Cosmological Inflation: The Canonic Theory

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Inflation hasn't won the race,
But so far it's the only horse
Andrei Linde.



Scalar Field In Cosmology

Problem 1

A scalar field $\varphi(\vec r,t)$ in a potential $V(\varphi)$ on flat background is described by Lagrangian \[L=\frac12\left(\dot\varphi^2-\nabla\varphi\cdot\nabla\varphi\right)-V(\varphi)\] Obtain the equation of motion (evolution) for this field from the least action principle.


Problem 2

Rewrite the action for free scalar field minimally coupled to gravitation \[S_\varphi=\int d^4x\sqrt{-g}\left(\frac12 g^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi\right)\] for the case of FLRW metric.


Problem 3

Using the action obtained in the previous problem, obtain the evolution equation for the scalar field in the expanding Universe.


Problem 4

Calculate the density and pressure of homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ in the FLRW metric.


Problem 5

Starting from the scalar field's action in the form \[ S = \int {d^4 x\sqrt { - g} \left[ {{1 \over 2}(\nabla \varphi )^2 - V(\varphi )} \right]} \] obtain the equation of motion for this field for the case of FLRW metric.



Problem 6

Construct the Lagrange function describing the dynamics of the Universe filled with a scalar field in potential $V(\varphi)$. Using the obtained Lagrangian, obtain the Friedman equations and the Klein--Gordon equation.



Problem 7

Obtain the equation of motion for a homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ starting from the conservation equation \[\dot\rho+3\frac{\dot a}{a}(\rho+p)=0.\]



Problem 8

Obtain the equation of motion for the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ using the analogy with Newtonian dynamics.



Problem 9

Express $V(\varphi)$ and $\varphi$ through the Hubble parameter $H$ and its derivative $\dot{H}$ for the Universe filled with quintessence.

Problem 10

Show that Friedman equations for the scalar field $\varphi(t)$ in potential $V(\varphi)$ can be presented in the form \[H^2=\frac{8\pi G}{3}\left(\frac12 \dot\varphi^2+V(\varphi)\right),\] \[\dot H=-4\pi G\dot\varphi^2.\]



Problem 11

Provided that the scalar field $\varphi(t)$ is a single--valued function of time, transform the second order equation for the scalar field into a system of first order equations.



Problem 12

Rewrite the equations for the scalar field in terms of conformal time.



Problem 13

Show that condition $\dot H>0$ cannot be realized for the scalar field with positively defined kinetic energy.



Problem 14

Show that the Klein--Gordon equation could be rewritten in dimensionless form $$ \varphi '' + \left( {2 - q} \right)\varphi ' = \chi ;\quad \chi \equiv - \frac{1}{H^2 }\frac{dV}{d\varphi }, $$ where prime denotes the derivative by $\ln a$, and $q = - {{a\ddot a} / {\dot a^2 }}$ is the deceleration parameter.



Problem 15

Represent the equation of motion for the scalar field in the form $$ \pm \frac{V_{,\varphi}}{V} = \sqrt {\frac{3(1 + w)}{\Omega _\varphi(a)}} \left[ 1 + \frac{1}{6}\frac{d\ln \left( x_{\varphi} \right)}{d\ln a} \right], $$ where $$ x_{\varphi}=\frac{1+w_{\varphi}}{1-w_{\varphi}}, \quad w_{\varphi}=\frac{\dot{\varphi}^2+2V(\varphi)}{\dot{\varphi}^2-2V(\varphi)}, $$ in the system of units such that $8\pi G=1.$


Problem 16

The term $3H\dot{\varphi}$ in the equation for the scalar field formally acts as friction that damps the inflation. Show that, nonetheless, this term does not lead to dissipative energy production.


Problem 17

Obtain the system of equations describing the scalar field dynamics in the expanding Universe containing radiation and matter in the conformal time.


Problem 18

Calculate the pressure of homogeneous scalar field in the potential $V(\varphi)$ using the obtained above energy density of the field and its equation of motion.


Problem 19

What condition should the homogeneous scalar field $\varphi(t)$ in potential $V(\varphi)$ satisfy in order to provide accelerated expansion of the Universe?


Problem 20

What conditions should the scalar field satisfy in order to provide expansion of the Universe close to the exponential one?

Introduction to Inflation

Problem 21

What considerations led A.Guth to name his theory describing the early Universe's dynamics as inflation theory?


Problem 22

A. Vilenkin in his cosmological bestseller Many world in one remembers: On a Wednesday afternoon, in the winter of 1980, I was sitting in a fully packed Harvard auditorium, listening to the most fascinating talk I had heard in many years. The speaker was Alan Guth, a young physicist from Stanford, and the topic was a new theory for the origin of the universe$\ldots$ The beauty of the idea was that in a single shot inflation explained why the universe is so big, why it is expanding, and why it was so hot at the beginning. A huge expanding universe was produced from almost nothing. All that was needed was a microscopic chunk of repulsive gravity material. Guth admitted he did not know where the initial chunk came from, but that detail could be worked out later. It's often said that you cannot get something for nothing. he said, but the universe may be the ultimate free lunch . Explain, why can this be.


Problem 23

Is energy conservation violated during the inflation?


Problem 24

Inflation is defined as any epoch for which the scale factor of the Universe has accelerated growth, i.e. $\ddot{a}>0$. Show that this condition is equivalent to requirement that the comoving Hubble radius decreases with time.


Problem 25

Show that in the process of inflation the curvature term in the Friedman equation becomes negligible. Even if that condition was not initially satisfied, the inflation quickly realizes it.


Problem 26

It is sometimes said, that the choice of $k = 0$ is motivated by observations: the density of curvature is close to zero. Is this claim correct?

Inflation in the Slow-Roll Regime

Problem 27

Obtain the evolution equations for the scalar field in expanding Universe in the inflationary slow-roll regime.


Problem 28

Find the time dependence of scale factor in the slow--roll regime for the case $V(\varphi)={m^2 \varphi^2 / 2}$.


Problem 29

Find the dependence of scale factor on the scalar field in the slow-roll regime.


Problem 30

Show that the conditions for realization of the slow--roll limit can be presented in the form: \[\varepsilon(\varphi)\equiv\frac{M^{*2}_{Pl}}{2}\left(\frac{V^\prime}{V}\right)^2\ll1; \ |\eta(\varphi)|\equiv\left|M^{*2}_{Pl}\frac{V^{\prime\prime}}{V}\right|\ll1; \ M^*_{Pl}\equiv(8\pi G)^{-1/2}.\]


Problem 31

Show that the condition $\varepsilon\ll1$ for the realization of the slow--roll limit obtained in the previous problem is also the sufficient condition for inflation.


Problem 32

Find the slow--roll condition for power law potentials.


Problem 33

Show that the condition $\varepsilon \ll\eta$ is satisfied in the vicinity of inflection point of the inflationary potential $V(\varphi)$.


Problem 34

Show that the inflation parameter $\varepsilon$ can be expressed through the state equation parameter $w$ for the scalar field.



Problem 35

Show that the second Friedman equation \[\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\] can be presented in the form \[\frac{\ddot{a}}{a}=H^2(1-\varepsilon).\]



Problem 36

Show that in the slow--roll regime $\varepsilon_H\rightarrow\varepsilon$ and $\eta_H\rightarrow\eta-\varepsilon$.



Problem 37

Show that the inflation parameters $\varepsilon_H,\ \eta_H$ can be presented in the following symmetric form \[\varepsilon_H=-\frac{d\ln H}{d\ln a};\ \eta_H=-\frac{d\ln H'}{d\ln a}.\]



Problem 38

Prove that condition $\ddot a>0$, which defines inflation, is equivalent to $\varepsilon_H<1$.



Problem 39

Show that inflation appears every time when the scalar field's value exceeds the Planck mass.



Problem 40

Find the energy momentum tensor for homogeneous scalar field in the slow--roll regime.

Solution of the Hot Big Bang Theory Problems

Problem 41

Show that in the inflation epoch the relative density $\Omega$ exponentially tends to unity.



Problem 42

Estimate the temperature of the Universe at the end of inflation.



Problem 43

Estimate the size of the Universe at the end of inflation.



Problem 44

Find the number $N_e$ of $e$-foldings of the scale factor during the inflation epoch.



Problem 45

Find the number $N_e$ of $e$-foldings of the scale factor for the inflation process near the inflection point.



Problem 46

Show that inflation transforms the unstable fixed point $x=0$ for the quantity \[x\equiv\frac{\Omega-1}{\Omega}\] into the stable one, therefore solving the problem of the flatness of the Universe.



Problem 47

Find the particle horizon in the inflationary regime, assuming $H\approx const$.



Problem 48

Find the solution of the horizon problem in the framework of inflation theory.



Problem 49

Did entropy change during the inflation period? If yes, then estimate what its change was.



Problem 50

Does the inflation theory explain the modern value of entropy of the Universe?



Problem 51

Find the solution of the monopole problem in the frame of inflation theory.