# LOG#111. Basic Cosmology (VI).

**Posted:**2013/06/23

**Filed under:**Cosmology, Physmatics |

**Tags:**10-folds, cosmological constant problem, e-folds, flatness problem, horizon problem, inflation, Issues of Big Bang Cosmology, negative pressure, phantom energy, quintessence, scalar field Leave a comment

The topic today: problems in the Standard Cosmological Model (LCDM), inflation and scalar fields!

## STANDARD COSMOLOGICAL MODEL: ISSUES

Despite the success of the Standard Cosmological model (or LCDM), today it is widely accepted that it is not complete, even if its main features and observables are known, there are some questions we can not understand in that framework.

Firstly, we have the **horizon problem. **For any comoving horizon, we obtain

Note that for a MD Universe and to for a RD Universe.

According to the CMB observations, today our Universe is very close to be isotropic, since the deviations are tiny

The issue, of course, is how can it be true?Recall that the largest scales observed today have entered theo horizon just recently, long after decoupling. Moreover, microscopic causal physics can not make it! So, where the above tiny anisotropy comes from? E.g., for a distant galaxy, we get

and this scale was outside the horizon in the past! Please, note that the entropy within a horizon volume is about

and so

Another problem is the **flatness problem. **The physical radius of curvature is given by the term

The total energy density as a function of the scale factor is given by

When the primordial nucleosynthesis happened, about , it gave and it gives . By the other hand, at the Planck time, i.e., if , it gives the value , and then . This large mismatch means that the Big Bang Universe requires VERY SPECIAL initial conditions, otherwise the Universe would not be (apparently) flat as we observe at current time. Note that if and at Planck time, then there are two options:

1) k>0: the Universe recollapse withim few .

2) k<0: the Universe reaches 3K at .

Therefore, the natural time scale for cosmology is the Planck time (at least in the very early Universe and before) . The current age of the Universe is about .

An important, yet unsolved and terrific, problem is the **cosmological constant problem. **The vacuum energy we observe today (in the form of dark energy) is given by

The equation of state for the vacuum energy is known to be , i.e., . It yields

i.e., the so called de Sitter space (dS) or de Sitter Universe. It is a maximally symmetric spacetime. In a dS Universe, we obtain

Observations provide (via dark energy) that

Theoretical (and “natural”) Quantum Field Theory (QFT) calculations give

or so( even the mismatch can be 122 or 123 orders of magnitude!!!!). This problem is far beyond our current knowledge of QFT. It (likely) requires new physics or to rethink QFT and/or the observed value of the vacuum energy density. It is a hint that our understanding of the Universe is not complete.

## INFLATION

One of the most “simple” and elegant solutions to the flatness problem and the horizon problem is the inflationary theory. What is inflation? Let me explain it here better. If the (early) Universe experienced an stage of “very fast” expansion, we can solve the horizon problem! However, there is a problem (you can call Houston if you want to…). If we do want a quick expansion in the early Universe, we need “negative pressure” to realize that scenario. Negative pressure can be obtained in an scalar field theory (there are some alternatives with a repulsive vector field and/or higher order tensors like a 3-form antisymmetric field, but the most simple solution is given by scalar fields).

The solution to the horizon problem in the inflationary theory proceeds as follows. Firstly, for the comoving horizon, we get

where the disance over which particles can travel in the course of one expansion time is the comoving Hubble radius , and it is encoded into the fraction

We have to make a distinction of the comoving horizon and the comoving Hubble radius . If we observe two particles whose comoving distance is r, then

1st. If , we can never have a communication between those particles.

2nd. If , then we can never communicate NOW using those two particles.

It shows that it is possible to have . A particle with ca not communicate today BUT they could be in causal contact early on. We only need that . That is, get contributions mostly from early epochs. Indeed, both in RD (Radiation Dominated) and MD(Matter Dominated) Universes, increase with time, so the latter epoch contributions dominate over cosmological time scales. Then, a solution for the horizon problem is that in the early Universe, in this “inflationary” (very fast) phase, for at least a brief amount of time (how much is not clear) the comoving Hubble radius DECREASED.

How can the scale factor evolve in order to solve the horizon problem?We do know that must decrease, so must increase! Therefore,

i.e., we get an (positively) accelerating expansion or **“inflation”. **

How can we understand quantitatively inflation? Firstly, suppose that the energy scale of inflation is about . It is about the Grand Unified Theory (GUT) energy scale, and close to the Quantum Gravity scale (the Planck scale) about . Obviously, it only matters at very high energies, very short distances or very tiny time scales after the Big Bang. Then,

For a RD Universe, and thus

During the inflation, the comoving Hubble radius had to decrease by, at least, 28 orders of magnitude. The most common way to build such inflationary model is to build a model where and

It gives and , where is the time when inflation ends. In fact, we obtain that

and thus can be guessed from the so-called “e-folds”, or exponential factors, in a very simple way: note that !!!!!!!!! Therefore, more than 64 “e-folds” (or the 64th power of the number “e”) provide the necessary 10-fold we were searching for.

**Remark:** The comoving horizon is very similar to an effective time parameter!

## NEGATIVE PRESSURE

In order to allow inflation, we require that the “weak energy condition” be violated. That is, we ask that (for inflation to be possible)

It implies that or that

Obviously, there is no particle/field in the Standard Model (at least until the discovery of the Higgs field in 2012) able to do that. So, if inflation happened, it is an unknown field/particle responsible for it! The simplest implementation of inflation uses, as we said before, some scalar field . Let us define a scalar field . Its lagrangian is generally given by

The energy-momentum tensor for this scalar field can be easily obtained by the well-known prescription

Thus, we get

Negative pressure is obtained whenever we have

Therefore:

1st. A scalar field with negative pressure is trapped into a “false vacuum”.

2nd. A scalar field “slow-rolling” toward its true vacuum provides a simple model for inflation.

The evolution of the scalar field in the expanding Universe is given by a simple equation

Models where the scalar field “slow-rolls” provide (slow varying in time as a scalar function!). The time variable implies that

and it becomes negative during the inflationary phase! The slow-roll parameters

where the dot means derivative with respect to . Moreover, we also get

**Remark:** for inflation and for RD Universes imply that is a definition of inflationary phase!

## Quintessence and phantom energy

For any scalar field, and the pressure and energy density given above, we can calculate the quantity:

In the case that the kinetic term is negligible, we obtain the dark energy fluid value . However, this equation is much more general. If we have a slow-rolling scalar field over cosmological times (a very slowly time dependent scalar field) it could mimic the cosmological constant behaviour ( we are ignoring some technical problems here). If the kinetic term is NOT negligible, the value can differ from the standard value of (dark energy/cosmological constant/vacuum energy). However, current observation support a pure cosmological constant term. Moreover, this general scalar field is commonly referred as **“quintessence”** if and as **“phantom energy”** if . Both, dark energy, quintessence or phantom energy models can affect to the future of the Universe. Instead of a Big Freeze (the thermal death of the Universe), the scalar dominated Universe can even destroy atoms/matter/galaxies at some point in the future (at least on very general grounds) and terminate the Universe in a Big (or Little) Rip. Thus, let me review the possible destinies of the Universe according to modern Cosmology:

**1) Big Crunch** (or recollapse of the Universe). The Universe recollapses until the initial singularity after some time, if it is massive enough. Current observations don’t favour this case.

**2) Big Freeze** (or thermal death of the Universe). The Universe expands forever cooling itself until it reaches a temperature close to the absolute zero. It was believed that it was the only possible option with the given curvature until the discovery of dark energy in 1998.

**3) Big Rip** (or Little Rip, depending on the nature of the scalar field and the concrete model). Vacuum energy expands the Universe with an increasing rate until it “rips” even fundamental particles/atoms/matter and galaxies apart from eath other. It is a new possibility due to the existence of scalar fields and/or dark energy, a mysterious energy that makes the Universe expanss with an increasing rate overseding the gravitational pull of galaxies and clusters!

A more exotic option is that the Universe suffers “oscillations” of positively accelerated expansion and negatively accelerated expansion (oscillatory/cyclic ethernal Universes)…But that is another story…

See you in my final basic cosmological post!

# LOG#057. Naturalness problems.

**Posted:**2012/12/02

**Filed under:**Physmatics, Quantum Gravity, The Standard Model: Basics |

**Tags:**CKM matrix, cosmic coincidence, cosmological constant, cosmological constant problem, critical energy density, curvature, dark energy, dark energy density, Dirac large number hypothesis, electroweak scale, energy, energy density, flatness problem, flavour problem, gauge hierarchy problem, Higgs boson, Higgs mechanism, Hubble constant, inflation, inflationary cosmologies, little hierarchy problem, mass, matter density, naturalness, naturalness problem, neutrino mass hierarchy, neutrino masses, neutrino oscillations, NO, NOSEX, parameter, parameter space, Planck era, Planck scale, PMNS matrix, QCD, QFT, quark-lepton complementarity, SM, Standard Cosmological Model, Standard Model, strong CP problem, theta term, types of naturalness, vacuum, vacuum energy, W boson, Z boson 8 Comments

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., Equivalently, imagine that the values of every fundamental and measurable physical quantity lies in the real interval . Then, 1 (or very close to this value) are “natural” values of the parameters while the two extrema or are “unnatural”. As we do know, in Physics, zero values are usually explained by some “fundamental symmetry” while extremely large parameters or even 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:

**REMEMBER: Naturalness** of X is, thus, being 1 or close to it, while values approaching 0 or are unnatural. Therefore, if some day you heard a physicist talking/speaking/lecturing about “naturalness” remember the triple 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 or even as an upper bound is quite likely to be true. By the other hand, NOSEX seems to say that there are two mass differences, and . 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

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

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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 . 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:

or more generally, dropping the factor

Why is the EW mass (energy) scale so small compared to Planck mass, i.e., why are the masses 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

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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

is estimated to be about 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 or in the framework of supersymmetric field theories, after SUSY symmetry breaking. Then, the problem is:

Why is ? 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,

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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

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

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

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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

There, is the total energy density contained in the whole Universe and is the so called critical density. The flatness problem arise when we deduce from cosmological data that:

At the Planck scale era, we can even calculate that

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

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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:

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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, . Why now? We do not know!

*“THERE* IS AS YET INSUFFICIENT *DATA* FOR A MEANINGFUL *ANSWER*.”

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

# LOG#056. Gravitational alpha(s).

**Posted:**2012/11/29

**Filed under:**Cosmology, Physmatics, Quantum Gravity, Relativity |

**Tags:**alpha, alpha strong, asymptotic freedom, atomic physics, confinement, cosmological constant, cosmological constant problem, cosmological gravitational alpha, cosmological parameter fitting, cosmological parameters, Cosmology, coupling constant, de Sitter radius, Einstein's field equations, energy density, energy density ratios, energy ratios, fine structure constant, gravitational alpha, gravitational constant, gravitational fine structure constant, Hubble parameter, Hubble's length, length ratios, naturalness problem, Planck's energy, Planck's length, QCD, QFT, quantum field theory, quantum theory, ratios, Relativity Leave a comment

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 . Its value is given by

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:

with being the electron elemental charge, the Planck’s constant divided by two pi, c is the speed of light and where we are using units with . Here is the Coulomb constant, generally with a value , 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 since ) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha 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:

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

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 :

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:

or equivalently

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** , or equivalently the squared value . Its value is given by:

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

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

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

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

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:

and from this, it can be also proved that

where we have introduced the experimentally deduced value 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 **:

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

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

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

The Planck energy density is defined as the energy density of Planck’s energy inside a Planck’s cube or side , i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume . Mathematically speaking, it is

It is an huge density energy!

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

wiht Pa denoting pascals () and where represents here matter (not energy) density ( with units in ). Of course, turning matter density into energy density requires a multiplication by . 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 .

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 is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density

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:

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

and the inverse ratio will be

So, we have obtained two additional really tiny and huge values for 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 and . The gravitational/cosmological analogue ratios follow the same rule and 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 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 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

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:

Note that . Since , 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:

and the inverse value

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:

and the inverse value

We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with . We get

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

since generally we define . We note that 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:

**Fascinating!** Isn’t it?** Stay tuned!!!**

*ADDENDUM:* After I finished this post, I discovered a striking (and interesting itself) connection between and . The relation or coincidence is the following relationship

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.

*ADDENDUM (II):
*

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.*