# 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#047. The Askaryan effect.

**Posted:**2012/10/17

**Filed under:**Physmatics, Relativity |

**Tags:**Askaryan effect, Askaryan radiation, cherenkov effect, cherenkov radiation, coherent radiation, cosmic rays, electromagnetism, electroweak theory, G. Askaryan, matter-radiation interactions, microwaves, neutrino detection, neutrino detectors, neutrino experiments, neutrino telescopes, neutrinos, New Physics, optical band of the spectrum, origin of cosmic rays, Physmatics, radio waves, Relativity, special relativity, Standard Model, superluminality, UHE neutrinos Leave a comment

I discussed and reviewed the important Cherenkov effect and radiation in the previous post, here:

https://thespectrumofriemannium.wordpress.com/2012/10/16/log046-the-cherenkov-effect/

Today we are going to study a relatively new effect ( new experimentally speaking, because it was first detected when I was an undergraduate student, in 2000) but it is not so new from the theoretical aside (theoretically, it was predicted in 1962). This effect is closely related to the Cherenkov effect. It is named Askaryan effect or Askaryan radiation, see below after a brief recapitulation of the Cherenkov effect last post we are going to do in the next lines.

We do know that charged particles moving faster than light through the *vacuum* emit Cherenkov radiation. How can a particle move faster than light? The *weak* speed of a charged particle can exceed the speed of light. That is all. About some speculations about the so-called tachyonic gamma ray emissions, let me say that the existence of superluminal energy transfer has not been established so far, and one may ask why. There are two options:

1) The simplest solution is that superluminal quanta just do not exist, the vacuum speed of light being the definitive upper bound.

2) The second solution is that the interaction of superluminal radiation with matter is very small, the quotient of tachyonic and electric fine-structure constants being . Therefore superluminal quanta and their substratum are hard to detect.

A related and very interesting question could be asked now related to the Cherenkov radiation we have studied here. What about neutral particles? Is there some analogue of Cherenkov radiation valid for chargeless or neutral particles? Because neutrinos are electrically neutral, conventional Cherenkov radiation of superluminal neutrinos does not arise or it is otherwise weakened. However neutrinos *do carry electroweak charge* and may emit certain *Cherenkov-like radiation* via weak interactions when traveling at superluminal speeds. The Askaryan effect/radiation is this Cherenkov-like effect for neutrinos, and we are going to enlighten your knowledge of this effect with this entry.

We are being bombarded by cosmic rays, and even more, we are being bombarded by neutrinos. Indeed, we expect that ultra-high energy (UHE) neutrinos or extreme ultra-high energy (EHE) neutrinos will hit us as too. When neutrinos interact wiht matter, they create some shower, specifically in dense media. Thus, we expect that the electrons and positrons which travel faster than the speed of light in these media or even in the air and they should emit (coherent) Cherenkov-like radiation.

## Who was Gurgen Askaryan?

Let me quote what wikipedia say about him: Gurgen Askaryan (December 14, 1928-1997) was a prominent Soviet (armenian) physicist, famous for his discovery of the self-focusing of light, pioneering studies of light-matter interactions, and the discovery and investigation of the interaction of high-energy particles with condensed matter. He published more than 200 papers about different topics in high-energy physics.

Other interesting ideas by Askaryan: the bubble chamber (he discovered the idea independently to Glaser, but he did not published it so he did not win the Nobel Prize), laser self-focussing (one of the main contributions of Askaryan to non-linear optics was the self-focusing of light), and the acoustic UHECR detection proposal. Askaryan was the first to note that the outer few metres of the Moon’s surface, known as the regolith, would be a sufficiently transparent medium for detecting microwaves from the charge excess in particle showers. The radio transparency of the regolith has since been confirmed by the Apollo missions.

If you want to learn more about Askaryan ideas and his biography, you can read them here: http://en.wikipedia.org/wiki/Gurgen_Askaryan

## What is the Askaryan effect?

The next figure is from the Askaryan radiation detected by the ANITA experiment:

The **Askaryan effect** is the phenomenon whereby a particle traveling faster than the phase velocity of light in a dense dielectric medium (such as salt, ice or the lunar regolith) produces a shower of secondary charged particles which contain a charge anisotropy and thus emits a cone of coherent radiation in the radio or microwave part of the electromagnetic spectrum. It is similar, or more precisely it is based on the Cherenkov effect.^{}

High energy processes such as Compton, Bhabha and Moller scattering along with positron annihilation rapidly lead to about a 20%-30% negative charge asymmetry in the electron-photon part of a cascade. For instance, they can be initiated by UHE (higher than, e.g.,100 PeV) neutrinos.

1962, Askaryan first hypothesized this effect and suggested that it should lead to strong coherent radio and microwave Cherenkov emission for showers propagating within the dielectric. Since the dimensions of the clump of charged particles are small compared to the wavelength of the radio waves, the shower radiates coherent radio Cherenkov radiation whose power is proportional to the square of the net charge in the shower. The net charge in the shower is proportional to the primary energy so the radiated power scales quadratically with the shower energy, .

Indeed, these radio and coherent radiations are originated by the Cherenkov effect radiation. We do know that:

from the charged particle in a dense (refractive) medium experimenting Cherenkov radiation (CR). Every charge emittes a field . Then, the power is proportional to . In a dense medium:

We have two different experimental and interesting cases:

A) **The optical case,** with . Then, we expect random phases and .

B) **The microwave case**, with . In this situation, we expect coherent radiation/waves with .

We can exploit this effect in large natural volumes transparent to radio (dry): pure ice, salt formations, lunar regolith,…The peak of this coherent radiation for sand is produced at a frequency around , while the peak for ice is obtained around .

The first experimental confirmation of the Askaryan effect detection were the next two experiments:

1) 2000 Saltzberg et.al., SLAC. They used as target silica sand. The paper is this one http://arxiv.org/abs/hep-ex/0011001

2) 2002 Gorham et.al., SLAC. They used a synthetic salt target. The paper appeared in this place http://arxiv.org/abs/hep-ex/0108027

Indeed, in 1965, Askaryan himself proposes ice and salt as possible target media. The reasons are easy to understand:

1st. They provide high densities and then it means a higher probability for neutrino interaction.

2nd. They have a high refractive index. Therefore, the Cerenkov emission becomes important.

3rd. Salt and ice are radio transparent, and of course, they can be supplied in large volumes available throughout the world.

The advantages of radio detection of UHE neutrinos provided by the Askaryan effect are very interesting:

1) Low attenuation: clear signals from large detection volumes.

2) We can observe distant and inclined events.

3) It has a high duty cycle: good statistics in less time.

4) I has a relative low cost: large areas covered.

5) It is available for neutrinos and/or any other chargeless/neutral particle!

Problems with this Askaryan effect detection are, though: radio interference, correlation with shower parameters (still unclear), and that it is limited only to particles with very large energies, about .

**In summary:**

*Askaryan effect = coherent Cerenkov radiation from a charge excess induced by (likely) neutral/chargeless particles like (specially highly energetic) neutrinos passing through a dense medium.*

## Why the Askaryan effect matters?

It matters since it allows for the detection of UHE neutrinos, and it is “universal” for chargeless/neutral particles like neutrinos, just in the same way that the Cherenkov effect is universal for charged particles. And tracking UHE neutrinos is important because they point out towards its source, and it is suspected they can help us to solve the riddle of the origin and composition of cosmic rays, the acceleration mechanism of cosmic radiation, the nuclear interactions of astrophysical objects, and tracking the highest energy emissions of the Universe we can observe at current time.

Is it real? Has it been detected? Yes, after 38 years, it has been detected. This effect was firstly demonstrated in sand (2000), rock salt (2004) and ice (2006), all done in a laboratory at SLAC and later it has been checked in several independent experiments around the world. Indeed, I remember to have heard about this effect during my darker years as undergraduate student. Fortunately or not, I forgot about it till now. In spite of the beauty of it!

Moreover, it has extra applications to neutrino detection using the Moon as target: GLUE (detectors are Goldstone RTs), NuMoon (Westerbork array; LOFAR), or RESUN (EVLA), or the LUNASKA project. Using ice as target, there has been other experiments checking the reality of this effect: FORTE (satellite observing Greenland ice sheet), RICE (co-deployed on AMANDA strings, viewing Antarctic ice), and the celebrated ANITA (balloon-borne over Antarctica, viewing Antarctic ice) experiment.

Furthermore, even some experiments have used the Moon (an it is likely some others will be built in the near future) as a neutrino detector using the Askaryan radiation (the analogue for neutral particles of the Cherenkov effect, don’t forget the spot!).

## Askaryan effect and the mysterious cosmic rays.

Askaryan radiation is important because is one of the portals of the UHE neutrino observation coming from cosmic rays. The mysteries of cosmic rays continue today. We have detected indeed extremely energetic cosmic rays beyond the scale. Their origin is yet unsolved. We hope that tracking neutrinos we will discover the sources of those rays and their nature/composition. We don’t understand or know any mechanism being able to accelerate particles up to those incredible particles. At current time, IceCube has not detected UHE neutrinos, and it is a serious issue for curren theories and models. It is a challenge if we don’t observe enough UHE neutrinos as the Standard Model would predict. Would it mean that cosmic rays are exclusively composed by heavy nuclei or protons? Are we making a bad modelling of the spectrum of the sources and the nuclear models of stars as it happened before the neutrino oscillations at SuperKamiokande and Kamikande were detected -e.g.:SN1987A? Is there some kind of new Physics living at those scales and avoiding the GZK limit we would naively expect from our current theories?