# LOG#123. Basic Neutrinology(VIII).

There are some indirect constraints/bounds on neutrino masses provided by Cosmology. The most important is the one coming from the demand that the energy density of the neutrinos should not be too high, otherwise the Universe would collapse and it does not happen, apparently…

Firstly, stable neutrinos with low masses (about $m_\nu\leq 1 MeV$) make a contribution to the total energy density of the Universe given by:

$\rho_\nu=m_{tot}n_\nu$

and where the total mass is defined to be the quantity

$\displaystyle{m_{tot}=\sum_\nu \dfrac{g_\nu}{2}m_\nu}$

Here, the number of degrees of freedom $g_\nu=4(2)$ for Dirac (Majorana) neutrinos in the framework of the Standard Model. The number density of the neutrino sea is revealed to be related to the photon number density by entropy conservation (entropy conservation is the key of this important cosmological result!) in the adiabatic expansion of the Universe:

$n_\nu=\dfrac{3}{11}n_\gamma$

From this, we can derive the relationship of the cosmic relic neutrino background (neutrinos coming from the Big Bang radiation when they lost the thermal equilibrium with photons!) or $C\nu B$ and the cosmic microwave background (CMB):

$T_{C\nu B}=\left(\dfrac{3}{11}\right)^{1/3}T_{CMB}$

From the CMB radiation measurements we can obtain the value

$n_\nu=411(photons)cm^{-3}$

for a perfect Planck blackbody spectrum with temperature

$T_{CMB}=2.725\pm0.001 K\approx 2.35\cdot 10^{-4}eV$

This CMB temperature implies that the $C\nu B$ temperature should be about

$T_{C\nu B}^{theo}=1.95K\approx 0.17meV$

Remark: if you do change the number of neutrino degrees of freedom you also change the temperature of the $C\nu B$ and the quantity of neutrino “hot dark matter” present in the Universe!

Moreover, the neutrino density $\Omega_\nu$ is related to the total neutrino density and the critical density as follows:

$\Omega_\nu=\dfrac{\rho_\nu}{\rho_c}$

and where the critical density is about

$\rho_c=\dfrac{3H_0^2}{8\pi G_N}$

When neutrinos “decouple” from the primordial plasma and they loose the thermal equilibrium, we have $m_\nu>>T$, and then we get

$\Omega_\nu h^2=10^{-2}m_{tot}eV$

with $h$ the reduced Hubble constant. Recent analysis provide $h\approx 67-71\cdot 10^{-2}$ (PLANCK/WMAP).

There is another useful requirement for the neutrino density in Cosmology. It comes from the requirements of the BBN (Big Bang Nucleosynthesis). I talked about this in my Cosmology thread. Galactic structure and large scale observations also increase evidence that the matter density is:

$\Omega_Mh^2\approx 0.05-0.2$

These values are obtained through the use of the luminosity-density relations, galactic rotation curves and the observation of large scale flows. Here, the $\Omega_M$ is the total mass density of the Universe as a fraction of the critical density $\rho_c$. This $\Omega_M$ includes radiation (photons), bayrons and non-baryonic “cold dark matter” (CDM) and “hot dark matter” (HDM). The two first components in the decomposition of $\Omega_M$

$\Omega_M=\Omega_r+\Omega_b+\Omega_{nb}+\Omega_{HDM}+\Omega_{CDM}$

are rather well known. The photon density is

$\Omega_rh^2=\Omega_\gamma h^2=2.471\cdot 10^{-5}$

The deuterium abundance can be extracted from the BBN predictions and compared with the deuterium abundances in the stellar medium (i.e. at stars!). It shows that:

$0.017\leq\Omega_Bh^2\leq 0.021$

The HDM component is formed by relativistic long-lived particles with masses less than about $1keV$. In the SM framework, the only HDM component are the neutrinos!

The simulations of structure formation made with (super)computers fit the observations ONLY when one has about 20% of HDM plus 80% of CDM. A stunning surprise certainly! Some of the best fits correspond to neutrinos with a total mass about 4.7eV, well above the current limit of neutrino mass bounds. We can evade this apparent contradiction if we suppose that there are some sterile neutrinos out there. However, the last cosmological data by PLANCK have decreased the enthusiasm by this alternative. The apparent conflict between theoretical cosmology and observational cosmology can be caused by both unprecise measurements or our misunderstanding of fundamental particle physics. Anyway observations of distant objects (with high redshift) favor a large cosmological constant instead of Hot Dark Matter hypothesis. Therefore, we are forced to conclude that the HDM of $\Omega_M$ does not exceed even $0.2$. Requiring that $\Omega_\nu <\Omega_M$, we get that $\Omega_\nu h^2\leq 0.1$. Using the relationship with the total mass density, we can deduce that the total neutrino mass (or HDM in the SM) is about

$m_\nu\leq 8-10 eV$ or less!

Mass limits, in this case lower limits, for heavy or superheavy neutrinos ($M_N\sim 1GeV$ or higher) can also be obtained along the same reasoning. The puzzle gets very different if the neutrinos were “unstable” particles. One gets then joint bounds on mass and timelife, and from them, we deduce limits that can overcome the previously seen limits (above).

There is another interesting limit to the density of neutrinos (or weakly interacting dark matter in general) that comes from the amount of accumulated “density” in the halos of astronomical objects. This is called the Tremaine-Gunn limit. Up to numerical prefactors, and with the simplest case where the halo is a singular isothermal sphere with $\rho\propto r^{-2}$, the reader can easily check that

$\rho=\dfrac{\sigma^2}{2\pi G_Nr^2}$

Imposing the phase space bound at radius r then gives the lower bound

$m_\nu>(2\pi)^{-5/8}\left(G_Nh_P^3\sigma r^2\right)^{-1/4}$

This bound yields $m_\nu\geq 33eV$. This is the Tremaine-Gunn bound. It is based on the idea that neutrinos form an important part of the galactic bulges and it uses the phase-space restriction from the Fermi-Dirac distribution to get the lower limit on the neutrino mass. I urge you to consult the literature or google to gather more information about this tool and its reliability.

Remark: The singular isothermal sphere is probably a good model where the rotation curve produced by the dark matter halo is flat, but certainly breaks down at small radius. Because the neutrino mass bound is stronger for smaller $\sigma r^2$, the uncertainty in the halo core radius (interior to which the mass density saturates) limits the reliability of this neutrino mass bound. However, some authors take it seriously! As Feynman used to say, everything depends on the prejudges you have!

The abundance of additional weakly interacting light particles, such as a light sterile neutrino $\nu_s$ or additional relativistic degrees of freedom uncharged under the Standard Model can change the number of relativistic degrees of freedom $g_\nu$. Sometimes you will hear about the number $N_{eff}$. Planck data, recently released, have decreased the hopes than we would be finding some additional relativistic degree of freedom that could mimic neutrinos. It is also constrained by the BBN and the deuterium abundances we measured from astrophysical objects. Any sterile neutrino or extra relativistic degree of freedom would enter into equilibrium with the active neutrinos via neutrino oscillations! A limit on the mass differences and mixing angle with another active neutrino of the type

$\Delta m^2\sin^2 2\theta\leq 3\cdot 10^{-6}eV^2$ should be accomplished in principle. From here, it can be deduced that the effective number of neutrino species allowed by neutrino oscillations is in fact a litle higher the the 3 light neutrinos we know from the Z-width bound:

$N_\nu (eff)<3.5-4.5$

PLANCK data suggest indeed that $N_\nu (eff)< 3.3$. However, systematical uncertainties in the derivation of the BBN make it too unreliable to be taken too seriously and it can eventually be avoided with care.

# 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!