LOG#106. Basic Cosmology (I).


The next thread is devoted to Cosmology. I will intend to be clear and simple about equations and principles of current Cosmology with a General Relativity background.

First of all…I will review the basic concepts of natural units I am going to use here. We will be using the following natural units:


We will take the Planck mass to be given by

M_P=\sqrt{8\pi G_N}\approx 1\mbox{.}2\cdot 10^{19}GeV

The solar mass is M_\odot=2\cdot 10^{30}kg and the parsec is given by the value

1pc=3\mbox{.}26lyr=3\mbox{.}1\cdot 10^{16}m

Well, current Cosmology is based on General Relativity. Even if I have not reviewed this theory with detail in this blog, the nice thing is that most of Cosmology can be learned with only a very little knowledge of this fenomenal theory. The most important ideas are: metric field, geodesics, Einstein equations and no much more…

In fact, newtonian gravity is a good approximation in some particular cases! And we do know that even in this pre-relativistic theory

\mbox{Gravitational force}=\mbox{Matter/Mass density}

via the Poisson’s equation

\nabla^2\phi =4\pi G_N\rho

This idea, due to the equivalence principle, is generalized a little bit in the general relativistic framework

\mbox{Spacetime geometry}=\mbox{Matter content/Energy-momentum}

The spacetime geometry is determined by the metric tensor g_{\mu\nu}(x). The matter content is given by the stress-energy-momentum tensor T_{\mu\nu}. As we know one of these two elements, we can know, via Eisntein’s field equations the another. That is, given a metric tensor, we can tell how energy-momentum “moves” in space-time. Given the energy-momentum tensor, we can know what is the metric tensor in spacetime and we can guess how the spacetime bends… This is the origin of the famous motto: “Spacetime says matter how to move, energy-momentum says spacetime how to curve”! Remember that we have “deduced” the Einstein’s field equations in the previous post. Without a cosmological constant term, we get

G_{\mu\nu}=\kappa^2T_{\mu\nu}=8\pi G_NT_{\mu\nu}

Given a spacetime metric g_{\mu\nu}, we can calculate the (affine/Levi-Civita) connection

\Gamma^\sigma_{\;\;\mu\nu}=\dfrac{1}{2}g^{\sigma\rho}\left(\partial_\mu g_{\nu\rho}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu}\right)

The Riemann tensor that measures the spacetime curvature is provided by the equation

R^\rho_{\;\; \sigma\mu\nu}=\partial_\mu \Gamma^\rho_{\;\;\mu\sigma}-\partial_\mu \Gamma^\rho_{\;\; \mu \sigma}+\Gamma^\rho_{\;\;\mu\lambda}\Gamma^\lambda_{\;\;\nu\sigma}-\Gamma^\rho_{\;\;\nu\lambda}\Gamma^\lambda_{\;\;\mu\sigma}

The Ricci tensor is defined to be the following “trace” of the Riemann tensor

R_{\mu\nu}=R^\lambda_{\;\;\mu\lambda \nu}

The Einstein tensor is related to the above tensors in the well-known manner


The Einstein’s equations can be derived from the Einstein-Hilbert action we learned in the previous post, using the action principle and the integral

\boxed{S_{EH}=\int d^4x \sqrt{-g} \left(\kappa^{-2}R+\mathcal{L}_M\right)}

The geodesic equation is the path of a freely falling particle. It gives a “condensation” of the Einstein’s equivalence principle too and it is also a generalization of Newton’s law of “no force”. That is, the geodesic equation is the feynmanity

\boxed{\dfrac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu _{\rho\sigma}\dfrac{dx^\rho}{d\tau}\dfrac{dx^\sigma}{d\tau}=0}

Finally, an important concept in General Relativity is that of isometry. The symmetry of the “spacetime manifold” is provided by a Killing vector that preserves transformations (isometries) of that manifold. Mathematically speaking, the Killing vector fields satisfy certain equation called the Killing equation

\boxed{\xi_{\mu ; \nu}+\xi_{\nu ; \mu}=0}

Maximally symmetric spaces have n(n+1)/2 Killing vectors in n-dimensional (nD) spacetime. There are 3 main classes or types of 2D maximally symmetric that can be generalized to higher dimensions:

1. The euclidean plane E^2.

2. The pseudo-sphere H^2. This is a certain “hyperbolic” space.

3. The spehre S^2. This is a certain “elliptic” space.

The Friedmann-Robertson-Walker Cosmology

Current cosmological models are based in General Relativity AND  a simplification of the possible metrics due to the so-called Copernican (or cosmological) principle: the Universe is pretty much the same “everywhere” you are in the whole Universe! Remarkbly, the old “perfect” Copernican (cosmological) principle that states that the Universe is the same “everywhere” and “every time” is wrong. Phenomenologically, we have found that the Universe has evolved and it evolves, so the Universe was “different” when it was “young”. Therefore, the perfect cosmological principle is flawed. In fact, this experimental fact allows us to neglect some old theories like the “stationary state” and many other “crazy theories”.

What are the observational facts to keep the Copernican principle? It seems that:

1st. The distribution of matter (mainly galaxies, clusters,…) and radiation (the cosmic microwave background/CMB) in the observable Universe is homogenous and isotropic.

2nd. The Universe is NOT static. From Hubble’s pioneer works/observations, we do know that galaxies are receeding from us!

Therefore, these observations imply that our “local” Hubble volume during the Hubble time is similar to some spacetime with homogenous and isotropic spatial sections, i.e., it is a spacetime manifold M=\mathbb{R}\times \Sigma. Here, \mathbb{R} denotes the time “slice” and \Sigma represents a 3D maximally symmetric space.

The geometry of a locally isotropic and homogeneous Universe is represented by the so-called Friedmann-Robertson-Walker metric


Here, a(t) is the called the scale factor.  The parameter k determines the geometry type (plane, hyperbolic or elliptical/spherical):

1) If k=0, then the Universe is “flat”. The manifold is E^3.

2) If k=-1, then the Universe is “open”/hyperbolic. The manifold would be H^3.

3) If k=+1, then the Universe is “closed”/spherical or elliptical. The manifold is then S^3.

Remark: The ansatz of local homogeneity and istoropy only implies that the spatial metric is locally one of the above three spaces, i.e., E^3,H^3,S^3. It could be possible that these 3 spaces had different global (likely topological) properties beyond these two properties.

Kinematical features of a FRW Universe

The first property we are interested in Cosmology/Astrophysics is “distance”. Measuring distance in a expanding Universe like a FRW metric is “tricky”. There are several notions of “useful” distances. They can be measured by different methods/approaches and they provide something called sometimes “the cosmologidal distance ladder”:

1st. Comoving distance. It is a measure in which the distance is “taken” by a fixed coordinate.

2nd. Physical distance. It is essentially the comoving distance times the scale factor.

3rd. Luminosity distance. It uses the light emitted by some object to calculate its distance (provided the speed of light is taken constant, i.e., special relativity holds and we have a constant speed of light)

4th. Angular diameter distance. Another measure of distance using the notion of parallax and the “extension” of the physical object we measure somehow.

There is an important (complementary) idea in FRW Cosmology: the particle horizon. Consider a light-like particle with ds^2=0. Then,




The total comoving distance that light have traveled since a time t=0 is equal to


It shows that NO information could have propagated further and thus, there is a “comoving horizon” with every light-like particle! Here, this time is generally used as a “conformal time” as a convenient tiem variable for the particle. The physical distance to the particle horizon can be calculated


There are some important kinematical equations to be known

A) For the geodesic equation, the free falling particle, we have




for the FRW metric and, moreover, the energy-momentum vector P^\mu=(E,\mathbf{p}) is defined by the usual invariant equation


This definition defines, in fact, the proper “time” \lambda implicitely, since


and the 0th component of the geodesic equation becomes


g_{\mu\nu}p^\mu p^\nu=-E^2+a^2\delta_{ij}p^ip^j=-m^2

EdE=a^2\vert \mathbf{p}\vert d\vert \mathbf{p}\vert

a^2 p\dfrac{dp}{dt}=-a\dot{a} p^2

\dfrac{1}{\vert \mathbf{p}\vert }\dfrac{d\vert \mathbf{p}\vert}{dt}+\dfrac{\dot{a}}{a}=0

Therefore we have deduced that \vert \mathbf{p}\vert \propto a^{-1}. This is, in fact, the socalled “redshift”.  The cosmological  redshift parameter is more generally defined through the equation


B) The Hubble’s law.

The luminosity distance measures the flux of light from a distant object of known luminosity (if it is not expanding). The flux and luminosity distance are bound into a single equation

\boxed{F=\dfrac{L}{4\pi d^2_L}}

If we use the comoving distance between a distant emitter and us, we get

\chi (a)=\int_t^{t_0}\dfrac{dt'}{a(t')}=\int_a^1\dfrac{da'}{a'^2 H(a')}

for a expanding Universe! That is, we have used the fact that luminosity itself goes through a comoving spherical shell of radius \chi (a). Moreover, it shows that

F=\dfrac{L (\chi)}{4 \pi \chi (a)^2 a_0^2}=\dfrac{L}{4\pi (\chi (a)/a)^2}

The luminosity distance in the expanding shell is

d_L=\dfrac{\chi (a)}{a}=\left(\dfrac{L}{4\pi F}\right)^{1/2}

and this is what we MEASURE in Astrophysics/Cosmology. Knowing a(t), we can express the luminosity distance in terms of the redshift. Taylor expansion provides something like this:


where higher order terms are sometimes referred as “statefinder parameters/variables”. In particular, we have




C) Angular diameter distance.

If we know that some object has a known length l, and it gives some angular “aperture” or separation \theta, the angular diameter distance is given by


The comoving size is defined as l/a, and the coming distance is again \chi (a). For “flat” space, we obtain that

\theta=\dfrac{l/a}{\chi (a)}

that is

d_A=a\chi (a)=\dfrac{\chi}{1+z}

In the case of “curved” spaces, we get

d_A=\dfrac{a}{H_0\sqrt{\vert \omega_k\vert}}\cdot\begin{cases}\sinh \left( \sqrt{\Omega_k}H_0\chi\right),\Omega_k>0\\ \sin \left( \sqrt{-\Omega_k}H_0\chi\right),\Omega_k<0\end{cases}

FRW dynamics

Gravity in General Relativity, a misnomer for the (locally) relativistic theory of gravitation, is described by a metric field, i.e., by a second range tensor (covariant tensor if we are purist with the nature of components). The metric field is related to the matter-energy-momentum content through the Einstein’s equations

G_{\mu\nu}=-\kappa^2 T_{\mu\ nu}

The left-handed side can be calculated for a FRW Universe as follows




The right-handed side is the energy-momentum of the Universe. In order to be fully consistent with the symmetries of the metric, the energy-momentum tensor MUST be diagonal and T_{11}=T_{22}=T_{33}=T. In fact, this type of tensor describes a perfect fluid with

T_{\mu\nu}=(\rho+p)U_\mu U_\nu+pg_{\mu\nu}

Here, \rho, p are functions of t (cosmological time) only. They are “state variables” somehow. Moreover, we have

U_\mu =(1,0,0,0)

for the fluid at rest in the comoving frame. The Friedmann equations are indeed the EFE for a FRW metric Universe

3\left(\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}\right)=\kappa^2\rho for the 00th compoent as “constraint equation.

2\dfrac{\ddot{a}}{a}+\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}=-\kappa^2p for the iith components.

Moreover, we also have


and this conservation law implies that


Therefore, we have got two independent equations for three unknowns (a, \rho, p). We need an additional equation. In fact, the equation of state for p=p(\rho) provides such an additional equation. It gives the “dynamics of matter”!

In summary, the basic equations for Cosmology in a FRW metric, via EFE, are the Friedmann’s equations (they are secretly the EFE for the FRW metric) supplemented with the energy-momentum conservations law and the equation of state for the pressure p=p(\rho):

1) \boxed{\dfrac{\dot{a}^2}{a^2}+\dfrac{k^2}{a^2}=\dfrac{\kappa^2}{3}\rho}

2) \boxed{\dot{\rho}+3\dfrac{\dot{a}}{a}(\rho+p)=0}

3) \boxed{p=p(\rho)}

There are many kinds of “matter-energy” content of our interest in Cosmology. Some of them can be described by a simple equation of state:

\boxed{p=\omega \rho}

Energy-momentum conservation implies that \rho\propto a^{-3(\omega +1)}. 3 special cases are used often:

1st. Radiation (relativistic “matter”). \omega=1/3 and thus, p=1/3\rho and \rho\propto a^{-4}

2nd. Dust (non-relativistic matter). \omega=0. Then, p=0 and \rho\propto a^{-3}

3rd. Vacuum energy (cosmological constant). \omega=-1. Then, p=-\rho and \rho=\mbox{constant}

Remark (I): Particle physics enters Cosmology here! Matter dynamics or matter fields ARE the matter content of the Universe.

Remark (II): Existence of a Big Bang (and a spacetime singularity). Using the Friedmann’s equation


if we have that (\rho+3p)>0, the so-called weak energy condition, then a=0 should have been reached at some finite time in the past! That is the “Big Bang” and EFE are “singular” there. There is no scape in the framework of GR. Thus, we need a quantum theory of gravity to solve this problem OR give up the FRW metric at the very early Universe by some other type of metric or structure.

Particles and the chemical equilibrium of the early Universe

Today, we have DIRECT evidence for the existence of a “thermal” equilibrium in the early Universe: the cosmic microwave background (CMB). The CMB is an isotropic, accurate and non-homogeneous (over certain scales) blackbody spectrum about T\approx 3K!

Then, we know that the early Universe was filled with a hot dieal gas in thermal equilibrium (a temperature T_e can be defined there) such as the energy density and pressure can be written in terms of this temperature. This temperature generates a distribution f(\mathbf{x},\mathbf{p}). The number of phase space elements in d^3xd^3p is


and where the RHS is due to the uncertainty principle. Using homogeneity, we get that, indeed, f(x,p)=f(p), and where we can write the volume d^3x=dV. The energy density and the pressure are given by (natural units are used)

\rho_i=g_i\int \dfrac{d^3p}{(2\pi)^3}f_i(p)E(p)

p_i=g_i\int \dfrac{d^3p}{(2\pi)^3}f_i (p)\dfrac{p^2}{3E(p)}

When we are in the thermal equilibrium at temperature T, we have the Bose-Einstein/Fermi-Dirac distribution

f(p)=\dfrac{1}{e^{(E-\mu)/T}\pm 1}

and where the + is for the Fermi-Dirac distribution (particles) and the - is for the Bose-Einstein distribution (particles). The number density, the energy density and the pressure are the following integrals

\boxed{\mbox{Number density}=n=\dfrac{N}{V}=\dfrac{g}{2\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{1/2}}{e^{(E-\mu)/T}\pm 1}dE}

\boxed{\mbox{Density energy}=\rho=\dfrac{E}{V}=\dfrac{g}{2\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{1/2}E^2}{e^{(E-\mu)/T}\pm 1}dE}

\boxed{\mbox{Pressure}=p=\dfrac{g}{6\pi^2}\int_m^\infty \dfrac{(E^2-m^2)^{3/2}}{e^{(E-\mu)/T}\pm 1}dE}

And now, we find some special cases of matter-energy for the above variables:

1st. Relativistic, non-degenerate matter (e.g. the known neutrino species). It means that T>>m and T>>\mu. Thus,

n=\left(\dfrac{3}{4}\right)\dfrac{\zeta (3)}{\pi^2}gT^3



2nd. Non-relativistic matter with m>>T only. Then,


\rho= mn+\dfrac{3}{2}p, and p=nT<<\rho

The total energy density is a very important quantity. In the thermal equilibrium, the energy density of non-relativistic species is exponentially smaller (suppressed) than that of the relativistic particles! In fact,

\rho_R=\dfrac{\pi^2}{30}g_\star T^4 for radiation with p_R=\dfrac{1}{3}\rho_R

and the effective degrees of freedom are


Remark: The factor 7/8 in the DOF and the variables above is due to the relation between the Bose-Einstein and the Fermi-Dirac integral in d=3 space dimensions. In general d, the factor would be


Entropy conservation and the early Universe

The entropy in a comoving volume IS a conserved quantity IN THE THERMAL EQUILIBRIUM. Therefore, we have that

\dfrac{\partial p_i}{\partial T}=g_i\int \dfrac{d^3p}{(2\pi)^3}\dfrac{df}{dT}\dfrac{p^2}{3E(p)}=g_i\int \dfrac{4\pi pE dE}{(2\pi)^3}\dfrac{df}{dE}\left(-\dfrac{E}{T}\right)\dfrac{p^2}{3E}

and then

\dfrac{\partial p_i}{\partial T}=\dfrac{g_i}{2\pi^2}\int \left(-\dfrac{d}{dE}\left(f\dfrac{p^3E}{3T}\right)+f\dfrac{d}{dE}\left(\dfrac{p^3E}{3T}\right)\right)dE


\dfrac{\partial p_i}{\partial T}=\dfrac{1}{T}(\rho_i+p_i)

Now, since

\dfrac{\partial \rho}{\partial t}+3\dfrac{\dot{a}}{a}(\rho+p)=0


\dfrac{\partial}{\partial t}\left(a^3(\rho+p)\right)-a^3\dfrac{\partial p}{\partial t}=0

\dfrac{1}{a^3}\dfrac{\partial (a^3(\rho +p))}{\partial t}-\dfrac{\partial \rho}{\partial t}=0

if we multiply by T and use the chain rule for \rho, we obtain

\dfrac{1}{a^3}\dfrac{\partial}{\partial t}\left(\dfrac{a^3(\rho+p)}{T}\right)=0

but it means that a^3s=\mbox{constant}, where s is the entropy density defined by

\boxed{s\equiv \dfrac{\rho+p}{T}}

Well, the fact is that we know that the entropy or more precisely the entropy density is the early Universe is dominated by relativistic particles ( this is “common knowledge” in the Stantard Cosmological Model, also called \Lambda CDM). Thus,

\boxed{s=\dfrac{2\pi^2}{45}g_\star T^3}

It implies the evolution of temperature with the redshift in the following way:

T\propto g_\star^{-1/3}a^{-1}

Indeed, since we have that n\propto a^{-3}, s\propto a^{-3}, the yield variable

Y_i\equiv \dfrac{n_i}{s}

is a convenient quantity that represents the “abundance” of decoupled particles.

See you in my next cosmological post!

LOG#057. Naturalness problems.


In this short blog post, I am going to list some of the greatest “naturalness” problems in Physics. It has nothing to do with some delicious natural dishes I like, but there is a natural beauty and sweetness related to naturalness problems in Physics. In fact, they include some hierarchy problems and additional problems related to stunning free values of parameters in our theories.

Naturalness problems arise when the “naturally expected” property of some free parameters or fundamental “constants” to appear as quantities of order one is violated, and thus, those paramenters or constants appear to be very large or very small quantities. That is, naturalness problems are problems of untuning “scales” of length, energy, field strength, … A value of 0.99 or 1.1, or even 0.7 and 2.3 are “more natural” than, e.g., 100000, 10^{-4},10^{122}, 10^{23},\ldots Equivalently, imagine that the values of every fundamental and measurable physical quantity X lies in the real interval \left[ 0,\infty\right). Then, 1 (or very close to this value) are “natural” values of the parameters while the two extrema 0 or \infty are “unnatural”. As we do know, in Physics, zero values are usually explained by some “fundamental symmetry” while extremely large parameters or even \infty can be shown to be “unphysical” or “unnatural”. In fact, renormalization in QFT was invented to avoid quantities that are “infinite” at first sight and regularization provides some prescriptions to assign “natural numbers” to quantities that are formally ill-defined or infinite. However, naturalness goes beyond those last comments, and it arise in very different scenarios and physical theories. It is quite remarkable that naturalness can be explained as numbers/contants/parameters around 3 of the most important “numbers” in Mathematics:

(0, 1, \infty)

REMEMBER: Naturalness of X is, thus, being 1 or close to it, while values approaching 0 or \infty are unnatural.  Therefore, if some day you heard a physicist talking/speaking/lecturing about “naturalness” remember the triple (0,1,\infty) and then assign “some magnitude/constant/parameter” some quantity close to one of those numbers. If they approach 1, the parameter itself is natural and unnatural if it approaches any of the other two numbers, zero or infinity!

I have never seen a systematic classification of naturalness problems into types. I am going to do it here today. We could classify naturalness problems into:

1st. Hierarchy problems. They are naturalness problems related to the energy mass or energy spectrum/energy scale of interactions and fundamental particles.

2nd. Nullity/Smallness problems. These are naturalness problems related to free parameters which are, surprisingly, close to zero/null value, even when we have no knowledge of a deep reason to understand why it happens.

3rd. Large number problems (or hypotheses). This class of problems can be equivalently thought as nullity reciprocal problems but they arise naturally theirselves in cosmological contexts or when we consider a large amount of particles, e.g., in “statistical physics”, or when we face two theories in very different “parameter spaces”. Dirac pioneered these class of hypothesis when realized of some large number coincidences relating quantities appearing in particle physics and cosmology. This Dirac large number hypothesis is also an old example of this kind of naturalness problems.

4th. Coincidence problems. This 4th type of problems is related to why some different parameters of the same magnitude are similar in order of magnitude.

The following list of concrete naturalness problems is not going to be complete, but it can serve as a guide of what theoretical physicists are trying to understand better:

1. The little hierarchy problem. From the phenomenon called neutrino oscillations (NO) and neutrino oscillation experiments (NOSEX), we can know the difference between the squared masses of neutrinos. Furthermore, cosmological measurements allow us to put tight bounds to the total mass (energy) of light neutrinos in the Universe. The most conservative estimations give m_\nu \leq 10 eV or even m_\nu \sim 1eV as an upper bound is quite likely to be true. By the other hand, NOSEX seems to say that there are two mass differences, \Delta m^2_1\sim 10^{-3} and \Delta m^2_2\sim 10^{-5}. However, we don’t know what kind of spectrum neutrinos have yet ( normal, inverted or quasidegenerated). Taking a neutrino mass about 1 meV as a reference, the little hierarchy problem is the question of why neutrino masses are so light when compared with the remaining leptons, quarks and gauge bosons ( excepting, of course, the gluon and photon, massless due to the gauge invariance).

Why is m_\nu << m_e,m_\mu, m_\tau, m_Z,M_W, m_{proton}?

We don’t know! Let me quote a wonderful sentence of a very famous short story by Asimov to describe this result and problem:


2. The gauge hierarchy problem. The electroweak (EW) scale can be generally represented by the Z or W boson mass scale. Interestingly, from this summer results, Higgs boson mass seems to be of the same order of magnitue, more or less, than gauge bosons. Then, the electroweak scale is about M_Z\sim M_W \sim \mathcal{O} (100GeV). Likely, it is also of the Higgs mass  order.  By the other hand, the Planck scale where we expect (naively or not, it is another question!) quantum effects of gravity to naturally arise is provided by the Planck mass scale:

M_P=\sqrt{\dfrac{\hbar c}{8\pi G}}=2.4\cdot 10^{18}GeV=2.4\cdot 10^{15}TeV

or more generally, dropping the 8\pi factor

M_P =\sqrt{\dfrac{\hbar c}{G}}=1.22\cdot 10^{19}GeV=1.22\cdot 10^{16}TeV

Why is the EW mass (energy) scale so small compared to Planck mass, i.e., why are the masses M_{EW}<<M_P so different? The problem is hard, since we do know that EW masses, e.g., for scalar particles like Higgs particles ( not protected by any SM gauge symmetry), should receive quantum contributions of order \mathcal{O}(M_P^2)


3. The cosmological constant (hierarchy) problem. The cosmological constant \Lambda, from the so-called Einstein’s field equations of classical relativistic gravity

\mathcal{R}_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}\mathcal{R}=8\pi G\mathcal{T}_{\mu\nu}+\Lambda g_{\mu\nu}

is estimated to be about \mathcal{O} (10^{-47})GeV^4 from the cosmological fitting procedures. The Standard Cosmological Model, with the CMB and other parallel measurements like large scale structures or supernovae data, agree with such a cosmological constant value. However, in the framework of Quantum Field Theories, it should receive quantum corrections coming from vacuum energies of the fields. Those contributions are unnaturally big, about \mathcal{O}(M_P^4) or in the framework of supersymmetric field theories, \mathcal{O}(M^4_{SUSY}) after SUSY symmetry breaking. Then, the problem is:

Why is \rho_\Lambda^{obs}<<\rho_\Lambda^{th}? Even with TeV or PeV fundamental SUSY (or higher) we have a serious mismatch here! The mismatch is about 60 orders of magnitude even in the best known theory! And it is about 122-123 orders of magnitude if we compare directly the cosmological constant vacuum energy we observe with the cosmological constant we calculate (naively or not) with out current best theories using QFT or supersymmetric QFT! Then, this problem is a hierarchy problem and a large number problem as well. Again, and sadly, we don’t know why there is such a big gap between mass scales of the same thing! This problem is the biggest problem in theoretical physics and it is one of the worst predictions/failures in the story of Physics. However,


4. The strong CP problem/puzzle. From neutron electric dipople measurements, theoretical physicists can calculate the so-called \theta-angle of QCD (Quantum Chromodynamics). The theta angle gives an extra contribution to the QCD lagrangian:

\mathcal{L}_{\mathcal{QCD}}\supset \dfrac{1}{4g_s^2}G_{\mu\nu}G^{\mu\nu}+\dfrac{\theta}{16\pi^2}G^{\mu\nu}\tilde{G}_{\mu\nu}

The theta angle is not provided by the SM framework and it is a free parameter. Experimentally,

\theta <10^{-12}

while, from the theoretical aside, it could be any number in the interval \left[-\pi,\pi\right]. Why is \theta close to the zero/null value? That is the strong CP problem! Once again, we don’t know. Perhaps a new symmetry?


5. The flatness problem/puzzle. In the Stantard Cosmological Model, also known as the \Lambda CDM model, the curvature of the Universe is related to the critical density and the Hubble “constant”:


There, \rho is the total energy density contained in the whole Universe and \rho_c=\dfrac{3H^2}{8\pi G} is the so called critical density. The flatness problem arise when we deduce from cosmological data that:

\left(\dfrac{1}{R^2}\right)_{data}\sim 0.01

At the Planck scale era, we can even calculate that

\left(\dfrac{1}{R^2}\right)_{Planck\;\; era}\sim\mathcal{O}(10^{-61})

This result means that the Universe is “flat”. However, why did the Universe own such a small curvature? Why is the current curvature “small” yet? We don’t know. However, cosmologists working on this problem say that “inflation” and “inflationary” cosmological models can (at least in principle) solve this problem. There are even more radical ( and stranger) theories such as varying speed of light theories trying to explain this, but they are less popular than inflationary cosmologies/theories. Indeed, inflationary theories are popular because they include scalar fields, similar in Nature to the scalar particles that arise in the Higgs mechanism and other beyond the Standard Model theories (BSM). We don’t know if inflation theory is right yet, so


6. The flavour problem/puzzle. The ratios of successive SM fermion mass eigenvalues ( the electron, muon, and tau), as well as the angles appearing in one gadget called the CKM (Cabibbo-Kobayashi-Maskawa) matrix, are roughly of the same order of magnitude. The issue is harder to know ( but it is likely to be as well) for constituent quark masses. However, why do they follow this particular pattern/spectrum and structure? Even more, there is a mysterious lepton-quark complementarity. The analague matrix in the leptonic sector of such a CKM matrix is called the PMNS matrix (Pontecorvo-Maki-Nakagawa-Sakata matrix) and it describes the neutrino oscillation phenomenology. It shows that the angles of PMNS matrix are roughly complementary to those in the CKM matrix ( remember that two angles are said to be complementary when they add up to 90 sexagesimal degrees). What is the origin of this lepton(neutrino)-quark(constituent) complementarity? In fact, the two questions are related since, being rough, the mixing angles are related to the ratios of masses (quarks and neutrinos). Therefore, this problem, if solved, could shed light to the issue of the particle spectrum or at least it could help to understand the relationship between quark masses and neutrino masses. Of course, we don’t know how to solve this puzzle at current time. And once again:


7. Cosmic matter-dark energy coincidence. At current time, the densities of matter and vacuum energy are roughly of the same order of magnitude, i.e, \rho_M\sim\rho_\Lambda=\rho_{DE}. Why now? We do not know!


And my weblog is only just beginning! See you soon in my next post! 🙂

LOG#056. Gravitational alpha(s).


The topic today is to review a beautiful paper and to discuss its relevance for theoretical physics. The paper is: Comment on the cosmological constant and a gravitational alpha by R.J.Adler. You can read it here: http://arxiv.org/abs/1110.3358

One of the most intriguing and mysterious numbers in Physics is the electromagnetic fine structure constant \alpha_{EM}. Its value is given by

\alpha_{EM}=7.30\cdot 10^{-3}

or equivalenty


Of course, I am assuming that the coupling constant is measured at ordinary energies, since we know that the coupling constants are not really constant but they vary slowly with energy. However, I am not going to talk about the renormalization (semi)group in this post.

Why is the fine structure constant important? Well, we can undertand it if we insert the values of the constants that made the electromagnetic alpha constant:

\alpha_{EM}=\dfrac{e^2}{\hbar c}

with e being the electron elemental charge, \hbar the Planck’s constant divided by two pi, c is the speed of light and where we are using units with K_C=\dfrac{1}{4\pi \varepsilon_0}=1. Here K_C is the Coulomb constant, generally with a value 9\cdot 10^9Nm^2/C^2, but we rescale units in order it has a value equal to the unit. We will discuss more about frequently used system of units soon.

As the electromagnetic alpha constant depends on the electric charge, the Coulomb’s electromagnetic constant ( rescaled to one in some “clever” units), the Planck’s constant ( rationalized by 2\pi since \hbar=h/2\pi) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha \alpha_{EM} is quantum and relativistic itself, and it also is related to elemental charges. Why alpha has the value it has is a complete mystery. Many people has tried to elucidate why it has the value it has today, but there is no reason of why it should have the value it has. Of course, it happens as well with some other constants but this one is particularly important since it is involved in some important numbers in atomic physics and the most elemental atom, the hydrogen atom.

In atomic physics, there are two common and “natural” scales of length. The first scale of length is given by the Compton’s wavelength of electrons. Usint the de Broglie equation, we get that the Compton’s wavelength is the wavelength of a photon whose energy is the same as the rest mass of the particle, or mathematically speaking:


Usually, physicists employ the “reduced” or “rationalized” Compton’s wavelength. Plugging the electron mass, we get the electron reduced Compton’s wavelength:

\boxed{\lambda_C=\dfrac{\lambda}{2\pi}=\dfrac{\hbar}{m_ec}=\dfrac{\hbar}{m_ec}=3.86\cdot 10^{-13}m}

The second natural scale of length in atomic physics is the so-called Böhr radius. It is given by the formula:

\boxed{a_B=\dfrac{\hbar^2}{m_e e^2}=5.29\cdot 10^{-11}m}

Therefore, there is a natural mass ratio between those two length scales, and it shows that it is precisely the electromagnetic fine structure constant alpha \alpha_{EM}:

\boxed{R_\alpha=\dfrac{\mbox{Reduced Compton's wavelength}}{\mbox{B\"{o}hr radius}}=\dfrac{\lambda_C}{a_B}=\dfrac{\left(\hbar/m_e c\right)}{\left(\hbar^2/m_ee^2\right)}=\dfrac{e^2}{\hbar c}=\alpha_{EM}=7.30\cdot 10^{-3}}

Furthermore, we can show that the electromagnetic alpha also is related to the mass ration between the electron energy in the fundamental orbit of the hydrogen atom and the electron rest energy. These two scales of energy are given by:

1) Rydberg’s energy ( electron ground minimal energy in the fundamental orbit/orbital for the hydrogen atom):


2) Electron rest energy:


Then, the ratio of those two “natural” energies in atomic physics reads:

\boxed{R'_E=\dfrac{\mbox{Rydberg's energy}}{\mbox{Electron rest energy}}=\dfrac{m_ee^4/2\hbar^2}{m_ec^2}=\dfrac{1}{2}\left(\dfrac{e^2}{\hbar c}\right)^2=\dfrac{\alpha_{EM}^2}{2}=2.66\cdot 10^{-5}}

or equivalently

\boxed{\dfrac{1}{R'_E}=37600=3.76\cdot 10^4}

R.J.Adler’s paper remarks that there is a cosmological/microscopic analogue of the above two ratios, and they involve the infamous Einstein’s cosmological constant. In Cosmology, we have two natural (ultimate?) length scales:

1st. The (ultra)microscopic and ultrahigh energy (“ultraviolet” UV regulator) relevant Planck’s length L_P, or equivalently the squared value L_P^2. Its value is given by:

\boxed{L_P^2=\dfrac{G\hbar}{c^3}\leftrightarrow L_P=\sqrt{\dfrac{G\hbar}{c^3}}=1.62\cdot 10^{35}m}

This natural length can NOT be related to any “classical” theory of gravity since it involves and uses the Planck’s constant \hbar.

2nd. The (ultra)macroscopic and ultra-low-energy (“infrared” IR regulator) relevant cosmological constant/deSitter radius. They are usualy represented/denoted by \Lambda and R_{dS} respectively, and they are related to each other in a simple way. The dimensions of the cosmological constant are given by

\boxed{\left[\Lambda \right]=\left[ L^{-2}\right]=(\mbox{Length})^{-2}}

The de Sitter radius and the cosmological constant are related through a simple equation:

\boxed{R_{dS}=\sqrt{\dfrac{3}{\Lambda}}\leftrightarrow R^2_{dS}=\dfrac{3}{\Lambda}\leftrightarrow \Lambda =\dfrac{3}{R^2_{dS}}}

The de Sitter radius is obtained from cosmological measurements thanks to the so called Hubble’s parameter ( or Hubble’s “constant”, although we do know that Hubble’s “constant” is not such a “constant”, but sometimes it is heard as a language abuse) H. From cosmological data we obtain ( we use the paper’s value without loss of generality):


This measured value allows us to derive the Hubble’s length paremeter

L_H=\dfrac{c}{H}=1.27\cdot 10^{26}m

Moreover, the data also imply some density energy associated to the cosmological “constant”, and it is generally called Dark Energy. This density energy from data is written as:

\Omega_\Lambda =\Omega^{data}_{\Lambda}

and from this, it can be also proved that

R_{dS}=\dfrac{L_H}{\sqrt{\Omega_\Lambda}}=1.46\cdot 10^{26}m

where we have introduced the experimentally deduced value \Omega_\Lambda\approx 0.76 from the cosmological parameter global fits. In fact, the cosmological constant helps us to define the beautiful and elegant formula that we can call the gravitational alpha/gravitational cosmological fine structure constant \alpha_G:

\boxed{\alpha_G\equiv \dfrac{\mbox{Planck's length}}{\mbox{normalized de Sitter radius}}=\dfrac{L_P}{\dfrac{R_{dS}}{\sqrt{3}}}=\dfrac{\sqrt{\dfrac{G\hbar}{c^3}}}{\sqrt{\dfrac{1}{\Lambda}}}=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}}

or equivalently, defining the cosmological length associated to the cosmological constant as

L^2_\Lambda=\dfrac{1}{\Lambda}=\dfrac{R^2_{dS}}{3}\leftrightarrow L_\Lambda=\sqrt{\dfrac{1}{\Lambda}}=\dfrac{R_{dS}}{\sqrt{3}}

\boxed{\alpha_G\equiv \dfrac{\mbox{Planck's length}}{\mbox{Cosmological length}}=\dfrac{L_P}{L_\Lambda}=\dfrac{\sqrt{\dfrac{G\hbar}{c^3}}}{\sqrt{\dfrac{1}{\Lambda}}}=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}=L_P\sqrt{\Lambda}=L_P\dfrac{R_{dS}}{\sqrt{3}}}

If we introduce the numbers of the constants, we easily obtaint the gravitational cosmological alpha value and its inverse:

\boxed{\alpha_G=1.91\cdot 10^{-61}\leftrightarrow \alpha_G^{-1}=\dfrac{1}{\alpha_G}=5.24\cdot 10^{60}}

They are really small and large numbers! Following the the atomic analogy, we can also create a ratio between two cosmologically relevant density energies:

1st. The Planck’s density energy.

Planck’s energy is defined as

\boxed{E_P=\dfrac{\hbar c}{L_P}=\sqrt{\dfrac{\hbar c^5}{G}}=1.22\cdot 10^{19}GeV=1.22\cdot 10^{16}TeV}

The Planck energy density \rho_P is defined as the energy density of Planck’s energy inside a Planck’s cube or side L_P, i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume V=L_P^3. Mathematically speaking, it is

\boxed{\rho_P=\dfrac{E_P}{L_P^3}=\dfrac{c^7}{\hbar G^2}=2.89\cdot 10^{123}\dfrac{GeV}{m^3}}

It is an huge density energy!

Remark: Energy density is equivalent to pressure in special relativity hydrodynamics. That is,

\mathcal{P}_P=\rho_P=\tilde{\rho}_P c^2=4.63\cdot 10^{113}Pa

wiht Pa denoting pascals (1Pa=1N/m^2) and where \tilde{\rho}_P represents here matter (not energy) density ( with units in kg/m^3). Of course, turning matter density into energy density requires a multiplication by c^2. This equivalence between vacuum pressure and energy density is one of the reasons because some astrophysicists, cosmologists and theoretical physicists call “vacuum pressure” to the “dark energy/cosmological constant” term in the study of the cosmic components derived from the total energy density \Omega.

2nd. The cosmological constant density energy.

Using the Einstein’s field equations, it can be shown that the cosmological constant gives a contribution to the stress-energy-momentum tensor. The component T^{0}_{\;\; 0} is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density

\boxed{\rho_\Lambda =T^{0}_{\;\; 0}=\dfrac{\Lambda c^4}{8\pi G}}

Using the previous equations for G as a function of Planck’s length, the Planck’s constant and the speed of light, and the definitions of Planck’s energy and de Sitter radius, we can rewrite the above energy density as follows:

\boxed{\rho_\Lambda=\dfrac{3}{8\pi}\left(\dfrac{E_P}{L_PR^2_{dS}}\right)=4.21 \dfrac{GeV}{m^3}}

Thus, we can evaluate the ration between these two energy densities! It provides

\boxed{R_\rho =\dfrac{\mbox{Planck's energy density}}{\mbox{CC energy density}}=\dfrac{\rho_P}{\rho_\Lambda}=\left( \dfrac{3}{8\pi}\right)\left(\dfrac{L_P}{R_{dS}}\right)^2=\left(\dfrac{1}{8\pi}\right)\alpha_G^2=1.45\cdot 10^{-123}}

and the inverse ratio will be

\boxed{\dfrac{1}{R_\rho}=6.90\cdot 10^{122}}

So, we have obtained two additional really tiny and huge values for R_\rho and its inverse, respectively. Note that the power appearing in the ratios of cosmological lengths and cosmological energy densities match the same scaling property that the atomic case with the electromagnetic alpha! In the electromagnetic case, we obtained R\sim \alpha_{EM} and R_E\sim \alpha_{EM}^2. The gravitational/cosmological analogue ratios follow the same rule R\sim \alpha_G and R_\rho\sim \alpha_G^2 but the surprise comes from the values of the gravitational alpha values and ratios. Some comments are straightforward:

1) Understanding atomic physics involved the discovery of Planck’s constant and the quantities associated to it at fundamental quantum level ( Böhr radius, the Rydberg’s constant,…). Understanding the Cosmological Constant value and the mismatch or stunning ratios between the equivalent relevant quantities, likely, require that \Lambda can be viewed as a new “fundamental constant” or/and it can play a dynamical role somehow ( e.g., varying in some unknown way with energy or local position).

2) Currently, the cosmological parameters and fits suggest that \Lambda is “constant”, but we can not be totally sure it has not varied slowly with time. And there is a related idea called quintessence, in which the cosmological “constant” is related to some dynamical field and/or to inflation. However, present data say that the cosmological constant IS truly constant. How can it be so? We are not sure, since our physical theories can hardly explain the cosmological constant, its value, and why it is current density energy is radically different from the vacuum energy estimates coming from Quantum Field Theories.

3) The mysterious value

\boxed{\alpha_G=\sqrt{\dfrac{G\hbar\Lambda}{c^3}}=1.91\cdot 10^{-61}}

is an equivalent way to express the biggest issue in theoretical physics. A naturalness problem called the cosmological constant problem.

In the literature, there have been alternative definitions of “gravitational fine structure constants”, unrelated with the above gravitational (cosmological) fine structure constant or gravitational alpha. Let me write some of these alternative gravitational alphas:

1) Gravitational alpha prime. It is defined as the ratio between the electron rest mass and the Planck’s mass squared:

\boxed{\alpha'_G=\dfrac{Gm_e^2}{\hbar c}=\left(\dfrac{m_e}{m_P}\right)^2=1.75\cdot 10^{-45}}

\boxed{\alpha_G^{'-1}=\dfrac{1}{\alpha_G^{'}}=5.71\cdot 10^{44}}

Note that m_e=0.511MeV. Since m_{proton}=1836m_e, we can also use the proton rest mass instead of the electron mass to get a new gravitational alpha.

2) Gravitational alpha double prime. It is defined as the ratio between the proton rest mass and the Planck’s mass squared:

\boxed{\alpha''_G=\dfrac{Gm_{prot}^2}{\hbar c}=\left(\dfrac{m_{prot}}{m_P}\right)^2=5.90\cdot 10^{-39}}

and the inverse value

\boxed{\alpha_G^{''-1}=\dfrac{1}{\alpha_G^{''}}=1.69\cdot 10^{38}}

Finally, we could guess an intermediate gravitational alpha, mixing the electron and proton mass.

3) Gravitational alpha triple prime. It is defined as the ration between the product of the electron and proton rest masses with the Planck’s mass squared:

\boxed{\alpha'''_G=\dfrac{Gm_{prot}m_e}{\hbar c}=\dfrac{m_{prot}m_e}{m_P^2}=3.22\cdot 10^{-42}}

and the inverse value

\boxed{\alpha_G^{'''-1}=\dfrac{1}{\alpha^{'''}_G}=3.11\cdot 10^{41}}

We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with \alpha_{EM}. We get

\alpha_G <\alpha_G^{'} <\alpha_G^{'''} < \alpha_G^{''}<\alpha_{EM}

\alpha_{EM}^{-1}<\alpha^{''-1}_G <\alpha^{'''-1}_G <\alpha^{'-1}_G < \alpha^{-1}_G

These inequations mean that the electromagnetic fine structure constant \alpha_{EM} is (at ordinary energies) 42 orders of magnitude bigger than \alpha_G^{'}, 39 orders of magnitude bigger than \alpha_G^{'''}, 36 orders of magnitude bigger than \alpha_G^{''} and, of course, 58 orders of magnitude bigger than \alpha_G. Indeed, we could extend this analysis to include the “fine structure constant” of Quantum Chromodynamics (QCD) as well. It would be given by:

\boxed{\alpha_s=\dfrac{g_s^2}{\hbar c}=1}

since generally we define g_s=1. We note that \alpha_s >\alpha_{EM} by 3 orders of magnitude. However, as strong nuclear forces are short range interactions, they only matter in the atomic nuclei, where confinement, and color forces dominate on every other fundamental interaction. Interestingly, at high energies, QCD coupling constant has a property called asymptotic freedom. But it is another story not to be discussed here! If we take the alpha strong coupling into account the full hierarchy of alphas is given by:

\alpha_G <\alpha_G^{'} <\alpha_G^{'''} < \alpha_G^{''}<\alpha_{EM}<\alpha_s

\alpha_s^{-1}<\alpha_{EM}^{-1}<\alpha^{''-1}_G <\alpha^{'''-1}_G <\alpha^{'-1}_G < \alpha^{-1}_G

Fascinating! Isn’t it? Stay tuned!!!

ADDENDUM: After I finished this post, I discovered a striking (and interesting itself) connection between \alpha_{EM} and \alpha_{G}. The relation or coincidence is the following relationship

\dfrac{1}{\alpha_{EM}}\approx \ln \left( \dfrac {1}{16\alpha_G}\right)

Is this relationship fundamental or accidental? The answer is unknown. However, since the electric charge (via electromagnetic alpha) is not related a priori with the gravitational constant or Planck mass ( or the cosmological constant via the above gravitational alpha) in any known way I find particularly stunning such a coincidence up to 5 significant digits! Any way, there are many unexplained numerical coincidences that are completely accidental and meaningless, and then, it is not clear why this numeral result should be relevant for the connection between electromagnetism and gravity/cosmology, but it is interesting at least as a curiosity and “joke” of Nature.


Some quotes about the electromagnetic alpha from wikipedia http://en.wikipedia.org/wiki/Fine-structure_constant

“(…)There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! (…)”. R.P.Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, p.129.

“(…) If alpha [the fine-structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.(…)” Max Born, in A.I. Miller’s book Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. p. 253. Publisher W.W. Norton & Co.(2009).

“(…)The mystery about α is actually a double mystery. The first mystery – the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.(…)” Malcolm H. Mac Gregor, M.H. MacGregor (2007). The Power of Alpha.