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energy to reach an infinitely small spatial volume.

The principle of measurement at the scale of fundamental particles is mainly based on three relations: the de Broglie relation between the momentum p and the wavelength λ, p=h/λ, which introduces the wave– particle duality for matter; the relation that links the energy E of a particle to its wave frequency or wavelength λ, such as proposed by Einstein to explain the photoelectric effect E = hc/λ; and the relation that links the energy E of a particle of rest mass m to its rest mass energy and to its kinetic energy associated with its momentum p=mv, E²= m²c⁴ + p²c², as mentioned in Einstein’s special theory of relativity. In the above formulas, v is the speed of the particle of mass m and c is the speed of light. The energy E can also be expressed by the formula E= γmc², where γ is given by γ =1/√(1-v²/c²). The speed of a particle is therefore given by v/c=√(1-(mc²/E)².

      In the LHC, the energy of a proton is 7 TeV (1.2 10^–6 J), far higher (by a factor of 7,500) than its rest energy, mc², which is 938 MeV. The formula for speed can then be rewritten as v/c = (1-(m²c⁴/2E²)), which is equal to 1 to the nearest 10^–8. Using the relation E= hc/λ, the resulting value of the wavelength is of the order of 10^–16 m, which gives the dimensions that can be reached in the LHC. The mass measured during two experiments at CERN in the LHC (8 TeV in 2012 and 13 TeV in 2015) is confirmed to the value of 125 GeV.

      To detect the Higgs boson, a particle of mass 125 GeV associated with the Higgs field, while the mass of a proton is 938 MeV, the proton is accelerated and consequently its kinetic energy is increased so that its energy given by E= γmc² significantly exceeds 938 MeV (8 TeV in 2012 and 13 TeV in 2015). The disintegration of colliding protons, each contributing an energy load of 8 TeV or 13 TeV, releases sufficient energy so that the Higgs boson can be expected to emerge during the recombination of subatomic particles. As the Higgs boson decays quasi-instantaneously after its emergence, the products of its decay must be analyzed to identify the excess energy and therefore the excess mass about 125 GeV.

      It is worth noting that at the Planck length, the required energies that cannot be expected in a particle accelerator would lead to the emergence of black holes.

The opposite dimensional extreme towards the infinitely large corresponds to the spatial extent of the Universe, whose estimated value according to cosmologists is 10²⁶ m. In cosmology, the observable Universe is a term used to describe the visible part of our Universe, the point from which light reaches us. It is a sphere whose limit is located at the cosmological horizon, having the Earth at its center. It is therefore a relative notion, as for other observers located somewhere else in the Universe, the observable sphere would not be the same (while its radius would be identical).

      In cosmology, distances are measured in light-years. A light-year is the distance that light travels in one year, which corresponds to approximately 9.5 x 10¹² m. The megaparsec, which is 3.26 million (3.26 x 10⁶) light-years, is another unit of distance that is also specific to extragalactic astrophysics. Finding the size of the Universe involves accurate measurements of fossil radiation, or of the cosmic microwave background (CMB) radiation that originated in the Big Bang and can be used to determine the volume filled by the Universe since its creation. Predicted for the first time by Ralph Alpher in 1948 in his thesis work, CMB was discovered by Arno Penzias and Robert Wilson at “Bell Telephone Laboratories” during the development of a new radio receiver following the interferences detected independently of the orientation of the antenna they were building. While in a first approximation CMB is isotropic, accurate measurements of this radiation lead to determining H₀, the Hubble constant, which indicates the rate of expansion of the Universe.

      In cosmology, detectors are above-ground telescopes. The WMAP (Wilkinson Microwave Anisotropy Probe) satellite launched in 2001 enabled the detection of CMB with good accuracy. Its intensity varies slightly in different directions of the sky and the fluctuations can be determined. Extremely accurate measurements of the WMAP in 2003 made it possible to calculate a value of H₀ of 70 kilometers per second and per megaparsec, which is within 5% in the hypothesis of a constant rate of expansion. Since the Universe is accelerating, during its expansion, the correction brought to H₀ made it possible to estimate the age of the Universe to 13.75 billion years, with a 0.1 billion margin of error. It is the scale fitting the domain to which corresponds the age of the Universe deduced from observations related to the Big Bang based on the inflationary model in an expanding Universe.

After the Big Bang, the elementary subatomic particles had no mass and could travel at the speed of light. After the expansion of the Universe and its cooling, the particles interacted with the Higgs field and consequently gained a mass.

      In the history of the Universe, the elementary particles interacted with the Higgs field, 10^–12 s after the Big Bang. The value of 125 GeV is considered as the critical value between a stable universe and a metastable universe. The “standard model of cosmology” elaborated at the beginning of this century, towards 2000, is probably at present the best model that enables the description of the evolution of the Universe, the significant stages in the history of the observable Universe as well as its current content, as revealed by astronomical observations. The standard model describes the Universe as an expanding homogeneous and isotropic space, on which large structures are overlaid as a result of the gravitational collapse of primordial inhomogeneities, which were formed during the inflation phase. There are still questions to be addressed, such as the nature of certain constituents of the Universe, black matter, and black energy and their relative abundance.

      The inflationary model relies on the hypothesis of the Universe expanding with an exponential acceleration R(t)=R₀exp(H(t)t), 10^–30 s after the Big Bang, where H(t) is the Hubble constant. This constant is measured from the Doppler effect, which explains the red shift of the light radiation emitted by a distant star that is receding from the point of observation. The inflationary model allows for a plausible interpretation of the CMB isotropy, with relative variations of the measured temperature of 10^–5. Based on the data provided by the Hubble, COBE (Cosmic Background Explorer) and WMAP (Wilkinson Microwave Anisotropy Probe) telescopes, as well as by the BOOMerang (Balloon Observations Of Millimetric Extragalactic Radiation ANd Geophysics) and MAXIMA (Millimeter Anisotropy eXperiment IMaging Array) experiments, scientists were able to determine the age of the Universe is 13.75 billion light-years.

The Universe is presently in accelerated expansion: if its speed is 70 km/s at 1 Megaparsec, it doubles at 2 Megaparsec, reaching 140 km/s and so on. Considering the Doppler shift or the red shift for the receding stars, and therefore the fact that not only are the stars receding, but also those that are twice farther recede twice faster, therefore considering the metrics applicable to the space that is stretching while galaxies are receding, the 13.8 billion years between the beginning of the rapid expansion of the Universe 10^–30 s after the Big Bang amount to 46.5 billion light-years, which is a radius of 93 billion light-years. Obviously, the light of stars that are at the periphery or at the cosmological horizon can no longer reach us, but as what we observe today goes back to the time needed for light to reach us while traveling a distance in a stretching space.

      These two examples show that at each dimensional scale, besides the appropriate experimental measurement techniques required for observation, we must have a good mastery of the theories adapted for the interpretation and analysis of the gathered data. At each scale, the engineer must acquire specific knowledge elaborated in the laboratories and develop the competences to enable the mastery of technologies and the implementation of innovations.

      This book which provides applications for Reliability of Multiphysical Systems Set Book 2 (Nanometer-scale Defect Detection Using Polarized Light), focuses on knowledge elaborated at the nanometer scale for applications in the field of engineering sciences. The subjects approached are related to simulation

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