Category:Quantum Cosmology

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Problem 1

problem id: QC_1

Consider two neutral particles of equal mass $M$ in flat space, whose motion obeys a Schrodinger equation. The stationary ground state of the system is the analog of an atom, but with gravitational binding instead of electricity. Estimate the size $R$ of the ground state wave function. (Craig J. Hogan, Quantum indeterminacy in local measurement of cosmic expansion, 1312.7797)

Problem 2

problem id: QC_2

Obtain the condition for the acceleration due to the gravitation of the bodies to be smaller than the cosmic acceleration, and their gravitational binding energy to be less than their cosmic expansion kinetic energy. Give an interpretation of the obtained inequality in terms of the gravitational free-fall time and the space-time curvature.

Problem 3

problem id: QC_3

The standard quantum uncertainty in position $x$ of a body of mass $M$ measured at two times separated by an interval $\tau$ is \begin{equation}\label{quantum} \Delta x_q(\tau)^2\equiv \langle (x(t)-x(t+\tau))^2\rangle> 2\hbar \tau/ M. \end{equation} Consider two bodies of identical mass $M$ in a state of minimal relative displacement uncertainty $\Delta x$. What value of their masses should be in order to make the uncertainty in position to be less than the change in their separation due to cosmic expansion.

Problem 4

problem id: QC_4

Estimate minimum size of a system needed for the uncertainty in position of its parts to be less than the change in their separation due to cosmic expansion.

Problem 5

problem id: QC_5

(see M.Gasperini, String theory and primordial cosmology, 1402.0101)

Like all classical theories, GR has a limited validity range. Because of those limits the standard cosmological model cannot be extrapolated to physical regimes where the energy and the space-time curvature are too high. Show that SCM is not applicable in the vicinity of the initial singularity (Big Bang).

Problem 6

problem id: QC_6

Construct Lagrangian and hamiltonian corresponding to the action \[S=\int\sqrt{-g}\left(\frac R{16\pi G}-\rho\right),\quad \rho=\sum\limits_i\rho_i.\]

Problem 7

problem id: QC_7

Using the lagrangian obtained in the previous problem, obtain the equation of motion for scale factor in closed FRLW Universe, filled with cosmological constant $\rho=\rho_\Lambda$ and show that it coincides with the first Friedmann equation (see Vilenkin, 9507018)

Problem 8

problem id: QC_8

Obtain and analyze the solution $a(t)$ of the equation obtained in the previous equation.

Problem 9

problem id: QC_9

Construct an effective Hamiltonian for the de Sitter model.

Problem 10

problem id: QC_10

Using result of the previous problem, obtain the Wheeler-DeWitt equation for the de Sitter model.

Problem 11

problem id: QC_11

Obtain Wheeler-de Witt equation for a Universe filled by cosmological constant and radiation. (1301.4569)

Introduction to vacuum fluctuations

(V. F. Mukhanov and S. Winitzki, Quantum Fields in Classical Backgrounds, Lecture notes --- 2004)

A vacuum fluctuations are fluctuation of an "empty" space. The word empty quoted because the quantum vacuum represents a collection of zero-point oscillations of quantum fields. We shall consider only the scalar fields. A free real massive classical scalar field $\varphi(x,t)$ (in a stationary Universe) satisfies the Klein-Gordon equation \[\ddot\varphi-\Delta\varphi+m^2\varphi=0.\] It is convenient to use the spatial Fourier decomposition, \[\varphi(\vec{x},t)=\frac1{(2\pi)^{3/2}}\int d^3k e^{-i\vec{k}\vec{x}}\varphi_{\vec{k}}(t).\] It is useful to consider a field $\varphi(\vec{x},t)$ in a box (a cube with sides $L$) of finite volume $V$ with the periodic boundary conditions imposed on the field $\varphi$ at the box boundary. In this case the Fourier decomposition can be written as \[\varphi(\vec{x},t)=\frac1{\sqrt V}\sum\limits_k \varphi_{\vec{k}}(t)e^{i\vec{k}\vec{x}},\] \[\varphi_{\vec{k}}(t)=\frac1{\sqrt V}\int d^3x \varphi(\vec{x},t)e^{i\vec{k}\vec{x}},\] where the sum goes over three-dimensional wave numbers $k$ with components of the form \[k_{(x,y,z)}=\frac{2\pi n_{(x,y,z)}}{L},\quad n_{(x,y,z)}=0,\pm1,\pm2,\ldots\] The Klein-Gordon equation for each $k$-component is \[\ddot\varphi_{\vec{k}}+(k^2+m^2)\varphi_{\vec{k}}=0.\] Each (in general) complex function $\varphi_{\vec{k}}(t)$ satisfies the harmonic oscillator equation with the frequency $\omega_k=(k^2+m^2)^{1/2}$. The functions $\varphi_{\vec{k}}(t)$ are called the modes of the field $\varphi$.
To quantize the field, each mode $\varphi_{\vec{k}}(t)$ is quantized as a separate harmonic oscillator. We replace the classical "coordinates" $\varphi_{\vec{k}}$ and momenta $\pi_{\vec{k}}\equiv\dot\varphi^*_{\vec{k}}$ by operators $\hat\varphi_{\vec{k}}$ and $\hat\pi_{\vec{k}}$ with the equal-time commutation relations \[[\hat\varphi_{\vec{k}},\hat\pi_{\vec{k}}]=i\delta(\vec{k}+\vec{k}').\]

Problem 12

problem id: QC_12

Construct the wave function of the vacuum state of a scalar field.

Problem 13

problem id: QC_13

Find the wave function, obtained in the previous problem, in the limit of quantization inside an infinitely large box ($V\to\infty$).

Problem 14

problem id:

Estimate the typical amplitude $\delta\varphi_{\vec{k}}$ of fluctuations in the mode $\varphi_{\vec{k}}$.

Problem 15

problem id: QC_15

Show, that if $\varphi_L$ is the average of $\varphi(t)$ over a volume $V=L^3$, the typical fluctuation $\delta\varphi_L$ of $\varphi_L$ is \[\langle\varphi_L^2\rangle\sim k^3\left(\delta\varphi_{\vec{k}}\right)_{k=L^{-1}}\].

Problem 16

problem id: QC_16

Analyze the expression for amplitude of the scalar field fluctuations, obtained in the previous problem, as a function of linear size $L$ of the region of the averaging.

Problem 17

problem id:

Obtain the modified Raychaudhuri equation in LQC.

Problem 18

problem id: QC_18

Consider a model of Universe filled with dust-like matter, i.e. $p=0$, to demonstrate the main distinction between the standard (Fridmannian) and the LQC (see A. Barrau1, T. Cailleteau, J. Grain4, and J. Mielczarek, Observational issues in loop quantum cosmology, arXiv:1309.6896 ).

Problem 19

problem id: QC_19

Estimate the typical wavelength of photons radiated by a black hole of mass M and compare it with the size of the black hole (the Schwarzschild radius).

Problem 20

problem id: QC_20

The temperature of a sufficiently small black hole can be high enough to efficiently produce baryons (e.g. protons) as components of the Hawking radiation. Estimate the required mass M of such black holes and compare their Schwarzschild radius with the size of the proton (its Compton length).

Problem 21

problem id: QC_21

(see V.Frolov, A.Zelnikov, Introduction to black hole physics, Oxford University Press, 2011)

GR allows the existence of black holes of arbitrary mass. Why do we not observe formation of small mass black holes in the laboratory or our everyday life?

Problem 22

problem id: QC_22

Black holes of small masses can be created in the early Universe when the matter density was high. Such black holes are called the primordial black holes (PBHs). The mass spectrum of the PBHs could span an enormous mass range. Determine the mass range of the PBHs, created during the radiation dominated epoch $10^{-43}sec<t<1sec$.

Problem 23

problem id: QC_23

(see V. F. Mukhanov and S. Winitzki, Introduction to Quantum Fields in Classical Backgrounds, Lecture notes - 2004)

A glass of water is moving with constant acceleration. Determine the smallest acceleration that would make the water boil due to the Unruh effect.

In the de Broglie-Bohm causal interpretation of quantum mechanics the Schrodinger equation for a single nonrelativistic particle \[i\hbar\frac{\partial\psi(x,t)}{\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+V(x)\right]\psi(x,t)\] with the substitution $\Psi=Re^{iS/\hbar}$ is transformed to the system of equations \begin{align} \nonumber \frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V-\frac{\hbar^2}{2m}\frac{\Delta R}R &=0,\\ \nonumber \frac{\partial R^2}{\partial t} + \nabla\left(R^2\frac{\nabla S}{m}\right) &=0. \end{align} The first equation is a Hamilton-Jacobi type equation for a particle submitted to an external potential which is the classical potential plus a new quantum potential \[Q=-\frac{\hbar^2}{2m}\frac{\Delta R}R\] The second equation can be treated as the continuity equation for a fluid with the density $\rho=R^2$. In a series of the problems below let us confine ourselves to the simple case when the Universe is supposed to be filled with only one component, namely, the nonrelativistic gas of point-like particles of the equal mass $m$. Then the energy-momentum tensor components have the following form: \[T^{\mu\nu}=(\rho_m+\rho_q+p_q)-p_qg^{\mu\nu}\] where $\rho_m$ is the classical energy density of nonrelativistic component (the corresponding classical pressure $p_m$ is equal to zero), while $\rho_q$ and $p_q$ represent quantum admixtures, generated by interaction with the quantum potential.

Problem 24

problem id: QC_24

Obtain relativistic generalization for the quantum potential.

Problem 25

problem id: QC_25

Find energy density $\rho_q$ of quantum admixture for the case when the background density $\rho_m$ represents a gas of point-like particles of the equal masses $m$.

Problem 26

problem id: QC_26

Calculate energy density and pressure generated by the quantum potential in Friedmannian Universe filled with non-relativistic matter.

Problem 27

problem id: QC_27

Show that in the limit $a\to\infty$ the de Broglie-Bohm model disregards quantum effects.

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