Universo Primitivo - ULisboaUniverso Primitivo 2020-2021 (1º Semestre) Mestrado em Física...

Universo Primitivo2020-2021 (1º Semestre)

Mestrado em Física - Astronomia

3. Thermodynamics in an expanding universe• Natural Units; • Classification and properties of elementary particles;• Thermal evolution at equilibrium:

• Density of states and macroscopic properties• Number density, energy density and pressure

• Ultra-relativistic limit• Non-relativistic limit

• Effective number of degrees of freedom• Internal degrees of freedom of particles according to the

standard model of particle physics• Evolution of relativistic degrees of freedom

• Entropy at equilibrium• Effective number of degrees of freedom in entropy;• Entropy conservation an its consequences;• Entropy and Temperature – time scaling for relativistic particles

• Key events in the thermal history of the Universe 2

Chapter 3

3

Ch. 2

Ch. 3

Ch. 2

Ch. 3 Ch. 4;

Sec. 2.3

References

Natural Units

4

In Particle Physics and Cosmology the expression “natural units” usually refers to setting the following fundamental constants equal to unity:

These are the speed of light, the Boltzmann constant and the Planck constant (ℏ = ℎ/2&).

8.6

2

As a consequence, the following fundamental properties (time; length, temperature and mass) can be written in units of energy (usually expressed in GeV, MeV, keV):

Where

Natural Units

5

To prove these, use the definitions of the following constants in the IS system and the definition of electron volt in Jules.

Example: of the mass of known particles in MeV:

Classification of elementary particles

6

The Standard Model of Particle Physics (SMPF) predicts various families of particles some of them are fundamental and other “composite” particles.

Fundamental particles are not know to have internal structure. Composite particles have internal structure (i.e. are made of other particles).

All particles of the SMPF can by classified in the following way:

Classification of elementary particles

7

Gauge Fields (exchange Bosons): Are fundamental particles that mediate interactions:

• Photon γ – electromagnetic;• 8 gluons " – strong interaction• # and $± − weak interaction• Graviton? (ℎ#$) – gravitational interaction

(quantum gravity)

Leptons:Are fundamental particles that interact via the electromagnetic and weak forces.

• Come in doublets with respect to the week force• Only distinguishable by the mass• Stable doublet: is the electron/electron neutrino

Hadrons:Have internal structure and interact via all types of forces.

• are made of quarks, confined in sets of 2 (Mesons) or 3 (Baryons) particles: up, down; charm, strange; top; bottom (', ), *, +, ,, -)

Scalar Higgs Boson • Higgs Field: The Higgs mechanism is believed to be the cause the Electroweak symmetry

breaking and describes the generation of the mass of all fermions and massive bosons

Thermal evolution at equilibrium

8

Fundamental assumptions about the primordial universe:• All fluid species are assumed to behave as ideal fluids.

• Thermal equilibrium of a fluid species may be established whenever the particles’ interaction rate, Γ(#), (expressed as the number of interaction events per unit of time) is larger than the expansion rate of the Universe, % # = (̇/(:

Γ # ≫ %(#)

• The best way to describe a fluid component is through its distribution function +(,, ., /, #).This gives the mean number of particle states in the position, , ± 1,, with momentum, . ± 1..

• In classical mechanics + is defined as the number of particles per phase space volume: 23 = + ,, ., /, # 2!4 2!5

• If space is homogeneous, the distribution function must be independent of ,. Moreover assuming isotropy, + must be a function of 5 = |.|, so + = + 5, /, # .


9

The phase-space of a species in Quantum physics:Uncertainty principle (1927): Δ" Δ# ≳ ℎ

Phase space smallest region of confinement:

One-dimension {!, #}:Δ" Δ# = 2(ℏ

Three-dimensions {$,%}:

*+ ,- = 2(ℏ !

Number of ”cells” in the Phase space:(natural units ℏ=1)

& '(')2*ℏ ! =

= ,2* !&'(')

Phase space density


10

The phase-space of a species in Quantum physics:In quantum mechanics the momentum operator ( ) eigenstates of a free particle inside a box of volume, V = (-, has a discrete spectrum of momentum/energy eigenstates, described by the (time-independent) Schrödinger equation:

where, and ) = ℏ*.

The 1D solution for the boundary condition + 0 = + ( = 0

is of the form + - = . sin(*.-), where:

The energy of each mode n is:

In 3D, the possible momentum and energy states are:

kn = n⇡/L, withn > 0<latexit sha1_base64="bFzom8uK8XAeyZQyACx0i8rmeM8=">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</latexit>

p2

2m = �~2r2

2m = E , r2 = �k2

<latexit sha1_base64="tEHLcZFZD1Jw4EwHN667XaawF5Q=">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</latexit>

k2 = 2mE/~2<latexit sha1_base64="sdl7ZR3YYBs+JTl61/4HxGbq2Uo=">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</latexit>

E~n =p2~n2m

=~2k2~n2m

=~22m

⇡2

L2

�n2x + n2

y + n2z

�<latexit sha1_base64="GtprrPYcGP0K5UnyPl2JYnNy7CU=">AAADSHicjVFNT9tAEB2bls+WBLiAerGIKlFVihyDRC9ICITEgUOQCCBhYq2XTbKKv7ReowYrJ/5d/0DFPyA3blVvzC6bqoD4WMvrN2/mPe/shFnEc+m6N5Y98eHj5NT0zOzcp8/zlerC4nGeFoKyFk2jVJyGJGcRT1hLchmx00wwEocROwn7uyp/cslEztPkSA4ydh6TbsI7nBKJVFC93gtK/5JRJxlu+R1BaJn9I9resPTiMe/3QiLaXv/1tKYMk3EVH+DmR6wj15LgZ9v7ngQDvV+1PV/wbk9+C6o1t+7q5TwHDQNqYFYzrf4GHy4gBQoFxMAgAYk4AgI5PmfQABcy5M6hRE4g4jrPYAizqC2wimEFQbaPexejM8MmGCvPXKsp/iXCV6DSga+oSbFOIFZ/c3S+0M6Kfcm71J7qbAP8hsYrRlZCD9m3dOPK9+pULxI68EP3wLGnTDOqO2pcCn0r6uTOf11JdMiQU/gC8wIx1crxPTtak+ve1d0Snb/VlYpVMTW1BYzUKXHAjafjfA6OvXpjve4dbtS2d8yop+ELrMIaznMTtmEfmtBC75FVsZatFfuXfWf/sf8+lNqW0SzBozVh3wOoNcGg</latexit>

En =p2n2m

=~2k2n2m

=~22m

⇡2

L2n

<latexit sha1_base64="i3gFzQfoCraLtzGKyo2skDoNO38=">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</latexit>

p̂ = i~r<latexit sha1_base64="ucXKbFdzZwvOf+hXCS0gi3f63gY=">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</latexit>

<latexit sha1_base64="zBwPccnpBIEytPAWxYnEJry3mrE=">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</latexit>

~p = ~ ⇡

L(nx, ny, nz)


11

The phase-space of a species in Quantum physics:Therefore the allowed momentum eigenstates in one octant of the 4 = (4/ , 40 , 41) space is ()⃗.2 = 27 8):

or So the number of points in this n-space octant is:

Generally, distribution function integrals are done over the whole9, : -space. That would lead to 8 times larger densities, so:

If particle species have g internal degrees of freedom the density of states in natural units in {9, :} is:

because and therefore ℎ = 2&.

~pn =~⇡L

~n<latexit sha1_base64="BAPE4AXycxfO1Qfh8tw/iztdtyo=">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</latexit>

~n =L

~⇡ ~pn<latexit sha1_base64="Qaz8hrqsO/XxsR5cPcw5UNFtLZ4=">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</latexit>

d3n =

✓L

~⇡

◆3

d3p<latexit sha1_base64="n+6VJkmQCnPnOYkP+cA33Gl73CE=">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</latexit>

Phase space density

d3n =1

8

✓L

⇡~

◆3

d3p =

✓L

h

◆3

d3p =1

h3d3x d3p

<latexit sha1_base64="EE9UbU5hkArMznTcoZR0TR7v7p4=">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</latexit>


12

From quantum states to microscopic properties:Under the assumptions of homogeneity and isotropy, the number of particles => = ? 9, :, 8, @ =-- =-) does not depend on x and is only a function of ) = |:|.

The number density of particle states is therefore:

Likewise, one can obtain the energy density of particles in real space by weighting the each momentum eigen-state by its energy, , an therefore:

The computation of the pressure of particles results in a similar way (This can be derived using statistical mechanics assuming a gas of weakly interacting particles, see next slides).


13

Derivation of (done in class),

Lets assume a gas of weakly interacting particles in statistical mechanics.

Consider the area element ./, in the figure on the left. Particles move with 0 1 .

The number of particles in the shaded volume .3 = 1 .4 ./" = 1 .4 .Ω6#is:

Not all particles in dV will hit dA.Only a fraction of this particles, with 71. 89 = :;< = , i.e. with the direction, >, will hit ./. So, assuming isotropy, the number of particles arriving on ./is:

(from Baumann Chap. 3.2)


14

(Derivation continuation…)

If these .?$ particles collide elastically at ./, each particle transfers a momentum 2|-. 7A| (because the particle is assumed to collide elastically, and is reflected with the same angle of impact).

So the pressure .B (defined as force / area = momentum / time / area) by these particles at ./ is:

where and the integration is made over the hemisphere of particles moving towards ./ (i.e. with )


15

Kinetic equilibriumIf particles exchange momentum and energy in an efficient way, the system is said to be in kinetic equilibrium. If the system achieves a maximum entropy state, then particles are distributed according to the Fermi-Dirac or Bose-Einstein distribution functions:

Where ! is the temperature of the system and " is the chemical potential defined as the change of energy with respect of the number of particles, at constant entropy, volume, and number other particle species.

or

At low temperature ! ≪ $ − " both distributions reduce to the Maxwell-Boltzmann distribution:

+ Fermions− Bosons

3, &!"#


16

Particle distribution functions

Undistinguishable particles

obeying to the Pauli’s Principle:

only one particle per state

semi-integer spins integer spins

Undistinguishable particles not subject

to the Pauli’s Principle: many particles can occupy one state

Examples: electron, proton, neutron… Examples: photon, gluon, mesons…


17

Chemical equilibrium• If a particle species, &, is in chemical equilibrium, then "C is related to the

other species chemical potential. For example if one has the following interaction (reaction) among species:

then

• Photons have chemical potential equal to zero, i.e. 'D = ), because the number of photons is not conserved. For example: double scattering interaction

• This implies that a particle, *, and its antiparticle, +*, ( ) have symmetric chemical potentials 'E = −'FE.

Thermal equilibrium• Thermal equilibrium is achieved for species which are both in kinetic

and chemical equilibrium. These species then share the same temperature, .

Thermal evolution at equilibrium:

18

Using the distribution functions one can compute the number and energy densities, and pressure from their expressions in slides 11, 12, with, :

• In general these expressions are solved numerically.

• However, for some cases of interest it is possible to derive analytical solutions.

• These are the cases of ultra-relativistic particles (m≪ :) and non-relativistic (; ≫ :) with vanishing chemical potential (< = 0)


19

Whenever the chemical potential is zero (photons) or negligible (e.g. electrons and protons) the number and energy densities can be written as:

Defining 9 ≡ E/F and ξ ≡ :/F these integrals can be written as

Which in some cases can be evaluated analytically using the Riemann-Zeta and Gama functions. In particular one has:


20

Ultra-relativistic limit: - → 0 (0 ≪ ! and " = 0)For 4→0 (7 ≪ H) on has for the integral part of the number density:

Bosons:I4 0 = J 2 + 1 Γ 2 + 1 = 2J 3 ≃ 2.4

Fermions:

I5 0 = I4 0 − 262-I4 0 =

-7 I4(0) =

-2 J(3)

For Fermions the integral is not directly related with the Riemann integrals. However one can use the mathematical equality,

and then apply the Riemann integral.


21

Ultra-relativistic limit: - → 0 (0 ≪ ! and " = 0 )So one obtains the following expressions for the number density:

Doing a similar computation for the 1± 0 , it is possible to derive the following expression for the energy density:

To compute the pressure for ultra-relativistic particles, 4→0, with <=0, it is straightforward to show that:


22

Non-relativistic limit: - ≫ 1 (0 ≫ ! and " = 0)For 4 ≫ 1 (7 ≫ H ) the number density integral gives the same expression for Fermions and Bosons:

Most of the contribution to this integral comes from ξ ≪ 4 . We can therefore expend the square root in a Taylor expansion to the lowest order in ξ to obtain:

This last integral is related with the Gamma Function integral with n = 2. So one gets:

Which leads to:


23

Non-relativistic limit: - ≫ 1 (0 ≫ ! and " = 0)The number density of non-relativistic particles

This tell us that massive particles are exponentially rare at low temperatures.

For the energy density, at low temperature one has

The energy density integral can be obtained using this expression:

The pressure can be also easily computed, giving


24

Non-relativistic limit: - ≫ 1 (0 ≫ ! and " = 0)From these expressions one concludes that:

• The densities and pressure of non-relativistic particles are strongly suppressed, by the exponential term ?"#/%, as temperature, !, drops bellow the particles mass, ;. This is known as Boltzmann suppression and is due to particle annihilations.

• These annihilations occur due to changes in the interactions involving the particle species. For example in the case of (particle-antiparticle pair production) at low temperature (typically below ~;), the thermal particle energies are not sufficient for pair production.

• Particle annihilations are typically associated with phase transitions,such as happens to the less massivequarks in the QCD phase transition.


25

Non-relativistic limit: - ≫ 1 (0 ≫ ! and " = 0)From these expressions one concludes that:

• The transition from relativistic to non-relativistic behaviour is not instantaneous (in fact about 80% of particle-antiparticle annihilations take place in the temperature range A ∈ [D/E,D]).

• When ; ≫ : the energy density and pressure of non-relativistic particles,

• G = H ; +!&: ≃ H;

• K = H: ≪ G = H;

• This means that non relativistic particles have in general negligible pressure. They behave as a “pressureless dust”, (i.e. as P=0 ‘matter’)

• Note that K = H: ⇔ KM = 3N': (in SI units) is the ideal gas law.

In a nutshell: decoupled non-relativistic particles behave as a gas of pressureless matter


26

Effective number of degrees of freedom of relativistic speciesFor a plasma of relativistic species, with bosons (labelled by i) and fermions (labelled by j) we have that:

The total energy density of relativistic species can therefore be written as:

where : = :( is the photons temperature and O∗is the effective number of degrees of freedom of the fluid at temperature ::

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27

Effective number of degrees of freedom of relativistic speciesThis expression allows that different species may not be in thermal equilibrium with the photon component. In fact we can distinguish two situations:

• For relativistic particles in thermal equilibrium with the photons we have:

when a species become non-relativistic, it is removed from the sums in O ∗*+.So, when : is away from the “mass thresholds” of particles O ∗*+ is independent of temperature

• For relativistic particles that are not in thermal equilibrium (or decoupling) from the photon fluid, O∗ varies with temperature:


28

Inventory of internal degrees of freedom of fundamental particles

Internal degrees of freedom of fundamental particles in the Standard Model of Particle Physics:

• Massless spin-1 (photons and gluons): 2 polarizations

• Massive spin-1 (8±, :#): 3 polarizations

• Massive spin-1/2 leptons (;±, <±, =±): 2 spins

• Massive spin-1/2 quarks: 2 spin and 3 colour states

• Neutrinos/anti-neutrinos: 1 helicity state

So the internal degrees of freedom for relativistic bosons and fermions in equilibrium are:

This gives:


29

Evolution of relativistic degrees of freedom (SMPP)

T~150 MeV:Quarks combine into baryons and mesons.Below ~30 MeV all

Hadrons except the Pionsbecome non-relativistic

T>100 GeV:All particles are

relativistic.By ~100 GeV the Higgs mechanism “gives mass”

to the electro-week (EW) mediators causing

the “EW phase transition”


30

Entropy at equilibriumFrom to the first law of thermodynamics (dR = H=S − T=U ; with V> = 0) one has:

From (3) one can derive that:

The Schwartz theorem applied to the thermodynamic variable Free Energy: allows one to write:

From (2) and the above equation on obtains:

✓@S

@⇢

◆

V

=V

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✓@S

@V

◆

⇢

=⇢+ P

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@P

@T=

@S

@V=

⇢+ P

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31

Entropy at equilibriumThis expression allows defining entropy and entropy density (or specific entropy), up to a constant, as:

The specific entropy of a relativistic boson species i can then be computed as (using the expressions of G,, K,, obtained earlier):

where the 1/3 term comes from the pressure K, = G,/3. A similar result holds for relativistic fermion species:

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Relativistic Bosons

Relativistic Fermions


32

Entropy at equilibriumFor a plasma of relativistic species, with bosons (labelled by i) and fermions (labelled by j) we have that:

The total energy density of relativistic species can therefore be written as:

where : = :( is the photons temperature and O∗is the effective number of degrees of freedom in entropy of the fluid at temperature ::

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33

Entropy at equilibriumOne should note that O∗- is a function of (:,/:)! whereas O∗, varies as (:,/:).This means that:

• Relativistic species in thermal equilibrium (:, = :): O∗- = O∗

• Non-relativistic decoupling species (:, ≠ :): O∗- ≠ O∗

In other words, if one writes

One has that for relativistic species in thermal equilibrium, andfor non-relativistic species in the process of decoupling from fluid.

Slide 29 shows both O∗- (dotted line) and O∗ (solid line). At high values of the degrees of freedom (i.e. higher temperatures) the curves appear on top of each other because the differences are small and only more visible at low :.


34

Conservation of EntropyA most important result about the evolution of the fluid in thermal equilibrium is that its entropy remains constant with the expansion (as opposed to its energy density that decreases with time). This can be proved by taking the time derivative of R:

= 0

• The first term vanishes, because

(FLRW continuity equation) and M = S!(!.• The second term also vanishes, because

WT

WH=WS

WU=(X + T)

H


35

Conservation of EntropyEntropy conservation has two important consequences:

• From S = ZU = [\4Z@. ⇒ Z ∝ _4-

In fact, whenever the number density 4 = >/U ∝ _4- (i.e. away particle massthresholds) one also has that 4/Z = >/S . Since S = const., one can set it to 1,to conclude that 4/Z = >. The same holds for individual species:

• Since and ZU = [\4Z@. one has:

Away from particle mass thresholds (`∗@ = [\4Z@.) one concludes that the temperatureof a relativistic fluid scales with the inverse of a(t). More generally (.> = `∗A

6/-H>_>):

g⇤S(T )T3a3 = g⇤S(Ti)T

3i a

3i = const.

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T = Ti

✓g⇤S(Ti)

g⇤S(T )

◆1/3 aia

= Ai g�1/3⇤S a�1

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36

Conservation of Entropy: Temperature – time dependenceWe can now combine this equation in energy density equation of relativistic particles and in the Friedmann equation to relate temperature with density and time. We have:

Plugging this result in the Friedman Equation (accounting only for relativistic particles) one obtains:

These results show that, whenever `∗ H and `∗@(H) are constants (i.e. away from particle mass thresholds) one obtains:

• the well know scaling for radiation aC ∝ b4D

• the solution of the Friedman equation with a = aC ∝ b4D is: b ∝ c,/E

At particle mass thresholds `∗ H and `∗@(H) are a function of temperature. The solution of the Friedmann equation is numerical and generally leads to deviations to the b ∝ c,/E scaling.

Plugging this in the temperature scaling, one finally obtains F ∝ d∗Fb4, ∝ c4,/E.

⇢r =⇡2

30g⇤T

4 =⇡2

30g⇤

⇣Ai g

�1/3⇤S a�1

⌘4=

⇡2

30A4

i

⇣g⇤g

�4/3⇤S

⌘a�4

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H2 =

8⇡G

3⇢r =

8⇡G

3

⇡2

30A

4i

⇣g⇤g

�4/3⇤S

⌘a�4

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37

Conservation of Entropy: Temperature – time dependenceDoing the maths, one can obtain the exact time dependence of the temperature of the relativistic fluid. Typically one obtains:

(which allows to write the rule of thumb: : ∼ 1MeV at about 1 second after the Big-Bang)

This expression allows one to establish a direct correspondence between a given energy scale of the relativistic fluid and time until the end of the radiation domination period. Beyond the radiation domination phase one needs to account for the other terms in the Friedman equation (see next slide).


38

Key events in the thermal history of the universe

The previous sets of equations allows one compute all thermodynamic properties of the primordial relativistic fluid and establish their dependence with time and redshifts.

All we need to know is of what the universe is made of and the physics of each of its components!


39


Baryogenesis: Quantum field theory requires the existence of anti-particles. This poses a problem: particle-antiparticle creation and annihilation (allowed by the Heisenberg principle) creates/destroys equal amounts of particle and anti-particles. However, we do observe an excess of matter (mostly baryons) over anti-matter! Models of baryogenesis attempt to describe this observational evidence using some dynamical mechanism (instead of assuming this particle-anti-particle asymmetry ab initio)

Electroweak phase transition:At ~100 GeV particles acquire mass through the Higgs mechanism. This leads to a drastic change of the weak interaction. The gauge bosons eG, f± become massive and soon after decouple from thermal equilibrium.

QCD phase transition:Above ~150 MeV quarks are asymptotically free (i.e. weakly interacting). Below this energy/mass threshold the strong force (mediated by the gluons) becomes more intense; the more massive quarks start to decouple from the fluid. The less massive become confined (with the gluons) inside the baryons (3 quarks + gluons) and mesons (quarks+anti-quark + gluons)


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Dark Matter freeze-out: Present observations indicates that dark matter is very-weakly interacting (or non-interacting). Depending on the mass of the dark matter candidates one should expect that it should decouple from the fluid early on. For example, if dark matter is made of WIMPs (weakly interactive massive particle), one should expect that their abundance should freeze around 1 MeV

Neutrino decoupling:Neutrinos only interact with the rest of the plasma through the weak force. They are expected to decouple from the fluid at ~0.8 MeV.

Electron-positron annihilation:Electrons and positron annihilate soon after the neutrinos. Positrons vanish, because electron-positron pair production is strongly suppressed below ~1MeV

Big Bang Nucleosynthesis:At ~0.1MeV (~3 minutes after the Big-Bang) protons and neutrons combine to form the first light nuclear elements.


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Recombination: At ~0.3 eV (260-380 kyr) free electrons combine with nuclei to form atoms. Predominantly Hydrogen: . Below this range of energies, this chemical reaction can no longer occur in the reverse order.

Photon decoupling:By ~0.23 eV (380 kyr) the primordial fluid is reduced to photons, that no longer interact with matter (free electrons). The Cosmic Microwave Background radiation propagates freely in the Universe.


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Brief history of the Universe

Universo Primitivo - ULisboaUniverso Primitivo 2020-2021 (1º Semestre) Mestrado em Física...

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