New Procedure for Delineating the Mass of a Higgs Boson, While Interpolating Properties of the Scalar Singlet Dark Matter Model

Author(s)
Andrew Walcott Beckwith

Affiliation(s)

Physics Department, College of Physics, Chongqing University, Huxi Campus, Chongqing, China.

Physics Department, College of Physics, Chongqing University, Huxi Campus, Chongqing, China.

ABSTRACT

We proceed to obtain a polynomial based iterative solution for early universe creation of the Higgs boson mass, using a derived polynomial of the form with the coefficients for derived through a series of specific integral formulations with the mass of a Higgs boson, and the construction of the coefficients derived as of the using a potential system, for the Higgs, largely similar to the usual Peskin and Schroeder quantum field theoretic treatment for a Higgs potential, which subsequently is modified, i.e. this is for the regime of space-time as up to the Electro-weak regime of cosmology, in terms of a spatial regime. The linkage to Dark matter is in terms of the Scalar Singlet Dark matter model proposed by Silvera and Zee, i.e. what we do is to use a procedure similar to the usual Standard model Higgs, but to find ways to iterate to isolate key inputs into electro weak symmetry breaking procedure for the creation of Dark Matter. Afterwards, we will use specific inputs into the Scalar Singlet Dark matter model which would isolate out input parameters which we think are amendable to experimental testing. We conclude with a discussion of entropy so generated, along the lines of a modification of the usual branching ratios used in Higgs physics, with spin offs we think are relevant to the Dark Matter problem. We also use it to critique some linkage between Dark Matter, and gravity, which may explain some of the findings of LIGO, which were reviewed in 2016 in one of our listed references.

We proceed to obtain a polynomial based iterative solution for early universe creation of the Higgs boson mass, using a derived polynomial of the form with the coefficients for derived through a series of specific integral formulations with the mass of a Higgs boson, and the construction of the coefficients derived as of the using a potential system, for the Higgs, largely similar to the usual Peskin and Schroeder quantum field theoretic treatment for a Higgs potential, which subsequently is modified, i.e. this is for the regime of space-time as up to the Electro-weak regime of cosmology, in terms of a spatial regime. The linkage to Dark matter is in terms of the Scalar Singlet Dark matter model proposed by Silvera and Zee, i.e. what we do is to use a procedure similar to the usual Standard model Higgs, but to find ways to iterate to isolate key inputs into electro weak symmetry breaking procedure for the creation of Dark Matter. Afterwards, we will use specific inputs into the Scalar Singlet Dark matter model which would isolate out input parameters which we think are amendable to experimental testing. We conclude with a discussion of entropy so generated, along the lines of a modification of the usual branching ratios used in Higgs physics, with spin offs we think are relevant to the Dark Matter problem. We also use it to critique some linkage between Dark Matter, and gravity, which may explain some of the findings of LIGO, which were reviewed in 2016 in one of our listed references.

Cite this paper

Beckwith, A. (2018) New Procedure for Delineating the Mass of a Higgs Boson, While Interpolating Properties of the Scalar Singlet Dark Matter Model.*Journal of High Energy Physics, Gravitation and Cosmology*, **4**, 96-122. doi: 10.4236/jhepgc.2018.41010.

Beckwith, A. (2018) New Procedure for Delineating the Mass of a Higgs Boson, While Interpolating Properties of the Scalar Singlet Dark Matter Model.

References

[1] Peskin, M. and Schroeder, D. (1995) An Introduction to Quantum Field Theory. Advanced Book Program, Perseus Books, Cambridge, MA.

[2] Halzen, F. and Martin, A. (1984) Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley and Sons, New York.

[3] Kleinert, H. (2016) Particles and Quantum Fields. World Scientific, Ltd, Singapore.

https://doi.org/10.1142/9915

[4] Maggiore, M. (2008) Gravitational Waves. Volume 1: Theory and Experiments. Oxford University Press, Oxford.

[5] Majumdar, D. (2015) Dark Matter: An Introduction. CRC Press, New York.

[6] Thorne, K. and Blandford, R. (2017) Modern Classical Physics (Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics). Princeton University Press, Princeton, NJ.

[7] Merhav, N. (2010) Information Theory and Statistical Physics—Lecture Notes. Technion, Haifa.

https://arxiv.org/abs/1006.1565

[8] Lloyd, S. (2002) Computational Capacity of the Universe. Physical Review Letters, 88, 237901.

https://doi.org/10.1103/PhysRevLett.88.237901

[9] Magill, J. and Galy, J. (2005) Radioactivity Radionuclides Radiation. Springer Verlag, Berlin.

[10] https://www.nucleonica.com/wiki/index.php?title=Branching_ratio

[11] Penrose, R. (1966) General Relativistic Energy Flux and Elementary Optics. In: Hoffmann, B., Ed., Perspectives in Geometry and Relativity: Essays in Honor of Valcav Hlavaty, Indiana University Press, Bloominton, IN, 259-274.

[12] Beckwith, A. (1927) History Lessons from the 5th Solvay Meeting. International Meeting, Kiev, September 2017, 1-54. http://vixra.org/abs/1708.0399

[13] Susskind, L. and Friedman, A. (2017) Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, New York.

[14] Roland, O. (1999) Understanding Quantum Mechanics. Princeton University Press, Princeton, NJ.

[15] Roland, O. (1994) The Interpretation of Quantum Mechanics. Princeton University Press, Princeton, NJ.

[16] Bacciagaluppi, G. and Valentini, A. (2009) Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press, Cambridge, MA.

[17] Gasiorowitz, S. (2003) Quantum Physics. 3rd Edition, Wiley Interscience, Hoboken, NJ.

[18] Kieffer, C. (2012) Quantum Gravity. 3rd Edition, Oxford Science Publications, Oxford University Press, Oxford.

[19] ‘t Hooft, G. (2002) Determinism Beneath Quantum Mechanics. Conf. Proceedings, Quo Vadis Quantum Mechanics, Philadelphia, 24-27 Sept 2002, 12 p.

https://arxiv.org/abs/quant-ph/0212095

[20] Ernest, A.D. (2004) A Quantum Approach to Dark Matter.

https://arxiv.org/ftp/astro-ph/papers/0406/0406139.pdf

[21] Beckwith, A. (2017) Quantum versus Classical Nature of the Early Universe and Its Consequences (Entanglement?). Bogoliubov Astroparticle and Quantum Foundations Conference, Kiev, Ukraine, September 2017, 1-24. http://vixra.org/abs/1710.0071

[22] Abbot, B.P., et al. (2016) Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116, Article ID: 061102.

https://doi.org/10.1103/PhysRevLett.116.061102

[23] Abbot, B.P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016) GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Physical Review Letters, 116, Article ID: 241103.

https://doi.org/10.1103/PhysRevLett.116.241103

[24] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://doi.org/10.1142/S0218271809015904

[25] Beckwith, A.W. (2011) Detailing Minimum Parameters As Far As Red Shift, Frequency, Strain, and Wavelength of Gravity Waves/Gravitons, and Possible Impact upon GW Astronomy. http://vixra.org/pdf/1103.0020v1.pdf

[26] Bird, S., Cholis, I., Muñoz, J.B., Ali-Haïmoud, Y., Kamionkowski, M., Kovetz, E.D., Raccanelli, A. and Riess, A.G. (2016) Did LIGO Detect Dark Matter? Physical Review Letters, 116, 201301.

https://doi.org/10.1103/PhysRevLett.116.201301

[27] Dildick, S., Kamon, T., Krutelyov, S., Pakhotin, Y., Rose, A., Safonov, A., Tatarinov, A., Bouhali, O., Hernandez, A.M.C. and for the CMS Collaboration (2015) Search for Non Standard Model Higgs Boson Decays in Events with Displaced Muon-Jets.

https://arxiv.org/abs/1510.02764

[28] Einhorn, M.B. and Wudka, J. (2013) Higgs-Boson Couplings beyond the Standard Model. Nuclear Physics B, 877, 792-806.

https://doi.org/10.1016/j.nuclphysb.2013.11.004

[29] Silveira, V. and Zee, A. (1985) Scalar Phantoms. Physics Letters B, 161, 136.

https://doi.org/10.1016/0370-2693(85)90624-0

[30] Krisztian, P. (2016) Prospects for beyond Standard Model Higgs Boson Searches at Future LHC Runs and Other Machines. cHarged2016, Uppsala, Sweden, 3-6 October 2016, 1-9.

https://arxiv.org/abs/1701.05124

[31] Khan, N. (2017) Exploring Extensions of the Scalar Sector of the Standard Model. PhD Thesis, Indian Institute of Technology, Indore.

https://arxiv.org/abs/1701.02205

[32] Beyer, W. (1987) Standard CRC Mathematical Tables. 28th Edition, CRC Press, Boca Raton, FL.

[33] https://en.wikipedia.org/wiki/Electroweak_epoch

[34] Kolb, E., Pi, S. and Raby, S. (1984) Phase Transitions in Super Symmetric Grand Unified Models. In: Fang, L. and Ruffini, R., Eds., Cosmology in the Early Universe, World Press Scientific, Pte. Ltd. Co, Singapore, 45-70.

[35] Beckwith, A. (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwartz Shield Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology, 2, 106-124.

https://doi.org/10.4236/jhepgc.2016.21012

[36] Gorbunov, D. and Rubakov, V. (2011) Introduction to the Theory of the Early Universe, Cosmological Perturbations and Inflationary Theory. World Scientific Publishing Pte. Ltd, Singapore.

[37] Unruh, W.G. (1986) Why Study Quantum Theory? Canadian Journal of Physics, 64, 128-130.

https://doi.org/10.1139/p86-019

[38] Unruh, W.G. (1986) Erratum: Why Study Quantum Gravity? Canadian Journal of Physics, 64, 1453. http://dx.doi.org/10.1139/p86-257

[39] Kolb, E. and Turner, S. (1994) The Early Universe. Westview Press, Chicago, IL.

[40] http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm

[41] Golub, G.H. and Welsch, J.H. (1969) Calculation of Gauss Quadrature Rules. Mathematics of Computation, 23, 221-230.

https://doi.org/10.1090/S0025-5718-69-99647-1

[42] Abramowitz, M. and Stegun, I.A. (1983) Chapter 25.4, Integration. In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55 (Ninth Reprint with Additional Corrections of Tenth Original Printing with Corrections (December 1972); First ed.), United States Department of Commerce, National Bureau of Standards; Dover Publications, Washington D.C., ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.

[43] http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx

[44] Liang, X.Q. (2003) Solving Roots of Polynomial Equation of Degree 4 with Real Coefficients. Formalized Mathematics, 11, 185-187. http://mizar.org/fm/2003-11/pdf11-2/polyeq_2.pdf

[45] Curtiss, D.R. (1918) Recent Extensions of Descartes’ Rule of Signs. Annals of Mathematics, 19, 251-278.

https://doi.org/10.2307/1967494

[46] Kostov, V.P. (2010) A Mapping Defined by the Schur-Szegö Composition. Comptes Rendus de l’Academie Bulgare des Sciences, 63, 943-952.

[47] https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

[48] https://people.richland.edu/james/lecture/m116/polynomials/zeros.html

[49] https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem

[50] Rüdinger, A. (2014) Strengthening the Gauss-Lucas Theorem for Polynomials with Zeros in the Interior of the Convex Hull. Preprint.

https://arxiv.org/abs/1405.0689

[51] Khalil, S. and Moretti, S. (2013) A Simple Symmetry as a Guide towards New Physics beyond the Standard Model.

https://arxiv.org/abs/1301.0144

[52] Gunion, J., Haber, H.E., Kane, G.L. and Dawson, S. (1990) The Higgs Hunters Guide. Addison-Wesley, Reading, MA.

[53] Craig, N., Galloway, J. and Thomas, S. (2013) Searching for Signs of the Second Higgs Doublet.

https://arxiv.org/abs/1305.2424

[54] Craig, N. and Thomas, S. (2012) Exclusive Signals of an Extended Higgs Sector. Journal of High Energy Physics, 1211, 83.

https://doi.org/10.1007/JHEP11(2012)083

[55] Branco, G.C., Ferreira, P.M., Lavoura, L., Rebelo, M.N., Sher, M. and Silva, J.P. (2012) Theory and Phenomenology of Two-Higgs-Doublet Models. Physics Reports, 516, 1-102.

https://doi.org/10.1016/j.physrep.2012.02.002

[56] Baez, J. (2015) Do Tachyons Exist?

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

[57] Green, M., Schwartz, J. and Witten, E. (1987) Superstring Theory. Volume 1: Introduction. Cambridge University Press, New York.

[58] Ohanian, H. and Ruffini, R. (2013) Gravitation and Space-Time. 3rd Edition, Cambridge University Press, New York.

https://doi.org/10.1017/CBO9781139003391

[59] Beckwith, A. (2014) Analyzing Black Hole Super-Radiance Emission of Particles/Energy from a Black Hole as a Gedanken-Experiment to Get Bounds on the Mass of a Graviton. Advances in High Energy Physics, 2014, Article ID: 230713.

https://doi.org/10.1155/2014/230713

[60] Penrose, R. (2011) Cycles of Time—An Extrardinary New View of the Universe. Alfred A. Knopf, New York.

[61] Plebasnki, J. and Krasinski, A. (2006) An Introduction to General Relativity and Cosmology. Cambridge University Press, Cambridge.

[62] Beckwith, A. (2017) How to Determine Initial Starting Time Step with an Initial Hubble Parameter H = 0 after Formation of Causal Structure Leading to Investigation of the Penrose Weyl Tensor Conjecture. http://vixra.org/abs/1706.0110

[63] Alves, M., Oswaldo, D., Miranda, O. and de Araujo, J. (2010) Can Massive Gravitons Be an Alternative to Dark Energy?

https://arxiv.org/abs/0907.5190

[64] Chiara, C., Durrer, R. and Servant, G. (2007) Gravitational Wave Generation from Bubble Collisions in First-Order Phase Transitions: An Analytic Approach.

http://fiteoweb.unige.ch/~durrer/papers/CDS_bubbles_vNEW.pdf

[65] Witten, E. (1984) Cosmic Separation of Phases. Physical Review D, 30, 272.

https://doi.org/10.1103/PhysRevD.30.272

[66] Hogan, C. (1986) Gravitational Radiation from Cosmological Phase Transitions. Monthly Notices of the Royal Astronomical Society, 218, 629.

https://doi.org/10.1093/mnras/218.4.629

[67] Abbott, B.P., et al. (2009) An Upper Limit on the Stochastic Gravitational-Wave Background of Cosmological Origin. Nature, 460, 990.

http://www.phys.ufl.edu/~tanner/PDFS/Abbott09Nature-Stochastic.pdf

https://doi.org/10.1038/nature08278

[68] Clarkson, C. and Seahra, S. (2007) A Gravitational Wave Window on Extra Dimensions. Classical and Quantum Gravity, 24, F33.

https://doi.org/10.1088/0264-9381/24/9/F01

[69] Corda, C. and Cuesta, H. (2010) Removing Black Hole Singularities with Non Linear Electrodynamics. Modern Physics A, 25, 2423-2429.

[70] Abbot, B.P., et al. (2017) IGO Scientific Collaboration and Virgo Collaboration.

http://www.ego-gw.it/ILIAS-GW/documents/N5AnnualReport2008/2007%20LSC-Virgo%20white%20paper%20on%20GW%20data%20analysis.pdf

[71] Abbot, B.P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2017) GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. PRL, 119, Article ID: 161101.

https://www.ligo.org/detections/GW170817/paper/GW170817-PRLpublished.pdf

[72] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://doi.org/10.1142/S0218271809015904

[1] Peskin, M. and Schroeder, D. (1995) An Introduction to Quantum Field Theory. Advanced Book Program, Perseus Books, Cambridge, MA.

[2] Halzen, F. and Martin, A. (1984) Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley and Sons, New York.

[3] Kleinert, H. (2016) Particles and Quantum Fields. World Scientific, Ltd, Singapore.

https://doi.org/10.1142/9915

[4] Maggiore, M. (2008) Gravitational Waves. Volume 1: Theory and Experiments. Oxford University Press, Oxford.

[5] Majumdar, D. (2015) Dark Matter: An Introduction. CRC Press, New York.

[6] Thorne, K. and Blandford, R. (2017) Modern Classical Physics (Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics). Princeton University Press, Princeton, NJ.

[7] Merhav, N. (2010) Information Theory and Statistical Physics—Lecture Notes. Technion, Haifa.

https://arxiv.org/abs/1006.1565

[8] Lloyd, S. (2002) Computational Capacity of the Universe. Physical Review Letters, 88, 237901.

https://doi.org/10.1103/PhysRevLett.88.237901

[9] Magill, J. and Galy, J. (2005) Radioactivity Radionuclides Radiation. Springer Verlag, Berlin.

[10] https://www.nucleonica.com/wiki/index.php?title=Branching_ratio

[11] Penrose, R. (1966) General Relativistic Energy Flux and Elementary Optics. In: Hoffmann, B., Ed., Perspectives in Geometry and Relativity: Essays in Honor of Valcav Hlavaty, Indiana University Press, Bloominton, IN, 259-274.

[12] Beckwith, A. (1927) History Lessons from the 5th Solvay Meeting. International Meeting, Kiev, September 2017, 1-54. http://vixra.org/abs/1708.0399

[13] Susskind, L. and Friedman, A. (2017) Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, New York.

[14] Roland, O. (1999) Understanding Quantum Mechanics. Princeton University Press, Princeton, NJ.

[15] Roland, O. (1994) The Interpretation of Quantum Mechanics. Princeton University Press, Princeton, NJ.

[16] Bacciagaluppi, G. and Valentini, A. (2009) Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press, Cambridge, MA.

[17] Gasiorowitz, S. (2003) Quantum Physics. 3rd Edition, Wiley Interscience, Hoboken, NJ.

[18] Kieffer, C. (2012) Quantum Gravity. 3rd Edition, Oxford Science Publications, Oxford University Press, Oxford.

[19] ‘t Hooft, G. (2002) Determinism Beneath Quantum Mechanics. Conf. Proceedings, Quo Vadis Quantum Mechanics, Philadelphia, 24-27 Sept 2002, 12 p.

https://arxiv.org/abs/quant-ph/0212095

[20] Ernest, A.D. (2004) A Quantum Approach to Dark Matter.

https://arxiv.org/ftp/astro-ph/papers/0406/0406139.pdf

[21] Beckwith, A. (2017) Quantum versus Classical Nature of the Early Universe and Its Consequences (Entanglement?). Bogoliubov Astroparticle and Quantum Foundations Conference, Kiev, Ukraine, September 2017, 1-24. http://vixra.org/abs/1710.0071

[22] Abbot, B.P., et al. (2016) Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116, Article ID: 061102.

https://doi.org/10.1103/PhysRevLett.116.061102

[23] Abbot, B.P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016) GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Physical Review Letters, 116, Article ID: 241103.

https://doi.org/10.1103/PhysRevLett.116.241103

[24] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://doi.org/10.1142/S0218271809015904

[25] Beckwith, A.W. (2011) Detailing Minimum Parameters As Far As Red Shift, Frequency, Strain, and Wavelength of Gravity Waves/Gravitons, and Possible Impact upon GW Astronomy. http://vixra.org/pdf/1103.0020v1.pdf

[26] Bird, S., Cholis, I., Muñoz, J.B., Ali-Haïmoud, Y., Kamionkowski, M., Kovetz, E.D., Raccanelli, A. and Riess, A.G. (2016) Did LIGO Detect Dark Matter? Physical Review Letters, 116, 201301.

https://doi.org/10.1103/PhysRevLett.116.201301

[27] Dildick, S., Kamon, T., Krutelyov, S., Pakhotin, Y., Rose, A., Safonov, A., Tatarinov, A., Bouhali, O., Hernandez, A.M.C. and for the CMS Collaboration (2015) Search for Non Standard Model Higgs Boson Decays in Events with Displaced Muon-Jets.

https://arxiv.org/abs/1510.02764

[28] Einhorn, M.B. and Wudka, J. (2013) Higgs-Boson Couplings beyond the Standard Model. Nuclear Physics B, 877, 792-806.

https://doi.org/10.1016/j.nuclphysb.2013.11.004

[29] Silveira, V. and Zee, A. (1985) Scalar Phantoms. Physics Letters B, 161, 136.

https://doi.org/10.1016/0370-2693(85)90624-0

[30] Krisztian, P. (2016) Prospects for beyond Standard Model Higgs Boson Searches at Future LHC Runs and Other Machines. cHarged2016, Uppsala, Sweden, 3-6 October 2016, 1-9.

https://arxiv.org/abs/1701.05124

[31] Khan, N. (2017) Exploring Extensions of the Scalar Sector of the Standard Model. PhD Thesis, Indian Institute of Technology, Indore.

https://arxiv.org/abs/1701.02205

[32] Beyer, W. (1987) Standard CRC Mathematical Tables. 28th Edition, CRC Press, Boca Raton, FL.

[33] https://en.wikipedia.org/wiki/Electroweak_epoch

[34] Kolb, E., Pi, S. and Raby, S. (1984) Phase Transitions in Super Symmetric Grand Unified Models. In: Fang, L. and Ruffini, R., Eds., Cosmology in the Early Universe, World Press Scientific, Pte. Ltd. Co, Singapore, 45-70.

[35] Beckwith, A. (2016) Gedanken Experiment for Refining the Unruh Metric Tensor Uncertainty Principle via Schwartz Shield Geometry and Planckian Space-Time with Initial Nonzero Entropy and Applying the Riemannian-Penrose Inequality and Initial Kinetic Energy for a Lower Bound to Graviton Mass (Massive Gravity). Journal of High Energy Physics, Gravitation and Cosmology, 2, 106-124.

https://doi.org/10.4236/jhepgc.2016.21012

[36] Gorbunov, D. and Rubakov, V. (2011) Introduction to the Theory of the Early Universe, Cosmological Perturbations and Inflationary Theory. World Scientific Publishing Pte. Ltd, Singapore.

[37] Unruh, W.G. (1986) Why Study Quantum Theory? Canadian Journal of Physics, 64, 128-130.

https://doi.org/10.1139/p86-019

[38] Unruh, W.G. (1986) Erratum: Why Study Quantum Gravity? Canadian Journal of Physics, 64, 1453. http://dx.doi.org/10.1139/p86-257

[39] Kolb, E. and Turner, S. (1994) The Early Universe. Westview Press, Chicago, IL.

[40] http://www.damtp.cam.ac.uk/lab/people/sd/lectures/nummeth98/integration.htm

[41] Golub, G.H. and Welsch, J.H. (1969) Calculation of Gauss Quadrature Rules. Mathematics of Computation, 23, 221-230.

https://doi.org/10.1090/S0025-5718-69-99647-1

[42] Abramowitz, M. and Stegun, I.A. (1983) Chapter 25.4, Integration. In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55 (Ninth Reprint with Additional Corrections of Tenth Original Printing with Corrections (December 1972); First ed.), United States Department of Commerce, National Bureau of Standards; Dover Publications, Washington D.C., ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.

[43] http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx

[44] Liang, X.Q. (2003) Solving Roots of Polynomial Equation of Degree 4 with Real Coefficients. Formalized Mathematics, 11, 185-187. http://mizar.org/fm/2003-11/pdf11-2/polyeq_2.pdf

[45] Curtiss, D.R. (1918) Recent Extensions of Descartes’ Rule of Signs. Annals of Mathematics, 19, 251-278.

https://doi.org/10.2307/1967494

[46] Kostov, V.P. (2010) A Mapping Defined by the Schur-Szegö Composition. Comptes Rendus de l’Academie Bulgare des Sciences, 63, 943-952.

[47] https://en.wikipedia.org/wiki/Descartes%27_rule_of_signs

[48] https://people.richland.edu/james/lecture/m116/polynomials/zeros.html

[49] https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem

[50] Rüdinger, A. (2014) Strengthening the Gauss-Lucas Theorem for Polynomials with Zeros in the Interior of the Convex Hull. Preprint.

https://arxiv.org/abs/1405.0689

[51] Khalil, S. and Moretti, S. (2013) A Simple Symmetry as a Guide towards New Physics beyond the Standard Model.

https://arxiv.org/abs/1301.0144

[52] Gunion, J., Haber, H.E., Kane, G.L. and Dawson, S. (1990) The Higgs Hunters Guide. Addison-Wesley, Reading, MA.

[53] Craig, N., Galloway, J. and Thomas, S. (2013) Searching for Signs of the Second Higgs Doublet.

https://arxiv.org/abs/1305.2424

[54] Craig, N. and Thomas, S. (2012) Exclusive Signals of an Extended Higgs Sector. Journal of High Energy Physics, 1211, 83.

https://doi.org/10.1007/JHEP11(2012)083

[55] Branco, G.C., Ferreira, P.M., Lavoura, L., Rebelo, M.N., Sher, M. and Silva, J.P. (2012) Theory and Phenomenology of Two-Higgs-Doublet Models. Physics Reports, 516, 1-102.

https://doi.org/10.1016/j.physrep.2012.02.002

[56] Baez, J. (2015) Do Tachyons Exist?

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

[57] Green, M., Schwartz, J. and Witten, E. (1987) Superstring Theory. Volume 1: Introduction. Cambridge University Press, New York.

[58] Ohanian, H. and Ruffini, R. (2013) Gravitation and Space-Time. 3rd Edition, Cambridge University Press, New York.

https://doi.org/10.1017/CBO9781139003391

[59] Beckwith, A. (2014) Analyzing Black Hole Super-Radiance Emission of Particles/Energy from a Black Hole as a Gedanken-Experiment to Get Bounds on the Mass of a Graviton. Advances in High Energy Physics, 2014, Article ID: 230713.

https://doi.org/10.1155/2014/230713

[60] Penrose, R. (2011) Cycles of Time—An Extrardinary New View of the Universe. Alfred A. Knopf, New York.

[61] Plebasnki, J. and Krasinski, A. (2006) An Introduction to General Relativity and Cosmology. Cambridge University Press, Cambridge.

[62] Beckwith, A. (2017) How to Determine Initial Starting Time Step with an Initial Hubble Parameter H = 0 after Formation of Causal Structure Leading to Investigation of the Penrose Weyl Tensor Conjecture. http://vixra.org/abs/1706.0110

[63] Alves, M., Oswaldo, D., Miranda, O. and de Araujo, J. (2010) Can Massive Gravitons Be an Alternative to Dark Energy?

https://arxiv.org/abs/0907.5190

[64] Chiara, C., Durrer, R. and Servant, G. (2007) Gravitational Wave Generation from Bubble Collisions in First-Order Phase Transitions: An Analytic Approach.

http://fiteoweb.unige.ch/~durrer/papers/CDS_bubbles_vNEW.pdf

[65] Witten, E. (1984) Cosmic Separation of Phases. Physical Review D, 30, 272.

https://doi.org/10.1103/PhysRevD.30.272

[66] Hogan, C. (1986) Gravitational Radiation from Cosmological Phase Transitions. Monthly Notices of the Royal Astronomical Society, 218, 629.

https://doi.org/10.1093/mnras/218.4.629

[67] Abbott, B.P., et al. (2009) An Upper Limit on the Stochastic Gravitational-Wave Background of Cosmological Origin. Nature, 460, 990.

http://www.phys.ufl.edu/~tanner/PDFS/Abbott09Nature-Stochastic.pdf

https://doi.org/10.1038/nature08278

[68] Clarkson, C. and Seahra, S. (2007) A Gravitational Wave Window on Extra Dimensions. Classical and Quantum Gravity, 24, F33.

https://doi.org/10.1088/0264-9381/24/9/F01

[69] Corda, C. and Cuesta, H. (2010) Removing Black Hole Singularities with Non Linear Electrodynamics. Modern Physics A, 25, 2423-2429.

[70] Abbot, B.P., et al. (2017) IGO Scientific Collaboration and Virgo Collaboration.

http://www.ego-gw.it/ILIAS-GW/documents/N5AnnualReport2008/2007%20LSC-Virgo%20white%20paper%20on%20GW%20data%20analysis.pdf

[71] Abbot, B.P., et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2017) GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. PRL, 119, Article ID: 161101.

https://www.ligo.org/detections/GW170817/paper/GW170817-PRLpublished.pdf

[72] Corda, C. (2009) Interferometric Detection of Gravitational Waves: The Definitive Test for General Relativity. International Journal of Modern Physics D, 18, 2275-2282.

https://doi.org/10.1142/S0218271809015904