# LOG#106. Basic Cosmology (I).

**Posted:**2013/05/26

**Filed under:**Cosmology, General Relativity, Physmatics |

**Tags:**Big Bang, Bose-Einstein distribution, cosmic microwave background, Cosmological principle, Cosmology, curvature parameter, curved Universe, dark energy, degrees of freedom, dust, early Universe, Einstein tensor, Einstein-Hilbert action, energy density, energy-momentum tensor, equivalence principle, Fermi-Dirac distribution, General Relativity, geodesic equation, geodesics, hot ideal gas, ideal gas, Killing equation, Killing vector, maximally symmetric space, natural units, neutrinos, number density, parsec, particle physics, perfect cosmological principle, perfect fluid, plane Universe, pressure, redshift, relativistic matter, Standard Cosmological Model, thermal equilibrium, yield Leave a comment

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

The solar mass is and the parsec is given by the value

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

via the Poisson’s equation

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

The spacetime geometry is determined by the metric tensor . The matter content is given by the stress-energy-momentum tensor . 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

Given a spacetime metric , we can calculate the (affine/Levi-Civita) connection

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

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

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

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

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

Maximally symmetric spaces have 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 .

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

3. The spehre . 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 . Here, denotes the time “slice” and 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, is the called the **scale factor. **The parameter determines the geometry type (plane, hyperbolic or elliptical/spherical):

1) If , then the Universe is “flat”. The manifold is .

2) If , then the Universe is “open”/hyperbolic. The manifold would be .

3) If , then the Universe is “closed”/spherical or elliptical. The manifold is then .

**Remark:** The ansatz of local homogeneity and istoropy only implies that the spatial metric is locally one of the above three spaces, i.e., . 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 . Then,

or

The total comoving distance that light have traveled since a time 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 is defined by the usual invariant equation

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

and the 0th component of the geodesic equation becomes

Therefore we have deduced that . 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

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

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

The luminosity distance in the expanding shell is

and this is what we MEASURE in Astrophysics/Cosmology. Knowing , 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

and

C) **Angular diameter distance.**

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

The comoving size is defined as , and the coming distance is again . For “flat” space, we obtain that

that is

In the case of “curved” spaces, we get

**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

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 . In fact, this type of tensor describes a perfect fluid with

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

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

for the 00th compoent as “constraint equation.

for the iith components.

Moreover, we also have

and this conservation law implies that

Therefore, we have got two independent equations for three unknowns . We need an additional equation. In fact, the equation of state for 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 :

1)

2)

3)

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:

Energy-momentum conservation implies that . 3 special cases are used often:

1st. **Radiation (relativistic “matter”).** and thus, and

2nd. **Dust (non-relativistic matter).** . Then, and

3rd.** Vacuum energy (cosmological constant).** . Then, and

**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 , the so-called weak energy condition, then 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 !

Then, we know that the early Universe was filled with a hot dieal gas in thermal equilibrium (a temperature can be defined there) such as the energy density and pressure can be written in terms of this temperature. This temperature generates a distribution . The number of phase space elements in is

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

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

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

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

1st. **Relativistic, non-degenerate matte**r (e.g. the known neutrino species). It means that and . Thus,

2nd. **Non-relativistic matter** with only. Then,

, and

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,

for radiation with

and the effective degrees of freedom are

**Remark:** The factor 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

and then

or

Now, since

then

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

but it means that , where is the entropy density defined by

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 ). Thus,

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

Indeed, since we have that , , the **yield** variable

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

See you in my next 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! 🙂