# LOG#107. Basic Cosmology (II).

Evolution of the Universe: the scale factor

The Universe expands, and its expansion rate is given by the Hubble parameter (not constant in general!)

$\boxed{H(t)\equiv \dfrac{\dot{a}(t)}{a(t)}}$

Remark  (I): The Hubble “parameter” is “constant” at the present value (or a given time/cosmological age), i.e., $H_0=H(t_0)$.

Remark (II): The Hubble time defines a Hubble length about $L_H=H^{-1}$, and it defines the time scale of the Universe and its expasion “rate”.

The critical density of matter is a vital quantity as well:

$\boxed{\rho_c=\dfrac{3H^2}{\kappa^2}\vert_{t_0}}$

We can also define the density parameters

$\Omega_i=\dfrac{\rho_i}{\rho_c}\vert_{t_0}$

This quantity represents the amount of substance for certain particle species. The total composition of the Universe is the total density, or equivalently, the sum over all particle species of the density parameters, that is:

$\boxed{\displaystyle{\Omega=\sum_i\Omega_i=\dfrac{\displaystyle{\sum_i\rho_i}}{\rho_c}}}$

There is a nice correspondence between the sign of the curvature $k$ and that of $\Omega-1$. Using the Friedmann’s equation

$\displaystyle{\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}=\dfrac{\kappa^2}{3}\sum_i\rho_i}$

then we have

$\dfrac{k}{H^2a^2}=\dfrac{\displaystyle{\sum_i\rho_i}}{\rho_c}-1=\Omega-1$

Thus, we observe that

1st. $\Omega>1$ if and only if (iff) $k=+1$, i.e., iff the Universe is spatially closed (spherical/elliptical geometry).

2nd. $\Omega=1$ if and only if (iff) $k=0$, i.e., iff the Universe is spatially “flat” (euclidean geometry).

3rd. $\Omega<1$ if and only if (iff) $k=-1$, i.e., iff the Universe is spatially “open” (hyperbolic geometry).

In the early Universe, the curvature term is negligible (as far as we know). The reason is as follows:

$k/a^2\propto a^{-2}<<\dfrac{\kappa\rho}{3}\propto a^{-3}(MD),a^{-4}(RD)$ as $a$ goes to zero. MD means matter dominated Universe, and RD means radiation dominated Universe. Then, the Friedmann’s equation at the early time is given by

$\boxed{H^2=\dfrac{\kappa^2}{3}\rho}$

Furthermore, the evolution of the curvature term

$\Omega_k\equiv \Omega-1$

is given by

$\Omega-1=\dfrac{k}{H^2a^2}\propto \dfrac{1}{\rho a^2}\propto a(MD),a^2(RD)$

and thus

$\vert \Omega-1\vert=\begin{cases}(1+z)^{-1}, \mbox{if MD}\\ 10^4(1+z)^{-2}, \mbox{if RD}\end{cases}$

The spatial curvature will be given by

$\boxed{R_{(3)}=\dfrac{6k}{a^2}=6H^2(\Omega-1)}$

and the curvature radius will be

$\boxed{R=a\vert k\vert ^{-1/2}=H^{-1}\vert \Omega-1\vert ^{-1/2}}$

We have arrived at the interesting result that in the early Universe, it was nearly “critical”. The Universe close to the critical density is very flat!

By the other hand, supposing that $a_0=1$, we can integrate the Friedmann’s equation easily:

$\boxed{\displaystyle{\left(\dfrac{\dot{a}}{a}\right)^2+\dfrac{k}{a^2}=\dfrac{\kappa^2}{3}\sum_i\rho_i=\dfrac{\kappa^2}{3}\sum_i\rho_i(0)a^{-3(1+\omega_i)}}}$

Then, we obtain

$\dot{a}^2=H_0^2\left[-\Omega_k+\sum_i\Omega_ia^{-1-3\omega_i}\right]$

We can make an analogy of this equation to certain simple equation from “newtonian Mechanics”:

$\dfrac{\dot{a}^2}{2}+V(a)=0$

Therefore, if we identify terms, we get that the density parameters work as “potential”, with

$\displaystyle{V(a)=\dfrac{1}{2}H_0^2\left[\Omega_k-\sum_i\Omega_ia^{-1-3\omega_i}\right]}$

and the total energy is equal to zero (a “machian” behaviour indeed!). In addition to this equation, we also get

$\boxed{\displaystyle{H_0t=\int_0^a\left[-\Omega_k+\sum_i\Omega_i\chi^{-1-3\omega_i}\right]^{-1/2}d\chi}}$

The age of the Universe can be easily calculated (symbolically and algebraically):

$\boxed{t_0=H_0^{-1}f(\Omega_i)}$

with

$f(\Omega_i)=\int_0^1\left[-\Omega_k+\sum_i\Omega_i\chi^{-1-3\omega_i}\right]^{-1/2}d\chi$

This equation can be evaluated for some general and special cases. If we write $p=\omega \rho$ for a single component, then

$a\propto t^{2/3(1+\omega)}$ if $\omega\neq -1$

Moreover, 3 common cases arise:

1) Matter dominated Universe (MD): $a\propto t^{2/3}$

2) Radiation dominated Universe (RD): $a\propto t^{1/2}$

3) Vacuum dominated Universe (VD): $e^{H_0t}$ ($w=-1$ for the cosmological constant, vacuum energy or dark energy).

THE MATTER CONTENT OF THE UNIVERSE

We can find out how much energy is contributed by the different compoents of the Universe, i.e., by the different density parameters.

Case 1. Photons.

The CMB temperature gives us “photons” with $T_\gamma=2\mbox{.}725\pm 0\mbox{.}002K$

The associated energy density is given by the Planck law of the blackbody, that is

$\rho_\gamma=\dfrac{\pi^2}{15}T^4$ and $\mu/T<9\cdot 10^{-5}$

or equivalently

$\Omega_\gamma=\Omega_r=\dfrac{2\mbox{.}47\cdot 10^{-5}}{h^2a^4}$

Case 2. Baryons.

There are four established ways of measuring the baryon density:

i) Baryons in galaxies: $\Omega_b\sim 0\mbox{.}02$

ii) Baryons through the spectra fo distant quasars: $\Omega_b h^{1\mbox{.}5}\approx 0\mbox{.}02$

iii) CMB anisotropies: $\Omega_bh^2=0\mbox{.}024\pm ^{0\mbox{.}004}_{0\mbox{.}003}$

iv) Big Bag Nucleosynthesis: $\Omega_bh^2=0\mbox{.}0205\pm 0\mbox{.}0018$

Note that these results are “globally” compatible!

Case 3. (Dark) Matter/Dust.

The mass-to-light ratio from galactic rotation curves are “flat” after some cut-off is passed. It also works for clusters and other bigger structures. This M/L ratio provides a value about $\Omega_m=0\mbox{.}3$. Moreover, the galaxy power spectrum is sensitive to $\Omega_m h$. It also gives $\Omega_m\sim 0\mbox{.}2$. By the other hand, the cosmic velocity field of galaxies allows us to derive $\Omega_m\approx 0\mbox{.}3$ as well. Finally, the CMB anisotropies give us the puzzling values:

$\Omega_m\sim 0\mbox{.}25$

$\Omega_b\sim 0\mbox{.}05$

We are forced to accept that either our cosmological and gravitational theory is a bluff or it is flawed or the main component of “matter” is not of baryonic nature, it does not radiate electromagnetic radiation AND that the Standard Model of Particle Physics has no particle candidate (matter field) to fit into that non-baryonic dark matter. However, it could be partially formed by neutrinos, but we already know that it can NOT be fully formed by neutrinos (hot dark matter). What is dark matter? We don’t know. Some candidates from beyond standard model physics: axion, new (likely massive or sterile) neutrinos, supersymmetric particles (the lightest supersymmetric particle LSP is known to be stable: the gravitino, the zino, the neutralino,…), ELKO particles, continuous spin particles, unparticles, preons, new massive gauge bosons, or something even stranger than all this and we have not thought yet! Of course, you could modify gravity at large scales to erase the need of dark matter, but it seems it is not easy at all to guess a working Modified Gravitational theory or Modified Newtonian(Einsteinian) dynmanics that avoids the need for dark matter. MOND’s, MOG’s or similar ideas are an interesting idea, but it is not thought to be the “optimal” solution at current time. Maybe gravitons and quantum gravity could be in the air of the dark issues? We don’t know…

Case 4. Neutrinos.

They are NOT observed, but we understand them their physics, at least in the Standard Model and the electroweak sector. We also know they suffer “oscillations”/flavor oscillations (as kaons). The (cosmic) neutrino temperature can be determined and related to the CMB temperature. The idea is simple: the neutrino decoupling in the early Universe implied an electron-positron annihilation! And thus, the (density) entropy dump to the photons, but not to neutrinos. It causes a difference between the neutrino and photon temperature “today”. Please, note than we are talking about “relic” neutrinos and photons from the Big Bang! The (density) entropy before annihilation was:

$s(a_1)=\dfrac{2\pi^2}{45}T_1^3\left[2+\dfrac{7}{8}(2\cdot 2+3\cdot 2)\right]=\dfrac{43}{90}\pi^2 T_1^3$

After the annihilation, we get

$s(a_2)=\dfrac{2\pi^2}{45}\left[2T_\gamma^3+\dfrac{7}{8}(3\cdot 2)T_\nu^3\right]$

Therefore, equating

$s(a_1)a_1^3=s(a_2)a_2^3$ and $a_1T_1=a_2T_\nu (a_2)$

$\dfrac{43}{90}\pi^2(a_1T_1)^3=\dfrac{2\pi^2}{45}\left[2\left(\dfrac{T_\gamma}{T_\nu}\right)^3+\dfrac{42}{8}\right](a_2T_\nu (a_2))^3$

$\dfrac{43}{2}\pi^2(a_1T_1)^3=2\pi^2\left[2\left(\dfrac{T_\gamma}{T_\nu}\right)^3+\dfrac{42}{8}\right](a_2T_\nu (a_2))^3$

and then

$\boxed{\left(\dfrac{T_\nu}{T_\gamma}\right)=\left(\dfrac{4}{11}\right)^{1/3}}$

or equivalently

$\boxed{T_\nu=\sqrt[3]{\dfrac{4}{11}}T_\gamma\approx 1\mbox{.}9K}$

In fact, the neutrino energy density can be given in two different ways, depending if it is “massless” or “massive”. For massless neutrinos (or equivalently “relativistic” massless matter particles):

I) Massless neutrinos: $\Omega_\nu=\dfrac{1\mbox{.}68\cdot 10^{-5}}{h^2}$

2) Massive neutrinos: $\Omega_\nu= \dfrac{m_\nu}{94h^2 \; eV}$

Case 5. The dark energy/Cosmological constant/Vacuum energy.

The budget of the Universe provides (from cosmological and astrophysical measurements) the shocking result

$\Omega\approx 1$ with $\Omega_M\approx 0\mbox{.}3$

Then, there is some missin smooth, unclustered energy-matter “form”/”species”. It is the “dark energy”/vacuum energy/cosmological cosntant! It can be understood as a “special” pressure term in the Einstein’s equations, but one with NEGATIVE pressure! Evidence for this observation comes from luminosity-distance-redshift measurements from SNae, clusters, and the CMB spectrum! The cosmological constant/vacuum energy/dark energy dominates the Universe today, since, it seems, we live in a (positively!) accelerated Universe!!!!! What can dark energy be? It can not be a “normal” matter field. Like the Dark Matter field, we believe that (excepting perhaps the scalar Higgs field/s) the SM has no candidate to explain the Dark Energy. What field could dark matter be? Perhaps an scalar field or something totally new and “unknown” yet.

In short, we are INTO a DARKLY, darkly, UNIVERSE! Darkness is NOT coming, darkness has arrived and, if nothing changes, it will turn our local Universe even darker and darker!

See you in the next cosmological post!

# 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:

$\hbar=c=k_B=1$

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

$G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}$

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

$\boxed{ds^2_{FRW}=-dt^2+a^2(t)\left[\dfrac{dr^2}{1-kr^2}+r^2\left(d\theta^2+\sin\theta^2d\phi^2\right)\right]}$

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,

$dt=a(t)\dfrac{1}{\sqrt{1-kr^2}}$

or

$\dfrac{dr}{\sqrt{1-kr^2}}=\dfrac{dt}{a(t)}$

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

$\boxed{\eta=\int_0^{r_H}\dfrac{dr}{\sqrt{1-kr^2}}=\int_0^t\dfrac{dt'}{a(t')}}$

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

$\boxed{d_H(t)=\int_0^{r_H}\sqrt{g_{rr}}dr=a(t)\int_0^t\dfrac{dt'}{a(t')}=a(t)\eta}$

There are some important kinematical equations to be known

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

$\Gamma^0_{ij}=\dfrac{\dot{a}}{a}\overline{g}_{ij}$

$\Gamma^i_{0j}=\Gamma^i_{j0}=\dfrac{\dot{a}}{a}\delta_{ij}$

$\Gamma^i_{jk}=\overline{\Gamma}^i_{jk}$

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

$P^\mu=\dfrac{dx^\mu}{d\lambda}$

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

$\dfrac{d}{d\lambda}=\dfrac{dx^0}{d\lambda}\dfrac{d}{dx^0}=E\dfrac{d}{dt}$

and the 0th component of the geodesic equation becomes

$E\dfrac{dE}{dt}=-\Gamma^0_{ij}p^ip^j=-\delta_{ij}a\dot{a}p^ip^j$

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

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

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

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

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

$\boxed{\dfrac{a(t_0)}{a(t)}=1+z=\dfrac{\lambda_0}{\lambda}}$

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:

$H_0d_L=z+\dfrac{1}{2}(1-q_0)z^2+\ldots$

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

$\boxed{H_0=\dfrac{\dot{a}_0}{a_0}}$

and

$\boxed{q_0=-\dfrac{a_0\ddot{a}_0}{\dot{a}_0^2}}$

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

$\boxed{d_A=\dfrac{l}{\theta}}$

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

$R_{00}=-3\dfrac{\ddot{a}}{a}$

$R_{ij}=(a\ddot{a}+2\dot{a}^2+2k)\overline{g}_{ij}$

$R=6\left(\dfrac{\ddot{a}}{a}+\dfrac{\dot{a}^2}{a^2}+\dfrac{k}{a^2}\right)$

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

$G_{\mu\nu}^{;\nu}=T_{\mu\nu}^{;\nu}=0$

and this conservation law implies that

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

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

$\dfrac{\ddot{a}}{a}=-\dfrac{\kappa^2}{6}(\rho+3p)$

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

$d^3xd^3p=\dfrac{d^3\mathbf{x}d^3\mathbf{p}}{(2\pi\hbar)^3}$

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,

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

$\rho=\left(\dfrac{7}{8}\right)\dfrac{\pi^2}{30}gT^4$

$p=\dfrac{1}{3}\rho$

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

$n=g\left(\dfrac{mT}{2\pi}\right)^{3/2}e^{-(m-\mu)/T}$

$\rho= mn+\dfrac{3}{2}p$, 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

$\displaystyle{\boxed{g_\star=\sum_{bosons}g_b+\dfrac{7}{8}\sum_{fermions}g_f}}$

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

$(1-\dfrac{1}{2^d})=\dfrac{2^d-1}{2^d}$

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

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

or

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

Now, since

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

then

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

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

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!

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 $q_{tach}^2/e^2<10^{-11}$. 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.

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.

## 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, $P_{RF}\propto E^2$.

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

$\dfrac{P_{CR}}{d\nu}\propto \nu d\nu$

from the charged particle in a dense (refractive) medium experimenting Cherenkov radiation (CR). Every charge emittes a field $\vert E\vert\propto \exp (i\mathbf{k}\cdot\mathbf{r})$. Then, the power is proportional to $E^2$. In a dense medium:

$R_{M}\sim 10cm$

We have two different experimental and interesting cases:

A) The optical case, with $\lambda <. Then, we expect random phases and $P\propto N$.

B) The microwave case, with $\lambda>>R_M$. In this situation, we expect coherent radiation/waves with $P\propto N^2$.

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 $5GHz$, while the peak for ice is obtained around $2GHz$.

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.

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 $E>10^{17}eV$.

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 $10^{20}eV$ 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?