical Review Letters, vol. 13, pp. 508-509, 1964.
- G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, “Global conservation laws and massless particles,” Physical Review Letters, vol. 13, no. 20, pp. 585–587, 1964.
- F. Englert and R. Brout, “Broken symmetry and the mass of gauge vector mesons,” Physical Review Letters, vol. 13, pp. 321–323, 1964.
- H. Pagels and E. Tomboulis, “Vacuum of the quantum Yang-Mills theory and magnetostatics,” Nuclear Physics B, vol. 143, no. 3, pp. 485–502, 1978.
- S. G. Matinyan and G. K. Savvidy, “Vacuum polarization induced by the intense gauge field,” Nuclear Physics, Section B, vol. 134, no. 3, pp. 539–545, 1978.
- Y. Zhang, “Inflation with quantum Yang-Mills condensate,” Physics Letters B, vol. 340, no. 1-2, pp. 18–22, 1994.
- R. Pasechnik, V. Beylin, and G. Vereshkov, “Dark energy from graviton-mediated interactions in the QCD vacuum,” Journal of Cosmology and Astroparticle Physics, vol. 2013, no. 6, article 011, 2013.
- R. Pasechnik, V. Beylin, and G. Vereshkov, “Possible compensation of the QCD vacuum contribution to the dark energy,” Physical Review D, vol. 88, no. 2, Article ID 023509, 2013.
- R. Pasechnik, G. Prokhorov, and O. Teryaev, “Mirror QCD and cosmological constant,” Universe, vol. 3, no. 2, article 43, 2017.
- P. Donà, A. Marcianò, Y. Zhang, and C. Antolini, “Yang-Mills condensate as dark energy: a nonperturbative approach,” Physical Review D, vol. 93, no. 4, Article ID 043012, 13 pages, 2016.
- A. Casher and L. Susskind, “Chiral magnetism (or magnetohadrochironics),” Physical Review D, vol. 9, no. 2, pp. 436–460, 1974.
- S. J. Brodsky and R. Shrock, “Condensates in quantum chromodynamics and the cosmological constant,” Proceedings of the National Academy of Sciences of the United States of America, vol. 108, no. 1, pp. 45–50, 2011.
- D. J. Fixsen, A. Kogut, S. Levin et al., “Arcade 2 measurement of the absolute sky brightness at 3–90 GHz,” The Astrophysical Journal, vol. 734, p. 5, 2011.
- R. Hofmann, “Low-frequency line temperatures of the CMB,” Annalen der Physik, vol. 18, no. 9, pp. 634–639, 2009.
- R. Hofmann, “Relic photon temperature versus redshift and the cosmic neutrino background,” Annalen der Physik, vol. 527, no. 3-4, pp. 254–264, 2015.
- S. Hahn and R. Hofmann, “SU(2)CMB at high redshifts and the value of H0,” Monthly Notices of the Royal Astronomical Society, vol. 469, no. 1, pp. 1233–1245, 2017.
- C. Patrignani, K. Agashe, G. Aielli et al., “Review of particle physics,” Chinese Physics C, vol. 40, no. 10, Article ID 100001, 2016.
- J.-L. Bernal, L. Verde, and A. G. Riess, “The trouble with H0,” Journal of Cosmology and Astroparticle Physics, vol. 10, article 019, 2016.
- W. Hu and N. Sugiyama, “Anisotropies in the cosmic microwave background: an analytic approach,” The Astrophysical Journal, vol. 444, p. 489, 1995.
- W. Hu and N. Sugiyama, “Small‐scale cosmological perturbations: an analytic approach,” The Astrophysical Journal, vol. 471, no. 2, pp. 542–570, 1996.
- A. G. Riess, L. M. Macri, S. L. Hoffmann et al., “A 2.4% determination of the local value of the hubble constant,” The Astrophysical Journal, vol. 826, no. 1, article 56, 2016.
- J. A. Frieman, C. T. Hill, A. Stebbins, and I. Waga, “Cosmology with ultralight pseudo nambu-goldstone bosons,” Physical Review Letters, vol. 75, no. 11, pp. 2077–2080, 1995.
- F. Giacosa and R. Hofmann, “A Planck-scale axion and SU(2) Yang-Mills dynamics: Present acceleration and the fate of the photon,” European Physical Journal C, vol. 50, no. 3, pp. 635–646, 2007.
- F. Giacosa, R. Hofmann, and M. Neubert, “A model for the very early Universe,” Journal of High Energy Physics, vol. 2008, no. 2, article 077, 2008.
- P. A. R. Ade, N. Aghanim, M. Arnaud et al., “Planck 2015 results,” Astronomy & Astrophysics, vol. 594, article A13, 63 pages, 2016.
- D. J. Fixsen, E. S. Cheng, J. M. Gales et al., “The cosmic microwave background spectrum from the full COBE FIRAS data set,” The Astrophysical Journal, vol. 473, p. 576, 1996.
- J. R. Bond and G. Efstathiou, “Cosmic background radiation anisotropies in universes dominated by nonbaryonic dark matter,” The Astrophysical Journal, vol. 285, p. L45, 1984.
- P. J. E. Peebles and D. T. Wilkinson, “Comment on the anisotropy of the primeval fireball,” Physical Review, vol. 174, no. 5, p. 2168, 1968.
- K. Fujikawa, “Path-integral measure for gauge-invariant fermion theories,” Physical Review Letters, vol. 42, no. 18, pp. 1195–1198, 1979.
- R. D. Peccei and H. R. Quinn, “Constraints imposed by CP conservation in the presence of pseudoparticles,” Physical Review D, vol. 16, no. 6, pp. 1791–1797, 1977.
- R. D. Peccei and H. R. Quinn, “CP conservation in the presence of pseudoparticles,” Physical Review Letters, vol. 38, no. 25, pp. 1440–1443, 1977.
- V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems,” Soviet Physics—JETP, vol. 34, no. 3, p. 610, 1972.
- J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” Journal of Physics C: Solid State Physics, vol. 6, no. 7, pp. 1181–1203, 1973.
- J. E. Gunn and B. A. Peterson, “On the density of neutral hydrogen in intergalactic space,” The Astrophysical Journal, vol. 142, p. 1633, 1965.
- R. H. Becker, X. Fan, R. L. White et al., “Evidence for reionization at z ~ 6: detection of a Gunn-Peterson trough in a z = 6.28 quasar,” The Astronomical Journal, vol. 122, p. 2850, 2001.
- R. Hofmann, “The fate of statistical isotropy,” Nature Physics, vol. 9, pp. 686–689, 2013.
Copyright © 2017 Steffen Hahn and Ralf Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.