Different Models of Inflation
There is a number of inflationary models. All of them deal with potentials of scalar fields which realize the slow-roll regime during sufficiently long period of evolution, then the inflation terminates and Universe enters the hot stage. It is worth noting that the models considered below are among the simplest ones and they do not exhaust all the possibilities, however they give the main idea about possible features of the evolution of scale factor in the slow-roll regime of inflation.
Contents
- 1 Chaotic Inflation (Inflation with Power Law Potential)
- 1.1 Problem 1
- 1.2 Problem 2
- 1.3 Problem 3
- 1.4 Problem 4
- 1.5 Problem 5
- 1.6 Problem 6
- 1.7 Problem 7
- 1.8 Problem 8
- 1.9 Problem 9
- 1.10 Problem 10
- 1.11 Problem 11
- 1.12 Problem 12
- 1.13 Problem 13
- 1.14 Problem 14
- 1.15 Problem 15
- 1.16 Problem 16
- 1.17 Problem 17
- 1.18 Problem 18
- 1.19 Problem 19
- 1.20 Problem 20
Chaotic Inflation (Inflation with Power Law Potential)
The chaotic inflation, or the inflation with high field, is considered as a rule with the power law potentials of the form $$V=g\varphi^n,$$ where $g$ is a dimensional constant of interaction: $$[g]=(\mbox{mass})^{4-n}$$ It should be noted that the slow-roll conditions for the given potential are always satisfied for sufficiently high values of the inflaton field $$\varphi\gg\frac{nM_{Pl}}{4\sqrt{3\pi}},$$ therefore the slow-roll takes place at field values which are great compared to Planck units.}
Problem 1
Consider inflation with simple power law potential
$$V=g\phi^n,$$
and show that there is wide range of scalar field values where classical Einstein equations are applicable and the slow-roll regime is realized too. Assume that the interaction constant $g$ is sufficiently small in Planck units.
Problem 2
Estimate the total duration of chaotic inflation in the case of power law potentials of second and forth order.
Problem 3
Express the slow-roll parameters for power law potentials in terms of $e$-folding number $N_e$ till the end of inflation.
Problem 4
Obtain the number $N$ of $e$-fold increase of the scale factor in the model $$V\left( \varphi \right) = \lambda \varphi ^4 \quad \left( {\lambda = 10^{ - 10} } \right).$$
Use the result of the previous problem to obtain for the power-law potential $V\left( \varphi \right) = \lambda \varphi ^n $ and $$ \varphi _i \gg \varphi _e $$ $$ N \sim \int_{\varphi _e }^{\varphi _i } \frac{\varphi d\varphi }{M_{Pl}^2 } \sim \frac{\varphi _i^2 }{M_{Pl}^2 }. $$ As $V(\varphi _i ) \sim M_{Pl}^4 $, in the considered model $\varphi _i \sim \lambda ^{ - 1/4} M_{Pl} $ and therefore $$ N \sim 10^5. $$
Problem 5
Estimate the range of scalar field values corresponding to the inflation epoch in the model $$V(\varphi ) = \lambda \varphi ^4 \left( {\lambda \ll 1} \right).$$
$$ V\left( {\varphi _0 } \right) \sim M_{Pl}^4 $$ For the considered potential $$ \phi _b \sim \lambda ^{ - 1/4} M_{Pl} \gg M_{Pl} $$ inflation will continue until the scalar field's amplitude decreases to a value of the order of Planck mass (see previous problem). Thus inflation period corresponds to the following interval of the scalar field's values $$ \lambda ^{ - 1/4} M_{Pl} < \phi < M_{Pl}. $$
Problem 6
Show that the classical analysis of the evolution of the Universe is applicable for the scalar field value $\varphi\gg M_{Pl}$, which allows the inflation to start.
For great values of scalar field $\varphi $ ($\varphi > M_{Pl} $), when the inflation process is possible, the energy density can still remain less than the Planck value, $M_{Pl}^4 $. Consider for example the potential $V\left( \varphi \right) = \lambda \varphi ^4 $, where $\lambda $ is a dimensionless coupling constant. Under the condition $\lambda \ll 1$ the energy density at $\varphi \sim M_{Pl} $ is less than the Planck one $V(\phi \sim M_{Pl} ) \sim \lambda M_{Pl}^4 \ll M_{Pl}^4.$
Problem 7
Find the time dependence for the scale factor in the inflation regime for potential $(1/n)\lambda\varphi^n$, assuming $\varphi\gg M_{Pl}$.
The condition $\left| \varphi \right| \gg M_{Pl} $ guarantees realization of the slow-roll regime, then $$ a(\varphi ) \simeq a_0 \exp \left( {8\pi G\int_\varphi ^{\varphi _0 } {\frac{V}[[:Template:V'(\varphi )]]d\varphi } } \right). $$ Therefore for any potential $V(\varphi ) = \left( {1/n} \right)\lambda \varphi ^n $ one obtains $$ a(\varphi (t)) \simeq a_0 \exp \left( {4\pi G/n\left( {\varphi _0^2 - \varphi ^2 (t)} \right)} \right). $$
Problem 8
The inflation conditions definitely break down near the minimum of the inflaton potential and the Universe leaves the inflation regime. The scalar field starts to oscillate near the minimum. Assuming that the oscillations' period is much smaller than the cosmological time scale, determine the effective state equation near the minimum of the inflaton potential.
Neglecting the expansion, represent the equation for scalar field $$ \ddot \varphi + 3H\dot \varphi + V'_\varphi = 0 $$ in the form $$ \frac{d}{dt}\left( {\varphi \dot \varphi } \right) - \dot \varphi ^2 + \varphi V'_\varphi = 0. $$ After averaging over the period of oscillations the first term turns to zero and therefore $$ \left\langle {\dot \varphi ^2 } \right\rangle \simeq \left\langle {\varphi V'_\varphi } \right\rangle . $$ The effective (averaged) equation of state reads $$ w \equiv \frac{p}{\rho } \simeq \frac{{\left\langle {\varphi V'_\varphi } \right\rangle - \left\langle {2V} \right\rangle }}{{\left\langle {\varphi V'_\varphi } \right\rangle + \left\langle {2V} \right\rangle }}. $$
Problem 9
Show that effective state equation for the scalar field, obtained in the previous problem for potential $V\propto\varphi^n$, in the case $n=2$ corresponds to non-relativistic matter and for $n=4$ - to the ultra-relativistic component (radiation).
Using the result of previous problem, for $V \propto \varphi ^n$ one obtains $$ w \simeq \frac{n - 2}{n + 2}. $$ The case $n = 2$ corresponds to $w = 0$ (non-relativistic matter), and $w = 1/3$ - to $n = 4$ (radiation).
Problem 10
Obtain the time dependence of scalar field near the minimum of the potential.
Let the potential have a minimum at $\varphi = 0$ and in the vicinity of the minimum $$ V(\varphi ) = \frac{m^2 \varphi ^2 }{2}. $$ The equation of motion for the scalar field in this potential reads $$ \ddot \varphi + 3H\dot \varphi + m^2 \varphi = 0. $$ Substitution $\varphi (t) = a^{ - 3/2} \chi (t)$ makes transition to an oscillator with variable frequancy and enables us to get rid of the first derivative $$ \ddot \chi + \left( {m^2 - \frac{3}{2}\frac{\ddot a}{a} - \frac{3}{4}\frac{\dot a^2 }{a^2 }} \right)\chi = 0. $$ In the regime of fast oscillations ($m^2 \gg H^2$) $$ \chi (t) = A\cos \left( {mt + \alpha } \right) $$ and the scalar field approaches the minimum as $$ \varphi(t) = Ca^{ - 3/2}\cos (mt + \alpha ). $$
Problem 11
Find the energy-momentum tensor of a homogeneous scalar field in the regime of fast oscillations near the potential's minimum.
Use the solutions $\varphi (t)$ obtained in the previous problem for potential \[V(\varphi ) = \frac{m^2 \varphi ^2 }{2}\] and the definitions $$ T_{00} = \frac{1}{2}\dot \varphi ^2 + V\left( \varphi \right) = \rho _\varphi ;\quad T_{ij} = \left( {\frac{1}{2}\dot \varphi ^2 - V\left( \varphi \right)} \right)\delta _{ij} = p_\varphi \delta _{ij} $$ to obtain $$ T_{00} = \rho _\varphi = a^{ - 3} \frac{m^2 C^2 }{2};\quad T_{ij} = p_\varphi \delta _{ij} = - a^{ - 3} \frac{m^2 C^2 }{2}\cos \left( {2mt + 2\alpha } \right). $$ Average over the oscillation period to find the energy-momentum tensor for non-relativistic matter: $\rho \propto a^{ - 3} ,~p = 0$.
Problem 12
Estimate the total duration of chaotic inflation in the case of power law potentials of second and forth order.
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Problem 13
Express the slow-roll parameters for power law potentials in terms of $e$-folding number $N_e$ till the end of inflation.
Problem 14
Obtain the number $N$ of $e$-fold increase of the scale factor in the model $$V\left( \varphi \right) = \lambda \varphi ^4 \quad \left( {\lambda = 10^{ - 10} } \right).$$
Use the result of the previous problem to obtain for the power-law potential $V\left( \varphi \right) = \lambda \varphi ^n $ and $$ \varphi _i \gg \varphi _e $$ $$ N \sim \int_{\varphi _e }^{\varphi _i } \frac{\varphi d\varphi }{M_{Pl}^2 } \sim \frac{\varphi _i^2 }{M_{Pl}^2 }. $$ As $V(\varphi _i ) \sim M_{Pl}^4 $, in the considered model $\varphi _i \sim \lambda ^{ - 1/4} M_{Pl} $ and therefore $$ N \sim 10^5. $$
Problem 15
Estimate the range of scalar field values corresponding to the inflation epoch in the model $$V(\varphi ) = \lambda \varphi ^4 \left( {\lambda \ll 1} \right).$$
$$ V\left( {\varphi _0 } \right) \sim M_{Pl}^4 $$ For the considered potential $$ \phi _b \sim \lambda ^{ - 1/4} M_{Pl} \gg M_{Pl} $$ inflation will continue until the scalar field's amplitude decreases to a value of the order of Planck mass (see previous problem). Thus inflation period corresponds to the following interval of the scalar field's values $$ \lambda ^{ - 1/4} M_{Pl} < \phi < M_{Pl}. $$
Problem 16
Show that the classical analysis of the evolution of the Universe is applicable for the scalar field value $\varphi\gg M_{Pl}$, which allows the inflation to start.
For great values of scalar field $\varphi $ ($\varphi > M_{Pl} $), when the inflation process is possible, the energy density can still remain less than the Planck value, $M_{Pl}^4 $. Consider for example the potential $V\left( \varphi \right) = \lambda \varphi ^4 $, where $\lambda $ is a dimensionless coupling constant. Under the condition $\lambda \ll 1$ the energy density at $\varphi \sim M_{Pl} $ is less than the Planck one $V(\phi \sim M_{Pl} ) \sim \lambda M_{Pl}^4 \ll M_{Pl}^4.$
Problem 17
Find the time dependence for the scale factor in the inflation regime for potential $(1/n)\lambda\varphi^n$, assuming $\varphi\gg M_{Pl}$.
The condition $\left| \varphi \right| \gg M_{Pl} $ guarantees realization of the slow-roll regime, then $$ a(\varphi ) \simeq a_0 \exp \left( {8\pi G\int_\varphi ^{\varphi _0 } {\frac{V}[[:Template:V'(\varphi )]]d\varphi } } \right). $$ Therefore for any potential $V(\varphi ) = \left( {1/n} \right)\lambda \varphi ^n $ one obtains $$ a(\varphi (t)) \simeq a_0 \exp \left( {4\pi G/n\left( {\varphi _0^2 - \varphi ^2 (t)} \right)} \right). $$
Problem 18
The inflation conditions definitely break down near the minimum of the inflaton potential and the Universe leaves the inflation regime. The scalar field starts to oscillate near the minimum. Assuming that the oscillations' period is much smaller than the cosmological time scale, determine the effective state equation near the minimum of the inflaton potential.
Neglecting the expansion, represent the equation for scalar field $$ \ddot \varphi + 3H\dot \varphi + V'_\varphi = 0 $$ in the form $$ \frac{d}{dt}\left( {\varphi \dot \varphi } \right) - \dot \varphi ^2 + \varphi V'_\varphi = 0. $$ After averaging over the period of oscillations the first term turns to zero and therefore $$ \left\langle {\dot \varphi ^2 } \right\rangle \simeq \left\langle {\varphi V'_\varphi } \right\rangle . $$ The effective (averaged) equation of state reads $$ w \equiv \frac{p}{\rho } \simeq \frac{{\left\langle {\varphi V'_\varphi } \right\rangle - \left\langle {2V} \right\rangle }}{{\left\langle {\varphi V'_\varphi } \right\rangle + \left\langle {2V} \right\rangle }}. $$
Problem 19
Show that effective state equation for the scalar field, obtained in the previous problem for potential $V\propto\varphi^n$, in the case $n=2$ corresponds to non-relativistic matter and for $n=4$ - to the ultra-relativistic component (radiation).
Using the result of previous problem, for $V \propto \varphi ^n$ one obtains $$ w \simeq \frac{n - 2}{n + 2}. $$ The case $n = 2$ corresponds to $w = 0$ (non-relativistic matter), and $w = 1/3$ - to $n = 4$ (radiation).
Problem 20
Obtain the time dependence of scalar field near the minimum of the potential.
Let the potential have a minimum at $\varphi = 0$ and in the vicinity of the minimum $$ V(\varphi ) = \frac{m^2 \varphi ^2 }{2}. $$ The equation of motion for the scalar field in this potential reads $$ \ddot \varphi + 3H\dot \varphi + m^2 \varphi = 0. $$ Substitution $\varphi (t) = a^{ - 3/2} \chi (t)$ makes transition to an oscillator with variable frequancy and enables us to get rid of the first derivative $$ \ddot \chi + \left( {m^2 - \frac{3}{2}\frac{\ddot a}{a} - \frac{3}{4}\frac{\dot a^2 }{a^2 }} \right)\chi = 0. $$ In the regime of fast oscillations ($m^2 \gg H^2$) $$ \chi (t) = A\cos \left( {mt + \alpha } \right) $$ and the scalar field approaches the minimum as $$ \varphi(t) = Ca^{ - 3/2}\cos (mt + \alpha ). $$