Enough Numbers to Build a Universe

What are the numbers that are used to build a universe? Well, we don't know for sure, but we do know that we can build our universe out of a handful of fundamental constants. Some are knowable in all possible universes, and some are only knowable in ours. Let's start with the mathematical constants.

D is the number of spatial dimensions. Space has three dimensions, which are commonly referred to as length, width, and height. These dimensions are perpendicular to each other, and they provide a framework for understanding the position and motion of objects in space.

$$
D = 3
$$

The universe also has a single temporal dimension that corresponds to the continuum of time. In spacetime, the temporal dimension is represented by a single number, known as a temporal coordinate, which describes the position of an object in time.

$$
T = 1
$$

Pi is a dimensionless mathematical constant that is defined as the ratio of the circumference of a circle to its diameter.

$$\pi \approx 3.141592653589793238$$

e is a dimensionless mathematical constant that arises in the study of exponential growth and decay. It arises naturally out of calculus.

$$e \approx 2.71828182845904523536$$

Now, physical constants. The complete standard model of physics requires 25 fundamental dimensionless constants. At present, their numerical values are not understood in terms of any overarching theory and are determined only from experimental measurement. These constants are:

• The fine structure constant
• The strong coupling constant
• The four parameters of the Cabibbo-Kobayashi-Maskawa matrix which describe how quarks oscillate between different forms
• The four parameters of the Pontecorvo-Maki-Nakagawa-Sakata matrix which describe how neutrinos oscillate between different forms
• Fifteen masses of the fundamental particles, which can expressed in terms of the Planck mass
  • six quarks
  • six leptons
  • the Higgs boson
  • the W boson
  • the Z boson

NIST has a table of all the physical constants that are used in the standard model of physics.

The Fine Structure Constant (denoted $\alpha$) is a fundamental physical constant that is used in the study of the interaction between electromagnetic radiation and matter. The value of the Fine Structure Constant is a dimensionless quantity.

$$\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$$

$$\alpha \approx \frac{1}{137}$$

$$\alpha = 0.0072973525693$$

The Planck constant is denoted by the letter h, and its value is $6.63 \cdot 10^{-34} J \cdot s$. This constant allows us to calculate the amount of energy associated with a particle, given its frequency. The equation for this relationship is given by the formula E = hf, where E is the energy of the particle, f is its frequency, and h is the Planck constant.

$$h = 6.62607015 \times 10^{-34} \quad J s$$

$$\hbar = \frac{h}{2 \pi}$$

The Planck mass is defined as the mass of a particle whose Compton wavelength is equal to the Planck length. The Compton wavelength of a particle is a measure of its size, and the Planck length is the smallest possible length that can be measured in our universe. Therefore, the Planck mass is a measure of the smallest possible mass that can exist in our universe under our current understanding.

$$1.22089 \times 10^{19} \quad GeV/c^2$$

The Cabibbo-Kobayashi-Maskawa matrix, also known as the CKM matrix, is a unitary matrix used in particle physics to describe the mixing of the six quarks that make up protons and neutrons. The CKM matrix describes the mixing of the six quarks, which are the up quark, the down quark, the charm quark, the strange quark, the top quark, and the bottom quark. These six quarks are thought to be different forms or "flavors" of the same fundamental particles, and they can oscillate or "switch" between these different flavors as they interact with other particles.

$$
\begin{bmatrix}
V_{ud}&V_{us}&V_{ub} \\
V_{cd}&V_{cs}&V_{cb} \\
V_{td}&V_{ts}&V_{tb}
\end{bmatrix}
$$

The Pontecorvo-Maki-Nakagawa-Sakata matrix, also known as the PMNS matrix, is a unitary matrix used in particle physics to describe the mixing of three generations of neutrinos.

The PMNS matrix describes the mixing of the three generations of neutrinos, which are the electron neutrino, the muon neutrino, and the tau neutrino. These three generations of neutrinos are thought to be different forms or "flavors" of the same fundamental particle, and they can oscillate or "switch" between these different flavors as they propagate through space.

The PMNS matrix is a 3x3 matrix with three rows and three columns, and it contains six real parameters that describe the mixing of the three generations of neutrinos. These six parameters are the three mixing angles and the three CP-violating phases, which are associated with the phenomenon of CP violation in particle physics.

$$
\begin{bmatrix}
U_{e1}&U_{e2}&U_{e3} \\
U_{\mu 1}&U_{\mu 2}&U_{\mu 3} \\
U_{\tau 1}&U_{\tau 2}&U_{\tau 3}
\end{bmatrix}
$$

The mass of the up quark is:

$$
1.4 × 10^{-22} – 2.7 × 10^{-22}
$$

The mass of the down quark is:

$$
3.4 × 10^{-22} – 4.8 × 10^{-22}
$$

The mass of the charm quark is:

$$
0.78 × 10^{-22} – 1.30 × 10^{-22}
$$

The mass of the charged quark is:

$$
1.4 × 10^{-22} – 2.7 × 10^{-22}
$$

The mass of the strange quark is:

$$
8.27 × 10^{-21}
$$

The mass of the top quark is:

$$
173.07 \quad GeV/c^2
$$

The mass of the bottom quark is:

$$
3.43 \times 10^{-19}
$$

The mass of the electron is:

$$
4.18546 × 10^{-23}
$$

The mass of the electron neutrino is:

$$
9.00978 \times 10^{-30}
$$

The mass of the muon is:

$$
8.65418 \times 10^{-21}
$$

The mass of the muon neutrino is:

$$
< 1.6 \times 10^{-28}
$$

The mass of the tau is:

$$
1.45535 \times 10^{-19}
$$

The mass of the tau neutrino is:

$$
1.6 \times 10^{-28}
$$

The mass of the W boson is:

$$
80.385 \quad GeV/c^2
$$

The mass of the Z boson is:

$$
91.1876 \quad GeV/c^2
$$

The mass of the Higgs boson is:

$$
125.35 \quad GeV
$$

The expectation value of the Higgs field is a measure of the average value of the Higgs field in a given region of space. It is defined as the weighted average of the Higgs field over all possible configurations of the field, where the weighting is determined by the probability of each configuration. The expectation value of the Higgs field is a measure of the average value of the Higgs field in a given region of space. It is defined as the weighted average of the Higgs field over all possible configurations of the field, where the weighting is determined by the probability of each configuration.

In gauge theories, the U(1) coupling constant is a parameter that describes the strength of the interaction between charged particles and the electromagnetic field. It is a measure of how strongly charged particles are affected by the electromagnetic field, and it is related to the charge of the particles and the intensity of the electromagnetic field.

The SU(2) coupling constant is a measure of the strength of the interaction between particles and the SU(2) gauge field. It is a fundamental constant of the Standard Model of particle physics, and it is related to the weak nuclear force. The SU(2) coupling constant is denoted by the symbol $\alpha_W$, and it is defined as the ratio of the coupling constant for the weak force to the coupling constant for the electromagnetic force.

The strong coupling constant is a measure of the strength of the interaction between quarks and gluons. It is a fundamental constant of the Standard Model of particle physics, and it is related to the strong nuclear force. The strong coupling constant is denoted by the symbol $\alpha_S$, and it is defined as the ratio of the coupling constant for the strong force to the coupling constant for the electromagnetic force.

The Weinberg angle is denoted by the symbol $\theta_{W}$, and it is defined as the angle between the weak force and the electromagnetic force in a certain reference frame. The value of the Weinberg angle is not fixed, but it is typically measured to be around 0.23. This value is important because it determines the strength of the weak force relative to the electromagnetic force, which in turn affects the behavior of particles on the atomic and subatomic scales.

$$
\theta_{W} = 0.22290
$$

The cosmological constant, also known as the vacuum energy, is a fundamental constant of the universe that is used in the study of cosmology. It is denoted by the symbol $\Lambda$, and it is defined as the energy density of empty space. The cosmological constant is related to the expansion of the universe, and it plays a crucial role in the study of the evolution of the universe over time. It is thought to be responsible for the acceleration of the expansion of the universe.

$$
\Lambda = 1.05 \times 10^{-52} \quad m^{-2}
$$

The speed of light is a fundamental constant of the universe that is used in the study of physics. It is denoted by the symbol $c$, and it is defined as the speed at which light travels in a vacuum. The speed of light is a constant value, and it is related to the properties of space and time.

$$
c = 299792458 \quad m/s
$$

The permeability of vacuum is a fundamental constant of the universe that is used in the study of physics. It is denoted by the symbol $\mu_0$, and it is defined as the magnetic permeability of a vacuum. The permeability of vacuum is related to the properties of space and time.

$$
\mu_0 = {4 \pi \alpha \hbar} / (e^2 c)
$$

The elementary charge is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron.

$$e = 1.60217663 \times 10^{-19} \text{coulombs} $$

The Coulomb constant is a fundamental constant of the universe that is used in the study of physics. It is denoted by the symbol $k_e$, and it is defined as the ratio of the electric force between two charged particles to the product of their charges. The Coulomb constant is related to the properties of space and time.

$$k_e = 1 / (4 \pi \epsilon_0)$$

The particle charges are the electric charges carried by the particles in the universe.

The particle rest masses are the masses of the particles in the universe.

The Rydberg constant is a physical constant that is used in the study of atomic and molecular structure. It is denoted by the symbol $R_{\infty}$, and it is defined as the limit of the Rydberg formula as the atomic number approaches infinity. The Rydberg constant is related to the energy levels of the hydrogen atom, and it is used to calculate the wavelengths of the spectral lines emitted by hydrogen atoms.

$$R_{\infty} = 10973731.568508 \quad m^{-1}$$

The Planck mass is a fundamental physical constant that is used in the study of physics. It is denoted by the symbol $m_P$, and it is defined as the mass of a particle whose Compton wavelength is equal to the Planck length. The Planck length is the smallest possible length that can be measured in the universe, and it is related to the properties of space and time.

$$m_P = 2.176434 \times 10^{-8} \quad kg$$

The Planck time is a fundamental physical constant that is used in the study of physics. It is denoted by the symbol $t_P$, and it is defined as the time it takes for a photon to travel one Planck length in a vacuum. The Planck time is related to the properties of space and time, and it is used to calculate the smallest possible time interval that can be measured in the universe.

$$t_P = 5.391247 \times 10^{-44} \quad s$$

The Planck length is a fundamental physical constant that is used in the study of physics. It is denoted by the symbol $l_P$, and it is defined as the length at which quantum effects become significant. The Planck length is related to the properties of space and time, and it is used to calculate the smallest possible length that can be measured in the universe.

$$l_P = 1.616255 \times 10^{-35} \quad m$$

The Planck temperature is a fundamental physical constant that is used in the study of physics. It is denoted by the symbol $T_P$, and it is defined as the temperature at which the thermal energy of a particle is equal to its rest mass energy. The Planck temperature is related to the properties of space and time, and it is used to calculate the highest possible temperature that can be measured in the universe.

$$T_P = 1.416 784 \times 10^{32} \quad K$$

The Newton constant, also known as the gravitational constant, is a physical constant that appears in Newton's law of universal gravitation. This law states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force is what we know as gravity.

The gravitational constant, denoted by the letter G, is a proportionality constant that appears in the mathematical expression for the gravitational force between two objects.

$$G = 6.67430 \cdot 10^{-11} \quad N m^2 / kg^2$$

Avogadro's number is defined as the number of atoms or molecules in a mole of a substance. A mole is a unit of measurement that is used in chemistry to represent the amount of a substance, and it is defined as the number of atoms or molecules in a given mass of the substance. The order of Avogadro's number is quite large, which reflects the fact that atoms and molecules are extremely small and numerous.

$$N = 6.02214076 \cdot 10^{23} \quad mol^{-1}$$

The Boltzmann constant is a physical constant that relates the average kinetic energy of particles in a gas to the temperature of the gas. This constant allows us to calculate the average kinetic energy of the particles in a gas, given its temperature. The equation for this relationship is given by the formula $E = (3/2)kT$, where E is the average kinetic energy of the particles, $T$ is the temperature of the gas in kelvins, and k is the Boltzmann constant.

$$k = 1.380649 \cdot 10^{-23} J/K$$

The Hubble constant is a measure of the expansion rate of the universe. The Hubble constant is typically denoted by the letter H, and it is defined as the ratio of the velocity with which a galaxy is moving away from us to its distance from us. This relationship is expressed mathematically as $H = v/d$, where $v$ is the velocity of the galaxy and d is its distance from us.

$$H_0 = 67.8 \quad km/s/\text{megaparsec}$$

And that's it! We have enough numbers to build our universe. There's probably a few more that will emerge as we develop a more complete understanding of physical laws, and it is likely that constants of spacetime itself may emerge out of some deeper theory but for now these are the 30 fundamental constants that underly, well, everything.