In my final post of this basic Cosmology thread, I am going to discuss scalar perturbations and to review some of the recent results about cosmological parameters by WMAP and PLANCK.
One of the predictions of the Standard Cosmological Model (LCDM or ΛCDM) is that we should expect some inhomogeneities and anisotropies in the Cosmic Microwave Background (CMB). Indeed, the “perturbations” of densities (matter, radiation,…) are closely related to these anisotropies and the easiest way to trigger them is by using “scalar perturbations”, i.e., perturbations induced by the scalar field in the early Universe! In fact, inflation should be understood at quantum level as a “quantum fluctuation” that triggers scalar perturbations of the so called “inflaton field”.
The scalar field perturbations are “shifts” or “fluctuations” in the scalar field, namely
Now, we have to work out a little bit the equation for , if we take into account that our Universe is seemingly a “smoothly” expanding Universe. To first order, we can neglect the “back reaction” on the metric field due to the scalar field . Therefore, the equation we get is really simple:
since the term can be neglected during inflation. Furthermore, we can write
for any Fourier mode “k” in the Fourier decomposition of the scalar field . If we define , then we can rewrite the above equation as follows
This last equation describes a harmonic oscillator equation with time-varying frequency! We can quantize using the formalism of creation and annihilation operators (, respectively):
and where U satisfies the equation of a harmonic oscillator as well
The variance of perturbations in are defined with the equations
and where we have defined the power spectrum in terms of correlation function as the quantity
We require a solution for the equation above for . During inflation, provides
Then, for the equation above for U becomes
And now, a subtle issue…Initial conditions! Any initial condition (with proper normalization) long before the horizon exit implies that . It implies that the term dominates and we can substitute the last equation by this one
and this equation has pretty simple solutions with proper normalization. Essentially, they are “plane waves” modulated with the Fourier mode k, i.e.,
A more precise proper solution is
Please, note that if , we recover the previous equation. After many e-folds, we get and
Thus, the power spectrum is deduced to be
For tensor perturbations (I have not discussed them here for simplicity), this result can be compared with
So, at least, scalar perturbations have the same order of magnitude and the same power of the Fourier modes! Indeed, the fluctuations are transfered to scalar perturbations or metric pertubations, since
After the horizon crossing, we obtain
Now, we can introduce the important concept of spectral index. For scalar fields, the perturbations are defined in terms of this spectral index as follows
For metric field perturbations, we write
and where the spectral index and its tensor analogue are defined through the simple expressions
Final conclusions, WMAP and PLANCK
The history of the Universe up to the period of BBN is (more or less) well understood in the framework of the Standard Cosmological Model (LCDM). It is well established and tested, BUT, please, do not forget we have to choose “the proper initial conditions” in order to obtain the right (measured/observed) results. In summary, up to the BBN we get
1. A homogenous and isotropic Universe, with small density perturbations. It is equivalent to a thermal bath with temperature above 1MeV, i.e., . Indeed, inflation seems to be the simplest and best solution to many problems of modern cosmology (e.g., the horizon problem or the flatness problem). However, the LCDM and the inflation itself can not answer what the inflaton field is and why the anisotropies are so tiny as we observed today. Moreover, the cosmological constant problem remains unanswered today (even when we have just discovered the Higgs field, and that some people are speculating that the Higgs field could be matched with the inflaton field, this idea has some technical problems).
2. The energy-matter content of the Universe…There is a wide agreement about the current composition of the Universe, based on the “concordance model” by several experiments, now confirmed by the WMAP/PLANCK probes. Generally speaking:
– Baryons are negligible since . In fact, only about a 5% is “matter” we do know through the Standard Model of particle physics. Most of that 5% is “light” (a.k.a., photons) and the remaining 1% are the elements we do know from the Periodic Table (and mainly hydrogen and helium, of course).
–(Cold) Dark Matter is about . There is no known particle that can be assigned to this dark matter stuff. It does not emit light, it is (likely) non-relativistic and uncharged under the gauge symmetry group of the SM, i.e., under . Of course, you could avoid the need for this stuff invoking a modified newtonian dynamics (MOND) and/or modified gravity (MOG). However, to my knowledge, any MOND/MOG theory generally ends by requiring some kind of stuff that is “like cold DM”. So, Occam’s razor here punches and says that cold DM is (likely) the most promising solution to the flat rotation curve of galaxies and this mysterious energy density. Remember: Dark Matter can not be any known particle belonging to the Standard Model we know today (circa 2013).
–Dark Energy (DE), a.k.a. vacuum energy or the cosmological constant. Reintroduced in 1998 to explain the SNae results and the high redshifts we measured from them, its existence is now fully established as well. Perhaps the dark energy name is “unfortunated” since it is, indeed, a negative pressure terms in the Einstein’s field equations. It is about the 70 or 80% of the current energy-matter budget of the Universe, i.e., it has an energy density about .
Let me draw again a picture I have showed before in this blog:
The main results for the cosmological parameters can be found here (in the case of the WMAP9 mission): http://arxiv.org/pdf/1212.5226v3.pdf
The main results for the cosmological parameter can be found here (in the case of the Planck mission): http://arxiv.org/pdf/1303.5076v1.pdf
We can compare the cosmological parameters coming from the WMAP9 mission
with those of the Planck mission
Cosmology is a fascinating subject. I have only sketched some elementary ideas here. I will be discussing more advanced topics in the near future, but you have to wait for it! :).
See you in my next blog post!
The topic today is to review a beautiful paper and to discuss its relevance for theoretical physics. The paper is: Comment on the cosmological constant and a gravitational alpha by R.J.Adler. You can read it here: http://arxiv.org/abs/1110.3358
One of the most intriguing and mysterious numbers in Physics is the electromagnetic fine structure constant . Its value is given by
Of course, I am assuming that the coupling constant is measured at ordinary energies, since we know that the coupling constants are not really constant but they vary slowly with energy. However, I am not going to talk about the renormalization (semi)group in this post.
Why is the fine structure constant important? Well, we can undertand it if we insert the values of the constants that made the electromagnetic alpha constant:
with being the electron elemental charge, the Planck’s constant divided by two pi, c is the speed of light and where we are using units with . Here is the Coulomb constant, generally with a value , but we rescale units in order it has a value equal to the unit. We will discuss more about frequently used system of units soon.
As the electromagnetic alpha constant depends on the electric charge, the Coulomb’s electromagnetic constant ( rescaled to one in some “clever” units), the Planck’s constant ( rationalized by since ) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha is quantum and relativistic itself, and it also is related to elemental charges. Why alpha has the value it has is a complete mystery. Many people has tried to elucidate why it has the value it has today, but there is no reason of why it should have the value it has. Of course, it happens as well with some other constants but this one is particularly important since it is involved in some important numbers in atomic physics and the most elemental atom, the hydrogen atom.
In atomic physics, there are two common and “natural” scales of length. The first scale of length is given by the Compton’s wavelength of electrons. Usint the de Broglie equation, we get that the Compton’s wavelength is the wavelength of a photon whose energy is the same as the rest mass of the particle, or mathematically speaking:
Usually, physicists employ the “reduced” or “rationalized” Compton’s wavelength. Plugging the electron mass, we get the electron reduced Compton’s wavelength:
The second natural scale of length in atomic physics is the so-called Böhr radius. It is given by the formula:
Therefore, there is a natural mass ratio between those two length scales, and it shows that it is precisely the electromagnetic fine structure constant alpha :
Furthermore, we can show that the electromagnetic alpha also is related to the mass ration between the electron energy in the fundamental orbit of the hydrogen atom and the electron rest energy. These two scales of energy are given by:
1) Rydberg’s energy ( electron ground minimal energy in the fundamental orbit/orbital for the hydrogen atom):
2) Electron rest energy:
Then, the ratio of those two “natural” energies in atomic physics reads:
R.J.Adler’s paper remarks that there is a cosmological/microscopic analogue of the above two ratios, and they involve the infamous Einstein’s cosmological constant. In Cosmology, we have two natural (ultimate?) length scales:
1st. The (ultra)microscopic and ultrahigh energy (“ultraviolet” UV regulator) relevant Planck’s length , or equivalently the squared value . Its value is given by:
This natural length can NOT be related to any “classical” theory of gravity since it involves and uses the Planck’s constant .
2nd. The (ultra)macroscopic and ultra-low-energy (“infrared” IR regulator) relevant cosmological constant/deSitter radius. They are usualy represented/denoted by and respectively, and they are related to each other in a simple way. The dimensions of the cosmological constant are given by
The de Sitter radius and the cosmological constant are related through a simple equation:
The de Sitter radius is obtained from cosmological measurements thanks to the so called Hubble’s parameter ( or Hubble’s “constant”, although we do know that Hubble’s “constant” is not such a “constant”, but sometimes it is heard as a language abuse) H. From cosmological data we obtain ( we use the paper’s value without loss of generality):
This measured value allows us to derive the Hubble’s length paremeter
Moreover, the data also imply some density energy associated to the cosmological “constant”, and it is generally called Dark Energy. This density energy from data is written as:
and from this, it can be also proved that
where we have introduced the experimentally deduced value from the cosmological parameter global fits. In fact, the cosmological constant helps us to define the beautiful and elegant formula that we can call the gravitational alpha/gravitational cosmological fine structure constant :
or equivalently, defining the cosmological length associated to the cosmological constant as
If we introduce the numbers of the constants, we easily obtaint the gravitational cosmological alpha value and its inverse:
They are really small and large numbers! Following the the atomic analogy, we can also create a ratio between two cosmologically relevant density energies:
1st. The Planck’s density energy.
Planck’s energy is defined as
The Planck energy density is defined as the energy density of Planck’s energy inside a Planck’s cube or side , i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume . Mathematically speaking, it is
It is an huge density energy!
Remark: Energy density is equivalent to pressure in special relativity hydrodynamics. That is,
wiht Pa denoting pascals () and where represents here matter (not energy) density ( with units in ). Of course, turning matter density into energy density requires a multiplication by . This equivalence between vacuum pressure and energy density is one of the reasons because some astrophysicists, cosmologists and theoretical physicists call “vacuum pressure” to the “dark energy/cosmological constant” term in the study of the cosmic components derived from the total energy density .
2nd. The cosmological constant density energy.
Using the Einstein’s field equations, it can be shown that the cosmological constant gives a contribution to the stress-energy-momentum tensor. The component is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density
Using the previous equations for G as a function of Planck’s length, the Planck’s constant and the speed of light, and the definitions of Planck’s energy and de Sitter radius, we can rewrite the above energy density as follows:
Thus, we can evaluate the ration between these two energy densities! It provides
and the inverse ratio will be
So, we have obtained two additional really tiny and huge values for and its inverse, respectively. Note that the power appearing in the ratios of cosmological lengths and cosmological energy densities match the same scaling property that the atomic case with the electromagnetic alpha! In the electromagnetic case, we obtained and . The gravitational/cosmological analogue ratios follow the same rule and but the surprise comes from the values of the gravitational alpha values and ratios. Some comments are straightforward:
1) Understanding atomic physics involved the discovery of Planck’s constant and the quantities associated to it at fundamental quantum level ( Böhr radius, the Rydberg’s constant,…). Understanding the Cosmological Constant value and the mismatch or stunning ratios between the equivalent relevant quantities, likely, require that can be viewed as a new “fundamental constant” or/and it can play a dynamical role somehow ( e.g., varying in some unknown way with energy or local position).
2) Currently, the cosmological parameters and fits suggest that is “constant”, but we can not be totally sure it has not varied slowly with time. And there is a related idea called quintessence, in which the cosmological “constant” is related to some dynamical field and/or to inflation. However, present data say that the cosmological constant IS truly constant. How can it be so? We are not sure, since our physical theories can hardly explain the cosmological constant, its value, and why it is current density energy is radically different from the vacuum energy estimates coming from Quantum Field Theories.
3) The mysterious value
is an equivalent way to express the biggest issue in theoretical physics. A naturalness problem called the cosmological constant problem.
In the literature, there have been alternative definitions of “gravitational fine structure constants”, unrelated with the above gravitational (cosmological) fine structure constant or gravitational alpha. Let me write some of these alternative gravitational alphas:
1) Gravitational alpha prime. It is defined as the ratio between the electron rest mass and the Planck’s mass squared:
Note that . Since , we can also use the proton rest mass instead of the electron mass to get a new gravitational alpha.
2) Gravitational alpha double prime. It is defined as the ratio between the proton rest mass and the Planck’s mass squared:
and the inverse value
Finally, we could guess an intermediate gravitational alpha, mixing the electron and proton mass.
3) Gravitational alpha triple prime. It is defined as the ration between the product of the electron and proton rest masses with the Planck’s mass squared:
and the inverse value
We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with . We get
These inequations mean that the electromagnetic fine structure constant is (at ordinary energies) 42 orders of magnitude bigger than , 39 orders of magnitude bigger than , 36 orders of magnitude bigger than and, of course, 58 orders of magnitude bigger than . Indeed, we could extend this analysis to include the “fine structure constant” of Quantum Chromodynamics (QCD) as well. It would be given by:
since generally we define . We note that by 3 orders of magnitude. However, as strong nuclear forces are short range interactions, they only matter in the atomic nuclei, where confinement, and color forces dominate on every other fundamental interaction. Interestingly, at high energies, QCD coupling constant has a property called asymptotic freedom. But it is another story not to be discussed here! If we take the alpha strong coupling into account the full hierarchy of alphas is given by:
Fascinating! Isn’t it? Stay tuned!!!
ADDENDUM: After I finished this post, I discovered a striking (and interesting itself) connection between and . The relation or coincidence is the following relationship
Is this relationship fundamental or accidental? The answer is unknown. However, since the electric charge (via electromagnetic alpha) is not related a priori with the gravitational constant or Planck mass ( or the cosmological constant via the above gravitational alpha) in any known way I find particularly stunning such a coincidence up to 5 significant digits! Any way, there are many unexplained numerical coincidences that are completely accidental and meaningless, and then, it is not clear why this numeral result should be relevant for the connection between electromagnetism and gravity/cosmology, but it is interesting at least as a curiosity and “joke” of Nature.
Some quotes about the electromagnetic alpha from wikipedia http://en.wikipedia.org/wiki/Fine-structure_constant
“(…)There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! (…)”. R.P.Feynman, QED: The Strange Theory of Light and Matter, Princeton University Press, p.129.
“(…) If alpha [the fine-structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.(…)” Max Born, in A.I. Miller’s book Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. p. 253. Publisher W.W. Norton & Co.(2009).
“(…)The mystery about α is actually a double mystery. The first mystery – the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.(…)” Malcolm H. Mac Gregor, M.H. MacGregor (2007). The Power of Alpha.