Shape of the inflaton potential and the

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Web End = Eur. Phys. J. C (2015) 75:437DOI 10.1140/epjc/s10052-015-3666-4

Regular Article - Theoretical Physics

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Web End = Shape of the inaton potential and the efciency of the universe heating

A. D. Dolgov1,2,3,a, A. V. Popov4,b, A. S. Rudenko1,5,c

1 Novosibirsk State University, Novosibirsk 630090, Russia

2 ITEP, Bol. Cheremushkinskaya ul., 25, Moscow 113259, Russia

3 Dipartimento di Fisica e Scienze della Terra, Universit degli Studi di Ferrara, Polo Scientico e Tecnologico Edicio C, Via Saragat 1, 44122 Ferrara, Italy

4 Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Troitsk, Moscow 142190, Russia

5 Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia

Received: 19 May 2015 / Accepted: 6 September 2015 / Published online: 21 September 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract It is shown that the efciency of the universe heating by an inaton eld depends not only on the possible presence of parametric resonance in the production of scalar particles but also strongly depends on the character of the inaton approach to its mechanical equilibrium point. In particular, when the inaton oscillations deviate from pure harmonic ones toward a succession of step functions, the production probability rises by several orders of magnitude. This in turn leads to a much higher temperature of the universe after the inaton decay, in comparison to the harmonic case. An example of the inaton potential is presented which creates a proper modication of the evolution of the inaton toward equilibrium and does not destroy the nice features of ination.

1 Introduction

Cosmological ination consisted, roughly speaking, of two epochs. The rst one was a quasi-exponential expansion, when the Hubble parameter, H, slowly changed with time and the universe expanded by a huge factor, eN, where

N = [integraldisplay] H dt 1. (1.1)

During this period the Hubble parameter exceeded the inaton mass or, rather, the square of the Hubble parameter was larger than the second derivative of the inaton potential:

H2 >

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

where U() is the potential of the inaton eld, . Due to the large Hubble friction (see Eq. (3.3)) during this epoch the eld remained almost constant, slowly moving in the direction of the force, U ().

The second stage began when H2 dropped below |U ()|

and continued till the inaton eld reached the equilibrium value where U (eq) = 0. It is usually assumed that eq = 0

and U(eq) = 0. The last condition is imposed to avoid

a nonzero vacuum energy. During this period oscillated around eq, producing elementary particles, mostly with masses smaller than the frequency of the inaton oscillations. This was a relatively short period which may be called a big bang, when the initial dark vacuum-like state exploded, creating hot primeval cosmological plasma.

The process of the universe heating was rst studied in Refs. [13] within the framework of perturbation theory. A non-perturbative approach was pioneered in Refs. [46], where the possibility of excitation of parametric resonance which might grossly enhance the particle (boson) production rate was mentioned. In the model of Ref. [4] parametric resonance could not be effectively induced because of the redshift and scattering of the produced particles which were dragged out of the resonance zone and the main attention in this work was set on non-perturbative production of fermions. However, the resonance may be effective if it is sufciently wide. In this case the particle production rate can be strongly enhanced [58].

As is well known, parametric resonance exists only in the process of the boson production. In quantum language it can be understood as Bose amplication of particle production due to the presence of identical bosons in the nal state; it is the same phenomenon as the induced radiation in laser. For bosons there could be another phenomenon leading to very fast and strong excitation of the bosonic eld coupled to ina-

d2U()

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

|U ()|, (1.2)

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

c e-mail: mailto:[email protected]

Web End [email protected]

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ton, if the effective mass squared of such eld became negative (a tachyonic situation) [911]. It happens for sufciently large and negative product g; see below Eqs. (2.1), (2.2).This is similar to the Higgs-like effect, when the vacuum state becomes unstable. However, in contrast to the Higgs phenomenon, this took place only during a negative half-wave of the inaton oscillations.

Both phenomena are absent in the case of fermion production. The imaginary mass of the fermions breaks the hermicity of the Lagrangian, so tachyons must be absent. As for parametric resonance, it is not present in the fermionic equations of motion. The latter property is attributed to the Fermi exclusion principle. A non-perturbative study of the fermion production shows that the production probability sharply grows when the effective mass of the fermions crosses zero [4,12].

The resonance amplication of the particle production by the inaton exists not only in the canonical case of harmonic inaton oscillations, i.e. for the potential m22/2, but for a very large class of inaton potentials. The explicit magnitude of the production probability depends, of course, upon the shape of the potential, but there is no big difference between different power law potentials, n. However, as we show in

this work, the production rate could be drastically enhanced for some special forms of the potential, if the potential noticeably deviates from a simple power law.

There are several more phenomena which might have an impact on the particle production probability. In Refs. [13 15] the effects of quantum or thermal noise on the inaton evolution have been studied. It was argued by these two groups that the resonance is not destroyed by noise. If the noise is not correlated temporally it would lead on the average to an increase in the rate of particle production [14,15]. This result is valid both for homogeneous and inhomogeneous noise. However, the cosmological expansion was neglected in Refs. [14,15] and it possibly means that the resonance, to survive in realistic cosmological situations, should be sufciently wide.

The efciency of the production depends also on the model of ination. In particular, in the case of multield ination the canonical parametric resonance is suppressed [16 18]. However, efcient (pre)heating is possible via tachyonic effects. To this end a trilinear coupling of the inaton to light scalars is necessary. If the tachyonic mechanism is not operative, the old perturbative approach [13] would be applicable.

A new mechanism of enhanced preheating after multield ination has been found in the recent paper [19], due to the presence of extra produced species which became light in the course of the multield inaton evolution.

The efciency of different earlier scenarios of the cosmological heating after ination is discussed in a large number of review papers, e.g. [20], where one can nd an extensive

list of literature. More recent development is described in reviews [21,22].

In this paper we study a parametric resonance excitation for different forms of the inaton potential, U(). A new effect is found: for some non-harmonic potentials of a single eld ination parametric resonance (not tachyonic) is excited considerably stronger than in the case of simple power law potentials, both in at space-time and in cosmology. Correspondingly, the cosmological particle production by the end of ination would be much more efcient, and the temperature of the created plasma would become noticeably higher. In Sect. 2 we consider this problem in at space-time to get a feeling for a proper choice of the inaton potential that could generate the signal (t) most efcient for the particle production. Consideration of the at space-time example clearly demonstrates the essence of the effect which is not obscured by the cosmological expansion. In Sect. 3 we study the evolution of the inaton eld in a cosmological background for different potentials U(). Based on the example considered in Sect. 2, we found a potential for which the inaton induces parametric resonance much more efciently than in the purely harmonic case, or other simple power law potentials. We also comment there on the properties of inationary cosmology with such modied inaton potentials. In this section the scalar particle production rate by such un-harmonic inaton is calculated. The results are compared to the particle production rate for the usual harmonic oscillations of the inaton. Section 4 is dedicated to an estimate of the effects of back reaction of the particle production on the inaton evolution. In Sect. 5 we draw conclusions.

We have chosen the sign and the amplitude of the initial inaton eld to avoid or to suppress tachyonic amplication of the produced eld .

2 Parametric resonance in at space-time

Let us consider at rst an excitation of parametric resonance in the classical situation, when the space-time curvature is not essential and the Fourier amplitude of the would-be resonating scalar eld satises the equation of motion

+ (m2 + k2 + g) = 0, (2.1) where m is the mass of , k is its momentum, and g is the coupling constant between and another scalar eld with the interaction

Lint =

2 g2. (2.2)

The classical eld is supposed to be homogeneous, =

(t) and to satisfy the equation of motion:

+ U () =

2 g2. (2.3)

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Below we mostly neglect the effects of the r.h.s. term in this equation. Its impact on the inaton evolution is relatively weak. It introduces the back reaction of the particle production and is discussed in Sect. 4.

Our task here is to determine U(), so that the parametric resonance for would be most efciently excited.An optimal meander form of the inaton eld is suggested by the phase parameter approach in the theory of parametric resonance [23]. Here we demonstrate that just a slight shift from the standard Mathieu model toward an optimal inaton potential leads to a drastic increase of the particle production rate. We compare the modied results with the standard case when the potential of is quadratic, U() = m22/2, and therefore Eq. (2.3) with zero r.h.s. has

a solution (t) = 0 cos(mt + ), where the amplitude 0

and the phase can be found from the initial conditions. One can choose the moment t = 0 in such a way that = 0, i.e.

(t) = 0 cos mt. Substituting this expression into Eq. (2.1),

we come to the well-known Mathieu equation:

+ 20(1 + h cos mt) = 0, (2.4) where 20 = m2 + k2 and h = g0/20. When h 1 and

the value of m is close to 20/n (where n is an integer), Eq. (2.4) describes a parametric resonance, i.e. the eld oscillates with an exponentially growing amplitude. For h

1 the solution of Eq. (2.4) can be represented as a product of a slowly (but exponentially) rising amplitude by a quickly oscillating function with frequency 0:

= 0(t) cos(0t + ). (2.5)

The amplitude 0 satises the equation

0 cos(0t + ) + 20

0 sin(0t + )= h200 cos mt cos(0t + ). (2.6)

Let us multiply Eq. (2.6) by sin(0t + ) and average over the

period of oscillations. The right hand side would not vanish on the average, if m = 20. In this case 0 would exponen

tially rise if = /4:

0 exp [parenleftbigg]

Let us note that the behavior of the solution for would dramatically change with rising h. For a large h the eigenfrequency squared of noticeably changes with time. It may approach zero and, if |h| > 1, it would even become nega

tive for a while; see Eq. (2.4). In the latter (tachyonic) case would rise much faster than in the case of classical parametric resonance. We postpone the study of tachyonic case for a future work, while here we conne ourselves to a nontachyonic situation.

To demonstrate an increase of the excitation rate for an anharmonic oscillation we choose, as a toy model, the potential for the would-be inaton eld satisfying the following conditions: at small it approaches the usual harmonic potential, U m22/2, while for it tends to a con

stant value. It is intuitively clear that in such a potential eld would live for a long time in the at part of the potential and quickly change sign near = 0. This behavior can be rather

close to a periodic succession of the step-functions which we mentioned above. As an example of such a potential we take

U() =

2m22

1 + 0(/mPl)2

1 + 2(/mPl)2 + 4(/mPl)4

, (2.9)

where m, 0, 2, 4 are some constant parameters, with m having dimension of mass, the j being dimensionless. Here mPl is the Planck mass. Observational data on the density perturbations induced by the inaton demand m 106 mPl

in the model with the inaton potential U() = m22/2,

so we use m = 106 mPl as a reference value throughout

the paper. Since the potential of our toy model is different from the harmonic one, the inationary density perturbations could also be different. With the particular choice of parameters j , which is presented below, the density perturbations require a somewhat smaller m, approximately by an order of magnitude. However, our aim here is not the construction of a realistic inationary model, but the demonstration of a new phenomenon of more efcient excitation of the parametric resonance for some forms of inaton oscillations. To this end we take 0 = 85, 2 = 4, 4 = 1. The plots of the potentials

are shown in Fig. 1. Here and below the red (or dashed) curves are for the quadratic potential U() = m22/2 and the blue

ones are for the potential (2.9). For our choice of parameters the plots cross each other at the points = 0, 9 mPl.

We did not look for a theoretical justication for the chosen form of the potential (2.9), bearing in mind the large freedom for possible forms of the scalar eld potential at large mass/energy scale, but it is noteworthy that a similar type of the potential which is quadratic near the minimum and is atter away from the minimum was studied in Refs. [24,25]. Such potentials were in turn derived in Refs. [2631]. Quoting Ref. [25]: This choice was motivated by monodromy and supergravity models of ination [2630] and a recent model of axion quintessence [31].

14h0t[parenrightbigg] . (2.7)

In this way we recovered the standard results of the parametric resonance theory.

The rise of the amplitude of is determined by the integral

2 +

2202 = g [integraldisplay] dt

, (2.8)

as one can see from Eq. (2.1). It can be shown that the maximum rate of the rise is achieved when is a quarter-period meander function (an oscillating succession of the step-functions with proper step duration); see Ref. [23].

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437 Page 4 of 12 Eur. Phys. J. C (2015) 75 :437

(0) =

(0)/2 0.745, where 2(0) = k2 + g(0). These ini

tial conditions correspond to the solution of Eq. (2.1) with constant : (t) = eit/2V , where V = 1 (in units

of m3) is the volume. As a result we obtain function (t) shown in Fig. 3. As expected, the amplitude of oscillations of increases faster in the case of the potential (2.9); e.g. at t = 450 m1 the amplitude ratio is approximately 2.

Since we are going to apply the results for the calculation of the particle production rate at the end of cosmological ination, it would be appropriate to present the number density of the produced -particles, nk(t), which is dened as

nk(t) = [parenleftBigg]

initial conditions (0) = 1/2(0) 0.67,

-10 -5 5 10

Fig. 1 The potential of the inaton U(). The red dashed line is the quadratic potential U() = m22/2, and the blue line is the potential

(2.9). The parameters are m = 106 mPl, 0 = 85, 2 = 4, 4 = 1.

The eld is measured in units of mPl and U() is measured in units of 1011 m4Pl

2 +

2k2 2

[parenrightBigg]

1k , (2.10)

where the expression in the brackets is the energy of the mode with momentum k, and k = [radicalbig]k2 +

g is the energy of one

-particle.

The number densities of the produced -particles for the harmonic and slightly step-like one are presented in Fig. 4 by red and blue curves, respectively.

Despite a decrease of particle number densities in some short time intervals, there is an overall exponential rise, which goes roughly as 2. Therefore, the ratio of particle numbers for the two types of the potential is approximately equal to the ratio of the amplitude squared, which is about 4 at t =

450 m1.

3 Resonance in expanding universe

The universes heating after ination was achieved due to coupling of the inaton eld to elementary particle elds. In this process mostly particles with masses smaller than the frequency of the inaton oscillations were produced. The decay of the inaton could create both bosons and fermions. The boson production might be strongly enhanced due to excitation of the parametric resonance in the production process [48]. Hence bosons were predominantly created initially. Later in the course of thermalization they gave birth to fermions. For a model description of the rst stage of this process we assume, as we have done in Sect. 2, that the inaton coupling to a scalar eld has the form g2/2, where

g > 0 is a coupling constant with the dimension of mass. We also assume that the initial value of is positive. With this choice of the parameters the tachyonic situation can be avoided. Otherwise would explosively rise even at inationary stage. As a result the contribution of to the total cosmological energy density would become non-negligible and should be taken into account in the Hubble parameter. This effect may inhibit ination. These problems will be studied elsewhere. Below we study a simpler situation of the initial

5 10 15 20 t

-2

-4

Fig. 2 Comparison of the numerical solution U (t) of Eq. (2.3) with the potential (2.9) (blue line) and the harmonic solution h(t) =

0 cos mt with 0 = 4.64 and m = 106 mPl (red dashed line). The

time t is measured in units of m1 and the eld is measured in units of mPl

Figure 1 for the potential (2.9) has a shape which is quite close to that depicted in Fig. 1 from Ref. [25]. The properties of the inationary model, which might be realized with the inaton potentials of the type (2.9) can be understood from the results of Refs. [24,25].

We solved Eq. (2.3) with zero r.h.s. and with the potential (2.9) numerically, using Mathematica (here as well as in the rest of the paper), and compared this solution, U (t), with the harmonic solution, h(t) = 0 cos mt. The results are

presented in Fig. 2 for the initial conditions (0) = 4.64 mPl, (0) = 0. The value of the amplitude 0 = 4.64 is chosen

because in this case the frequencies of U (t) and h(t) are approximately equal.

For the chosen shape of the potential (2.9), the function U (t) differs from a cosine toward the step function. Therefore, one can expect that parametric resonance would be excited stronger for the potential (2.9) than for the quadratic one.

Now we can numerically solve Eq. (2.1) with the computed (t) and with the chosen values for this example: m = 0, k = 1, g = 5 108 (in units of m), and the

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100 200 300 400 t

-5

-10

-20

-10

Fig. 3 Oscillations of the eld (t). The left panel corresponds to harmonic potential of and the right panel corresponds to the potential (2.9). The time t is measured in units of m1

100 200 300 400 t

n t

300

200

100

Fig. 4 The number density of produced -particles nk(t). The time t is measured in units of m1

stage of heating when the energy density of the produced particles is small in comparison with the energy density of the inaton, so the back reaction is not of much importance. The effects of the back reaction of particle production on the inaton evolution is discussed in Sect. 4.

The equation of motion of the Fourier mode of with conformal momentum k in the FLRW metric has the form

+ 3H + [parenleftbigg]

in the

equation of motion for (t) (3.3) strongly modies the evolution of and it is necessary that the resonance should be generated faster than signicantly dropped down.

We nd numerical solutions of Eq. (3.3) with the same potentials as above, i.e. the harmonic one and U() presented in Eq. (2.9). As initial conditions we take

(0) = 0

and two different values (0) = 4.64 mPl and (0) = 9 mPl,

which we consider in parallel. The rst one, (0) = 4.64 mPl,

is equal to the value which we took in the at universe case (see Fig. 2). However, in this case the initial energies, U((0)), are very different for the two potentials (see Fig. 1). So we consider also the initial condition (0) = 9 mPl for

which the two potentials U((0)) have equal magnitudes. The results of a numerical solution of Eq. (3.3) are shown in Fig. 5.

At the stage of ination the inaton eld, (t), rolls toward the minimum of potential quite slowly, due to the large friction H. For successful ination one needs the condition

[integraltext]

k2a2 + g[parenrightbigg] = 0, (3.1)

where a = a(t) is the cosmological scale factor, H = a/a is

the Hubble parameter, and eld is taken for simplicity to be massless, m = 0. We assume that the universe is 3D-at

and that the cosmological energy density is dominated by the inaton eld, so H is expressed through as

H = [radicalbigg]

8 3

2/2 + U()

mPl , (3.2)

where U() is the potential of the inaton, mPl 1.2

1019 GeV is the Planck mass, and it is assumed, as usually, that the inaton eld is homogeneous, = (t). Corre

spondingly the equation of motion for has the form

+ 3H + U () = 0, (3.3)

where U = dU/d.

We study here the particle production by , which evolves in the potential (2.9), described in the previous section, where

it has been shown that the particle production in at space-time is strongly enhanced in comparison with the particle production by with the potential U() = m22/2. We do

the same thing here.

In expanding universe the liquid friction term 3H

teti H(t) dt > 70 to be satised, where ti is the time of the beginning and te is the time of the end of ination. It can easily be seen in Fig. 6 that this condition is fullled for both potentials. The scale factor a(t) grows exponentially during ination (see Fig. 7).

After (t) reaches the minimum of the potential, it does not have enough energy to climb high back because of the energy loss due to the friction, so starts to oscillate with decreasing amplitude. The moment t0, when (t) crosses zero for the rst time, can be considered as the end of ination, the onset of the oscillations, and the universes heating. The Hubble parameter becomes quite small by this moment, H [lessorsimilar] m, and continues to decrease, so during the oscillation period one can neglect H in comparison with m.

Equation (3.3) is simplied by the substitution (t) =

(t)/a3/2(t), and in the case of quadratic potential it has a solution (t) = 0(t) cos mt, where 0(t) a3/2(t).

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437 Page 6 of 12 Eur. Phys. J. C (2015) 75 :437

0.04

0.02

30 40 50 60 70 80t

-0.02

-0.04

-0.06

-0.08

20 40 60 80 100 t

t t

0.04

0.02

100 200 300 400 500t

-0.02

-0.04

-0.06

-0.08

100 200 300 400 500 t

Fig. 5 Inaton eld, (t), in the expanding universe. The time t is measured in units of m1 and eld is measured in units of mPl. In the left panels the evolution (t) is shown starting from t = 0. In the right

panels the oscillations of (t) are presented in more detail, starting from

the moment when = 0 for the rst time. The upper plots correspond

to the case (0) = 4.64 mPl, and the lower ones correspond to the case

(0) = 9 mPl

H t

17.5

12.5

7.5

2.5

20 40 60 80 t

100 200 300 400 500 t

Fig. 6 Hubble parameter H(t). The time t is measured in units of m1 and H is measured in units of m. The left plot corresponds to the case (0) = 4.64 mPl, and the right one corresponds to the case (0) = 9 mPl

ln a t

5000

400

4000

300

3000

200

2000

100

1000

20 40 60 80 t

100 200 300 400 500 t

Fig. 7 Logarithm of the scale factor a(t). The time t is measured in units of m1. The initial value is a(0) = 1. The left plot corresponds to

(0) = 4.64 mPl, and the right one corresponds to (0) = 9 mPl

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k a t

0.9

1.2

0.8

0.7

0.6

0.8

0.5

0.6

0.4

0.3

1000 3000 5000 7000t

Fig. 8 The physical momentum k/a(t) (in units of m). The black horizontal lines cross the functions k/a(t) in the resonance points, k/a(t) = m/2.

The left plot corresponds to (0) = 4.64 mPl, and the right one corresponds to (0) = 9 mPl

Therefore, Eq. (3.1) turns into

+ 3H +

n(t) = [parenleftbigg]

2 +

[parenrightbigg]

1 . (3.11)

It is reasonable to impose the initial conditions for the eld (t) at the moment t0, which is the moment of the onset of the inaton oscillations. We choose the initial conditions as (t0) = 1/2(t0)V (t0),

(t0) = (t0)/2V (t0), where

V (t0) = 1 in units of m3. These conditions correspond

to vanishing initial density of the -particles. To avoid the tachyonic situation we put k/a(t0) = 50 m and g = 5

104 m, therefore we ensure that 2 is always positive during the interesting time interval.

At large time the amplitude of the oscillation becomes small and the potential (2.9) closely approaches the quadratic one; therefore the condition of parametric resonance (3.5) holds for both potentials. Thus, the resonance occurs in the narrow region when k/a(t) is near m/2. In Fig. 8 one can see at what time the resonance occurs for the two potentials of with the chosen parameters.

We have assumed here that the inaton eld gives the dominant contribution to the cosmological energy density. Therefore, when the energy density of the produced particles, , becomes comparable to =

2/2 + U(), the

model stops to be self-consistent and we should modify the calculations. The easiest way is to take the energy density of the produced particles at this moment as an ultimate one and to estimate the cosmological heating temperature on the basis of this result. A more precise way is to take into account the back reaction of the produced particles on the damping of the inaton oscillations and to include the contribution of the created particles into the Hubble parameter. The rst simplied approach, which gives a correct order of magnitude estimate of the temperature, is sufcient for our purposes.

Resonance particle production could induce specic features in the primordial spectrum of density perturbations [33], which might be potentially observable. We thank the referee for mentioning this effect. This could be the subject of a separate study.

The energy densities of the produced particles, (t), for (0) = 4.64 mPl and (0) = 9 mPl are presented in Fig. 9.

k2 cos mt[parenrightbigg] = 0, (3.4)

which is the Mathieu equation with a friction term, so the condition of parametric resonance [32] is

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]m

k2 a2

1 + g0a2

2k a

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[radicalBigg][parenleftbigg]

g0a

2 9H2. (3.5)

Let us consider now Eq. (3.1) in the general case. It is convenient to make the substitution (t) = X(t)/a3/2(t), so

one obtains

X + [parenleftbigg]

[parenrightbigg] X = 0. (3.6)

Equation (3.6) has the form of the free oscillator equation

X + 2X = 0 with frequency depending on time. During

the oscillations the terms H2 and a/a are relatively small, so

one can take

(t) [radicalBigg]

k2a2(t) + g(t). (3.7)

If we neglect the time dependence of a(t) and take a small g, then would be almost constant and the solution of (3.6) is X(t) eit/2, which corresponds to (t) =

eit/2V , where V = a3 is the comoving volume.

The energy and number of produced particles in a comoving volume are, respectively,

E(t) =

k2a2 + g

4 H2

2X2

2 +

2 , (3.8)

N(t) = [parenleftbigg]

2 +

2X2

[parenrightbigg]

1 . (3.9)

The scale factor a(t) changes much slower with time than X(t), therefore

X/a3/2. Thus the energy and the number

densities of the produced -particles would be

(t) =

2 +

2 , (3.10)

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1.2

30000

25000

0.8

20000

0.6

15000

0.4

10000

0.2

5000

4000 5000 6000 7000t

1600 1700 1800 1900 2000 t

t t

20000

15000

10000

5000

4000 5000 6000 7000t

2150 2200 2250 2300 t

Fig. 9 Energy density, (t), of the produced particles . The time t is measured in units of m1 and (t) is measured in units of m4. The upper plots correspond to (0) = 4.64 mPl, and the lower ones correspond to (0) = 9 mPl. Black lines in the right panels denote

One can see that for the quadratic potential of the parametric resonance is quite weak and the energy density of particles produced during the resonance is much less than , which is about 3 103 m4 at the moment of the resonance emer

gence. On the contrary, in the potential (2.9) increases very quickly and becomes comparable to at t 2050 and

t 2330 in the cases (0) = 4.64 mPl and (0) = 9 mPl,

respectively.

The number densities n(t) calculated according to Eq. (3.11) and the total numbers N(t) = n(t) a3(t) of the

produced particles are presented in Figs. 10 and 11, respectively. These plots can be understood as follows. During some time after the beginning of the oscillations the conditions of parametric resonance are not fullled and therefore -particles are not produced in a considerable amount. Due to the universes expansion the term (k/a(t))2 drops down and at some moment the mode with conformal momentum k enters the resonance region. It is manifested as an exponential growth of (t), n(t), and N(t). Then the resonance conditions stop to be satised once again and -particle production is almost terminated, so their total number tends to a constant value, while their number density, n(t), decreases as a3.

Let us suppose that -particles reach thermal equilibrium very quickly. The energy density of relativistic particles is simply related to the temperature:

30 gT 4, (3.12)

where g 100 is the number of relativistic particle species

in the thermalized plasma. Therefore, from Fig. 9 in the case of the potential (2.9) one can estimate the temperature by the moment when as

T 5 m 105 mPl 1014 GeV. (3.13)

It is instructive to present a qualitative explanation of the obtained results. The initial moment, when the particle production is initiated, is the moment of the onset of the inaton oscillations. They started when the Hubble friction term in the inaton equation of motion (3.3) becomes small in comparison with the potential term. It took place when H2 dropped down below U (). For the harmonic potential Uh = m22/2 it happened after reached the boundary

value 2h = 3m2Pl/4. For our potential (2.9) the oscilla

tion regime is reached at a 6 times larger value of , i.e. 2 = 9m2Pl/2. This result is obtained in the case of 0 1,

as has been chosen for the potential (2.9).

Discussing particle production we considered the initial physical momentum of the produced particles p = k/ain =

50 m to avoid a tachyonic excitation, and we took initial conditions for corresponding to the vacuum state, i.e. to the state where the real -quanta were absent. In the course of the cosmological expansion this momentum evolved down to the resonance value k/ares = m/2, both in the cases of har

monic and non-harmonic evolution. Since the Hubble parameter in the modied theory was larger than that in the harmonic case, the former reached the resonance value faster; see Fig. 8. However, this is not directly essential, because in

123

Eur. Phys. J. C (2015) 75 :437 Page 9 of 12 437

n t

12000

10000

1.5

8000

6000

4000

0.5

2000

1000 3000 5000 7000t

1600 1700 1800 1900 2000 t

n t

20000

15000

10000

5000

1000 3000 5000 7000t

2150 2200 2250 2300 t

Fig. 10 The number density of the produced -particles n(t). The time t is measured in units of m1 and number density is measured in units of m3. The upper plots correspond to (0) = 4.64 mPl and the lower ones correspond to (0) = 9 mPl

1000 3000 5000 7000t

N t

2.5106

1.41010

1.21010

2106

11010

1.5106

8109

1106

6109

4109

500000

2109

1600 1700 1800 1900 2000 t

N t

2.5107

21010

2107

1.51010

1.5107

11010

1107

5106

5109

1000 3000 5000 7000t

2150 2200 2250 2300 t

Fig. 11 The total number of produced -particles N(t). The time t is measured in units of m1. The upper plots correspond to (0) = 4.64 mPl,

and the lower ones correspond to (0) = 9 mPl

both cases p reaches the resonance value at the same red-shift relative to the initial state. An important factor which determines the effectivity of the resonance particle production is the amplitude of the inaton eld at the resonance.Initial amplitudes differed by the factor 6, but this is not the end of the story. The amplitude of the harmonic inaton dropped down as 1/a3/2, while the evolution of the inaton living in the potential (2.9) is considerably slower.For a purely quartic potential, 4, the amplitude of the

inaton eld drops as 1/a, so it comes to the resonance with the amplitude 10 times larger than the harmonic inaton.

In the case considered in this paper the modied potential is not purely quartic, moreover, it approaches the harmonic form at small , when the resonance is excited. Nevertheless, the amplitude of during the harmonic regime is much larger than in the purely harmonic case, though not by such a large factor.

123

437 Page 10 of 12 Eur. Phys. J. C (2015) 75 :437

According to the equations presented in Sect. 2 the amplitude of at the resonance rises as exp(grest/2m), where

res is the inaton amplitude at the resonance. This explains the difference in efciency of the particle production between the purely harmonic and modied cases.

Finally, it should be noted that production of -particles is possible also due to the usual non-resonant decay .

However, the width of such a decay is very small. Indeed, it follows from Eq. (2.2) that = g2m/32 for neutral

massless . Therefore, for our choice of the interaction constant, g = 5 104 m, the typical time of the decay is

= 1/ 4 108 m1, which is much longer than the

time when the resonance occurs (see Fig. 8). Thus the contribution of the non-resonant decay is negligible.

The inationary model based on the potential (2.9) is similar to the well-known new inationary scenario or ination at a small eld ; see e.g. Ref. [34]. Correspondingly, the slow roll parameters satisfy the condition ||. The deni

tion of the parameters and their relation to scalar and tensor perturbations can be found in Refs. [3437]. The inequality mentioned above means that the amplitude of the gravitational waves in the model considered here must be very small. However, this conclusion is model dependent and is not necessarily true.

4 Back reaction

The tremendous rate of particle production makes its back reaction on the inaton evolution non-negligible quite soon.The simplest way to estimate this back reaction is to use the density balance condition: the energy loss by the inaton must be equal to the energy carried away by the produced particles. Such a comparison is done in Fig. 9. Before this moment we can neglect the related decrease of the inaton amplitude. This is similar to the well-known instant decay approximation, which usually works pretty well. So as regards the order of magnitude we can rely on the results obtained with the neglected back reaction. However, in the instant decay approximation the decay rate remains constant, while the parametric resonance rate is proportional to the amplitude of the inaton, and so with decreasing the resonance production drops down and it may happen that the remaining part of the inaton energy would decay much more slowly. Keeping in mind that usually the inaton makes non-relativistic matter, while produced particles are relativistic, we can conclude that the relative contribution of the inaton to the cosmological energy density would grow and ultimately the dominant part of the (re)heated cosmic plasma would be created by slow (perturbative) inaton decay. However, this is not a subject of the present work.

For a more accurate treatment of the back reaction we can use the equation of motion of the inaton with quan-

tum effects induced by particle production in the one loop approximation [38,39]. The corresponding equation is an integro-differential one, which for the coupling of the produced quantum eld of the form (2.2) can be written as

c + V (c)

g2162 c ln

g2 162

[integraldisplay]

[c(t ) c(t)]

g2 162

[integraldisplay]

0 ttin

c(t ). (4.1)

Here we use the notation c for the classical inaton eld to keep track with Ref. [38]. The integrals in the r.h.s. are ultraviolet nite. The logarithmically innite contribution in the l.h.s., related to the ultraviolet cut-off 0, is taken out

by the mass renormalization, so with a possible bare mass term in the potential, Vm0 = m202c/2 (here m0 is the bare

mass of c), we obtain

c + m2(t1)c + [bracketleftbig]V

g2 162

[integraldisplay]

[c(t ) c(t)]

g2 162

[integraldisplay]

0 ttin

c(t ), (4.2)

where t1 is an arbitrary normalization point and the running mass is m2(t1) = m2(t2) (g2/162) ln(t1/t2). In Eq. (4.2)

we explicitly separated the massive part, Vm0, in the potential, so that the term in the square brackets vanishes for the harmonic potential, V () = m202/2.

The equation governing the evolution of c can be grossly simplied if the particle production goes in the resonant mode. In this case the occupation numbers of the eld become very large and the eld can be treated as a classical one.

We assume, as is usually done in studies of parametric resonance, that is a real eld. Since the resonance is rather narrow, the universes expansion can be neglected. Correspondingly, the k-mode of the massless eld satises the following equation of motion:

k(t) + k2k(t) + gc(t)k(t) = 0. (4.3) In what follows we omit the subindices k and c.

This equation can be transformed into the integral equation:

(t) = 0 g [integraldisplay]

R(t t )(t )(t ), (4.4) where 0 is the initial value of , which is assumed for simplicity to be zero (it does not rise exponentially, so can be neglected anyhow) and

GR(t) = [braceleftbigg]

sin kt/k, if t > 0,0, if t < 0, (4.5)

is the retarded Greens function of Eq. (4.3).

0 dt G

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Eur. Phys. J. C (2015) 75 :437 Page 11 of 12 437

Log t

1800 1900 2000 t

2100 2200 2300 t

-5

-10

-15

Fig. 12 Energy density of the produced -particles for (0) = 4.64 mPl (left panel) and for (0) = 9 mPl (right panel) in a normal logarithm

scale

Making the ansatz = A exp( t) sin(kt + ), and

neglecting oscillating terms in the integral (4.4), we nd that this ansatz is self-consistent, i.e. indeed rises exponentially, if = 0 sin mt, and k = m/2, = /2:

= A

and the characteristic time of the oscillation damping is

2 =

g04k et cos kt. (4.6)

It is assumed here that both A and the amplitude of the inaton oscillations, 0, slowly change with time. For self-consistency one needs to impose the condition g0/4k =

1. It leads to the canonical expression for presented in Eq. (2.7).

Let us turn now to Eq. (2.3). The impact of the -particle production originating from the term g2/2 can be esti

mated as follows. We substitute expression (4.6) for and assume that evolves as = 0(t) sin mt, where 0(t) is

(in comparison with frequency m) a slowly varying function of time. In this way we obtain the equation:

0 cos mt =

2 g A2e2t cos2 kt

mg0,res . (4.10)

According to our calculations 0,res 104 mPl = 100 m,

and the parameter A can be estimated as A 0.1 m (the

initial value of ). Using also g = 5 104 m, we nd that

d 20/m.

In Fig. 12 the energy density of the produced -particles is presented, as usually, for (0) = 4.64 mPl (left panel)

and for (0) = 9 mPl (right panel). The exponential rise is

clearly observed with the exponent quite close to the above calculated one.

The characteristic lapse of time during which disappears down to zero can be estimated from Eq. (4.9) and is equal to

2 ln

16m 0,res

g A2

820,res

m g0,res ln

4 e2t(1 + cos 2kt). (4.7)

The rst term in the brackets in the r.h.s. describes the tadpole contribution and should be disregarded. The second term is consistent with the l.h.s. if k = m/2, as expected, and ulti

mately we arrive at the equation

0 =

A2 . (4.11)

With the chosen above parameter values we have

300/m. In reality it would be somewhat longer because with the decreasing amplitude of the parametric resonance exponent drops down proportionally to .

5 Conclusion

We have shown that even a small modication of the shape of the inaton oscillations could lead to a signicant increase of the probability of particle production by the inaton and consequently to a higher universe temperature after ination. The particle (boson) production by the inaton oscillating in a harmonic potential was compared to the particle production in a toy inationary model with a at inaton potential at innity. It was found that the parametric resonance in the latter case is excited much more efciently.

The inationary model which is considered above is not necessarily realistic. We took it as a simple example to demonstrate the efciency of the particle production and the heating of the universe. The impact of the energy density of the produced particles on the cosmological expansion may

g A2

8m e2t. (4.8)

0 is always negative, the amplitude of the inaton oscillation 0 constantly decreases. Thus, if we neglect variations of and take = g0,res/2m, where 0,res is the value of

0 at the beginning of resonance, we obtain the lower limit on the time of the oscillation damping. Indeed, in such a case the amplitude of the inaton oscillation can easily be found:

0 = 0,res

g A216m e2t, (4.9)

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437 Page 12 of 12 Eur. Phys. J. C (2015) 75 :437

noticeably change the phenomenological properties of the underlying inationary model.

We skip on purpose a possible tachyonic amplication of -excitement to avoid the deviation from the main path of the work on amplication of particle production due to variation of the shape of the inaton oscillations.

In Ref. [40] a similar study of the impact of the anharmonic corrections to the inaton oscillations was performed, but the effect is opposite to that advocated in our paper: instead of amplication the anharmonicity leads to a damping of particle production. However, in this paper the form of the inaton oscillations is different from ours. It shows that the shape of the signal is indeed of crucial importance to the efciency of the particle production. We thank M. Amin for the indication of Ref. [40]. Recently there have appeared a few more papers [4143] in which the efciency of inationary heating was studied. However, the mechanism considered there is different from ours.

Acknowledgments AD and AS acknowledge the support of the grant of the Russian Federation government 11.G34.31.0047.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

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Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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It is shown that the efficiency of the universe heating by an inflaton field depends not only on the possible presence of parametric resonance in the production of scalar particles but also strongly depends on the character of the inflaton approach to its mechanical equilibrium point. In particular, when the inflaton oscillations deviate from pure harmonic ones toward a succession of step functions, the production probability rises by several orders of magnitude. This in turn leads to a much higher temperature of the universe after the inflaton decay, in comparison to the harmonic case. An example of the inflaton potential is presented which creates a proper modification of the evolution of the inflaton toward equilibrium and does not destroy the nice features of inflation.

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Title

Shape of the inflaton potential and the efficiency of the universe heating

Author

Dolgov, A D; Popov, A V; Rudenko, A S

Pages

1-12

Publication year

2015

Publication date

Sep 2015

Publisher

Springer Nature B.V.

ISSN

14346044

e-ISSN

14346052

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1140/epjc/s10052-015-3666-4

ProQuest document ID

1727603527

SIF and Springer-Verlag Berlin Heidelberg 2015

Shape of the inflaton potential and the efficiency of the universe heating

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