1 Introduction

## Abstract

It is shown that the inflationary era in early universe is realized due to the effect of backreaction of quantized matter fields. In fact we start by quantizing a free scalar field in the Friedmann-Robertson-Walker space-time, and the field is fluctuating quantum mechanically around the bottom of the mass potential. We then obtain the vacuum expectation value of the energy density of the scalar field as a functional of the scale factor of the universe. By plugging it into the Einstein equation, a self-consistent equation is established in which the matter fields determine the time evolution of the universe. We solve this equation by setting few conditions and find the following solution: the universe expands à la de Sitter with e-folding number and then it turns to shrink with a decreasing Hubble parameter which rapidly goes to zero.

KUNS-2521

OIQP-14-9

A new mechanism of realizing inflationary universe

with recourse to backreaction of quantized free fields

– Inflation without inflaton –

Yoshinobu Habara, Hikaru Kawai and Masao Ninomiya

Okayama Institute for Quantum Physics, 1-9-1 Kyoyama, Okayama 700-0015, Japan

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

E-mail: habara@yukawa.kyoto-u.ac.jp,
hkawai@gauge.scphys.kyoto-u.ac.jp,
msninomiya@gmail.com

PACS numbers: 04.62.+v, 98.80.Cq, 11.25.-w, 11.25.Mj, 98.80.Qc

Keywords: inflation, string theory, cosmic microwave background

## 1 Introduction

It is believed by most astro-cosmologists as well as astro-particle physicists that the only solution to resolve the flatness and horizon problems in the universe is to think of exponential expansion called inflation [1] right after birth of our universe. In recent years several observational results to confirm the inflation were reported [2, 3, 4]. In particular, the most important evidence in the present days to provide data is the temperature fluctuation of the cosmic microwave background radiation (CMB) which shows us quantitatively that there was the era of the inflation in the early universe. Using this data we can investigate the mechanism of the inflationary expansion in detail. The CMB is the photon emitted in the recombination era of which the wavelength is very long so that CMB reaches us without much scattering afterwards. Therefore CMB carries to us the information of the early universe. The most peculiar properties of CMB is that it is homogeneous and uniform with very high accuracy and furthermore the spectrum agrees very well with the black body radiation with the temperature 2.725 [K] while the temperature fluctuation is of the order of . This temperature fluctuation is believed to come from the gravitational potential through Sachs-Wolfe effect  [5]. In the usual inflation models the gravitational potential is believed to be created when the hypothetical scalar particle(s) called inflaton(s) fluctuates quantum mechanically during rolling down classically along with the potential. There are many models proposed for creation mechanism, such as slow-roll inflation [6], chaotic inflation [7] etc.

In our previous papers [8], we show that the temperature fluctuation of CMB can be explained as the vacuum fluctuation of a large number of matter fields. Here the matter fields are originated from the string excited states and/or the Kaluza-Klein modes on the compactified space. There appear very many kinds of fields in the exponentially expanding universe, where the Hubble parameter plays the role of temperature. These fields fluctuate around the bottom of the mass potential, and generate the energy density of fluctuation. Finally the fluctuation creates the gravitational potential through the Einstein equation. Although each fluctuation due to single field is very small, but enormous number of species of fields contribute and it turns out to be summed up to a huge fluctuation compatible to the present observational value. Furthermore it is possible to determine the radii of the compactified spaces, as well as the string coupling constant by comparison with the observational values of CMB. The very merit of our theory is in the fact that we do not need to put any assumption or fine-tuning on the form of the matter fields potential, but we simply consider free fields. However in our above described theory there still are several problems to be resolved. In the first place the spectral index which indicates the distance dependence of the fluctuation or, in other words, the 2-point correlation function of the fields exceeds one, i.e. in our model, although the observed value of the Planck is  [3]. As was pointed out in our foregoing papers [8] the reason why we obtained is in the fact that we assumed the background metric is exactly the de Sitter space-time. In order to explain the experimental value of the spectral index, we may have to introduce a time dependent Hubble parameter , something like .

So far we have not discussed the origin of the inflation, and simply assumed the de Sitter space-time. In this paper we point out the possibility that the inflation is caused by the vacuum energy of the many fields in the expanding universe.

It is the purpose of the present article that we present a self-consistent theory of the time evolution of the universe by incorporating the backreaction of the quantized matter fields1. The main idea of the present paper follows our pervious articles [8], except that we do not assume de Sitter background and incorporate the backreaction of the matter fields. In the early universe we consider species of the free scalar fields. The time evolution of the early universe is assumed to follow the classical Einstein equation,

 Rμν−12gμνR=−8πGN∑i=1⟨T(i)μν⟩. (1.1)

On the right hand side the energy-momentum tensor of the species of the matter fields is regarded as the vacuum expectation value , in the metric . When one expresses by a functional of arbitrary metric , , the Einstein equation (1.1) describes the time evolution of caused by the backreaction from the matter fields. Here we take the vacuum of the Bunch-Davies type in the calculation of . As we will see is proportional to . Therefore in the early universe, where is large, we can ignore the initial state dependence of the energy-momentum tensor and simply take .

We mainly use the Eikonal approximation to make solutions of the equation of motion of the matter fields. Then we perform the 1-loop momentum integration to calculate the correlation functions. By so doing the eq. (1.1) becomes a second order differential equation for .

The present paper is organized as follows. In section 2, we review briefly the Eikonal approximation. Then we investigate the solution of matter field of mass in two cases, that is, for the cases of and , and evaluate for each cases. In section 3 we assume that there exist both light and heavy particles in the universe and search for de Sitter-like expanding solutions of eq. (1.1), where is almost constant. Then we examine the solutions when the Hubble parameter is very small compared to that of the de Sitter-like expanding era. From this analysis we will see that the Hubble parameter becomes zero and the de Sitter era ends for a certain region of the initial conditions. Section 4 is devoted to conclusion and outlook. In Appendix A we describe in detail the calculation by using the Airy function around boundary between UV and IR regimes in the Eikonal approximation. In Appendix B we show the calculation of the correlation functions using the solution in the IR regime in the Eikonal approximation.

## 2 Scalar field in the Friedmann-Robertson-Walker space-time and the Eikonal approximation

It is legitimate to assume that the background metric is homogeneous and uniform as are indicated from CMB data. We thus start with investigating the one species of the scalar field described by the action

 S=∫dnxan−12{˙ϕ2−1a2(∇ϕ)2−m2ϕ2} (2.1)

in the Friedmann-Robertson-Walker space-time

 gμν(x)=⎛⎜ ⎜ ⎜ ⎜⎝10000−a2(t)0000−a2(t)0000−a2(t)⎞⎟ ⎟ ⎟ ⎟⎠. (2.2)

For the later convenience to apply the dimensional regularization, the space-time dimension in eq. (2.1) is taken to be with . In terms of the co-moving time

 dτ=dta, (2.3)

we write the differentiation as

and then the Fourier mode of the rescaled scalar field

 χ(τ,→x)≡an−22(t)ϕ(t,→x) (2.5)

obeys the following equation of motion:

 χ′′k+(k2−V[a])χk=0. (2.6)

Here the potential is defined by

 V[a] ≡(1−ϵ2)a′′a−ϵ(12−ϵ4)(a′a)2−m2a2 (2.7)

where is the Hubble parameter

 H(t)=˙a(t)a(t). (2.8)

The quantity we are interested in is the -component of the energy-momentum tensor as a functional of (or ),

 ⟨T00⟩[a] =μ4−n2⟨˙ϕ2+1a2(∇ϕ)2+m2ϕ2⟩ (2.9)

Here a mass scale is introduced to keep the mass dimension of the energy-momentum tensor.

We solve the equation of motion (2.6) with recourse to the Eikonal approximation by putting

 χk=Ae−if. (2.10)

Then by using the boundary condition

 χk∼1√ke−ikτ,(τ→−∞) (2.11)

and the assumption of the Eikonal approximation

 (f′)2≫A′′A, (2.12)

(2.6) has the solution

 χk=1√2f′e−if,f′=√k2−V[a]. (2.13)

Here the Wronskian is used to normalize the solution:

 χk(χ∗k)′−χ∗kχ′k=i. (2.14)

The general solution for is given by

 χ(τ,→x)=∫d3−ϵ→k(2π)3−ϵ{1√2f′(k,τ)e−if(k,τ)−i→k⋅→xa→k+c.c.} (2.15)

in the oscillatory region,

 k2−V>0. (2.16)

We then quantize the field canonically by using the commutation relation

 [χ(τ,→x),χ′(τ,→y)]=iδ3−ϵ(→x−→y). (2.17)

The vacuum state is defined as usual

 a→k|0⟩=0 (2.18)

so that the vacuum expectation value of an operator is given by .

In the following we consider the situation where and are much smaller than itself,

 ˙HH2≪1and¨HH3≪1, (2.19)

which means that and vary much slower than the time evolution of the universe.

### 2.1 Energy density for the case of a large Hubble parameter: H(t)≫m

In this subsection we evaluate the energy density of a scalar field when Hubble parameter is much larger than mass; . It is well known that if is constant is given by

 ρ=⟨T00⟩=332π2H4{1+O(m2H2)}. (2.20)

Note that this result is obtained without using the Eikonal approximation.

When and , the potential (2.7) is always positive, i.e.

 V[a]>0. (2.21)

Then the eq. (2.16) is not always satisfied, and we should analyze the following three momentum regions to continue the solution in the ultra-violet limit (equivalently ) into the infrared region:

1. In this region the solution is oscillatory as

 χk(τ)=1√214√k2−V(τ)e−i∫ττkdτ′√k2−V(τ′). (2.22)

Here is determined from .

2. The solution is expressed in terms of the Airy function

 χk(τ)=e−iπ4√2B−16{iAi(B13(τ−τk))+Bi(B13(τ−τk))},B≡V′(τk). (2.23)
3. The damping and enhancing solutions given by

 χk(τ)=e−iπ4√21√[4]V(τ)−k2{i2e−∫ττkdτ′√V(τ′)−k2+e+∫ττkdτ′√V(τ′)−k2}. (2.24)

In the above (ii) and (iii), the boundary is chosen to satisfy the validity condition of the Eikonal approximation (2.12). In fact,

 (f′)2≫A′′A⇒(1f′)′≪1⇒k≪ ⎷V−(V′2)23≡k0. (2.25)

Here is obtained from (2.4) and (2.7),

 V′=(1−ϵ2)a3(4H3+6H˙H+¨H)−ϵ(1−ϵ2)a3(H3+H˙H)−2m2a3H. (2.26)

We next evaluate the vacuum expectation value of the energy density in each region. In the region (i) it is given by

 ⟨T00⟩H≫m =132π2(−m4+2m2H2−6H2˙H−2H¨H+˙H2)1ϵ −1384π212H2+˙H−m2 ×[6{43H4−2(log4π−γ+13)m2H2+(log4π−γ+32)m4}(2H2−m2) +(log4π−γ−52)m4}˙H +12{+(log4π−γ−1)˙H+(log4π−γ−53)(2H2−m2)}H¨H +6{4(log4π−γ−74)H2+(log4π−γ−52)m2}˙H2 −6(log4π−γ−32)˙H3+¨H2] −164π2(−m4+2m2H2−6H2˙H−2H¨H+˙H2)log2H2+˙H−m2μ2+O(ϵ). (2.27)

Only in this region there appears an ultra-violet divergence which should be renormalized. Indeed in (2.27), the term gives the divergence in dimensional regularization. There are five terms in the coefficient of . The and terms are renormalized into the cosmological constant term and the Einstein-Hilbert action , respectively. The rest three terms are , , and and from the dimensional analysis their counter terms are given by and . In the case of FRW metric (2.2), these two actions give the following additional terms to the Euler-Lagrange equation:

 ∫d4x√−gR2⇒ 18(−6H2˙H−2H¨H+˙H2), ∫d4x√−gRμνRμν⇒ 6(−6H2˙H−2H¨H+˙H2). (2.28)

Therefore the three coefficients of the term can be renormalized by these two counter terms. In principle we may introduce finite renormalization for the three terms , and . However, since we have to make the same renormalization to the two cases and of the single theory, we relegate to argue this problem after the following subsection 2.2 for the case of a small Hubble parameter .

Then we make use of the minimal subtraction, and expand (2.27) for large assuming and :

 ⟨T00⟩H≫m ≃164π2{−43H4−6(log4π−γ−199)H2˙H−2(log4π−γ−53)H¨H +(log4π−γ−176)˙H2} −164π2(−6H2˙H−2H¨H+˙H2)log2H2μ2+⋯. (2.29)

For the regions (ii) and (iii) the contributions to are described in the appendices A and B respectively. We here show the combined result of (ii) and (iii) calculated numerically under the conditions , and :

 ⟨T00⟩H≫m ≃0.01365H4+0.01333H2˙H+0.0005572H¨H+0.02139˙H2+⋯. (2.30)

Note that the coefficient of does not remain stationary when we vary the artificial parameter , which was introduced to separate two regions (ii) and (iii). However this coefficient is very small compared to the other terms and does not affect the physical result.

### 2.2 Energy density for the case of a small Hubble parameter: H(t)≪m

In this subsection we derive the energy density in the case of a small Hubble parameter, . In the same manner as the previous subsection 2.1 we calculate the vacuum expectation value of the energy density. Again we assume that the change of the Hubble parameter is fairly slow,

 ˙HH2≪1,¨HH3≪1.

In this case the potential term (2.7) of the equation of motion (2.6) is always negative,

 V[a]<0. (2.31)

Then the solution of the Eikonal approximation (2.13) is expressed as

 f′=√k2+(−V[a]).

Apparently any turning point does not exist so that the solution (2.13) is valid in the whole region of .

Thus we obtain the energy density as

 ⟨T00⟩H≪m =132π2(−m4+2m2H2−6H2˙H−2H¨H+˙H2)1ϵ +1384π21m2−2H2−˙H ×[−6{(log4π−γ+32)m4−2(log4π−γ+13)m2H2+43H4}(m2−2H2) +6{(log4π−γ−52)m4−8(log4π−γ−54)m2H2 +12(log4π−γ−2)H4}˙H +12{−(log4π−γ−53)(m2−2H2)+(log4π−γ−1)˙H}H¨H +6{(log4π−γ−52)m2+4(log4π−γ−74)H2}˙H2 −6(log4π−γ−32)˙H3+¨H2] −164π2(−m4+2m2H2−6H2˙H−2H¨H+˙H2)logm2−2H2−˙Hμ2+O(ϵ). (2.32)

We then make the renormalization by using the minimal subtraction, and the expansion assuming and gives the result,

 ⟨T00⟩H≪m ≃164π2{−(log4π−γ+32)m4+2(log4π−γ−23)m2H2 +23H4−6(log4π−γ−199)H2˙H−2(log4π−γ−53)H¨H +(log4π−γ−2)˙H2} −164π2(−m4+2m2H2−6H2˙H−2H¨H+˙H2)logm2μ2+⋯. (2.33)

Hereby we next make use of the finite renormalization for the cosmological constant term and the Einstein-Hilbert action in such a way that and terms and and terms are set to zero. The reason for so doing is because the energy density for small values of should be zero. Although we should add the same counter terms to (2.29), such terms are negligible when is much larger than . Furthermore we may make a finite renormalization for and which generates an additional term to (2.33).

## 3 Solution of the Einstein equation including backreaction

So far we have obtained the energy density of a single scalar field for two cases. The explicit form for the case of a large Hubble parameter is given by

 ⟨T00⟩H≫m =164π2(−43H4+383H2˙H+103H¨H−176˙H2) +164π2(log4π−γ−log2H2μ2+λ)(−6H2˙H−2H¨H+˙H2) +0.01365H4+0.01333H2˙H+0.0005572H¨H+0.02139˙H2 (3.1)

which is the sum of (2.29) and (2.30). For the case of a small Hubble parameter we have

 ⟨T00⟩H≪m =164π2(23H4+383H2˙H+103H¨H−2˙H2) +164π2(log4π−γ−logm2μ2+λ)(−6H2˙H−2H¨H+˙H2). (3.2)

We next consider the Einstein equation assuming that there are many fields with various masses. Then the total energy density is written as

 (3.3)

where and stand for the numbers of the fields whose masses are lighter and heavier than the Hubble parameter, respectively. The ratio of to may vary as the Hubble parameter evolves in time. However, the coefficients of each terms in (3.1) and (3.2) are roughly the same, so that we may replace and with their average. Then (3.3) becomes

 ⟨T00⟩=N2(⟨T00⟩H≫m+⟨T00⟩H≪m),

where is the total number of the species. Then the -component of the Einstein equation reads

 (3.4)

Rewriting this equation after plugging (3.1) and (3.2), we obtain the equation of describing the time evolution of the Hubble parameter :

 −3H2 =−8πGN×12{164π2(−23H4+763H2˙H+203H¨H−296˙H2) +132π2(log4π−γ−log√2Hmμ2+λ)(−6H2˙H−2H¨H+˙H2) +0.01365H4+0.01333H2˙H+0.0005572H¨H+0.02139˙H2}, (3.5)

where is the typical mass of the fields heavier than the Hubble parameter. A comment is in order: The mass scale is originally arbitrary, but as a cut-off scale in the general relativity, field theories and/or string theory, it may be natural to choose Planck scale or string scale .

### 3.1 Inflationary solution

In this subsection we analytically solve the above eq. (3.5) by using linear approximation around the de Sitter solution. It is clear that the eq. (3.5) has the solution, where is a constant,

 H(t)=1√8πGN3×12{−164π223+0.01365}≃4.353×mpl√N≡H0, (3.6)

which is nothing but the de Sitter solution. We next consider a fluctuation around this solution ,

 H=H0+~H. (3.7)

In the linear approximation the eq. (3.5) becomes

 r¨~H+q√p˙~H+2~H=0. (3.8)

where and are constants given by

 p =8πN3m2pl×12{−164π223+0.01365}, q =8πN3m2pl×12{164π2763−632π2(log4π−γ−log√2H0mμ2+λ)+0.01333}, r =8πN3m2pl×12{164π2203−232π2(log4π−γ−log√2H0mμ2+λ)+0.0005572}. (3.9)

 H=H0(1−C1eh1t−C2eh2t),C1,C2≪1, h1,h2=−q√p±√q2p−8r2r. (3.10)

Let us turn to investigate the behavior of the solution (3.10) by using observational values. If we assume that the matter field potential is about the forth power of the GUT scale,  [4], we obtain

 H20=8πG3V⇒H0≃1014 [GeV]. (3.11)

Consequently, from (3.6), the number of species is evaluated as

 N≃1011. (3.12)

Let us, for example, choose and in (3.5) as

 μ=ms≃1018 [GeV], m=1016 [GeV], (3.13)

where is the string scale. In the case of , in which we do not make an additional finite renormalization to the minimal subtraction, the solution (3.10) becomes

 H(t) ≃H0(1−C1e−12.48mpl√Nt−C2e0.4527mpl√Nt) ≃H0(1−C1e−2.866H0t−C2e0.1040H0t). (3.14)

If satisfies

 C2≲10−4,

the duration of the inflation becomes

 Δt≳60H0,

i.e. the e-folding number exceeds . In Fig. 3.1(a) we depict the time evolution of the Hubble parameter with the choices and . The present approximation and is valid in the region

 0≲mpl√Nt≲14. (3.15)

In view of (3.6) this is nothing but the region

 0≲H0t≲60. (3.16)

However, if we introduce an additional finite renormalization in the form of

 λ64π2(−6H2˙H−2H¨H+˙H2),

we can relax the condition of by choosing appropriate . For instance if we choose , the solution (3.10) reads

 H(t) ≃H0(1−C1e−12.85mpl√Nt−C2e0.1530mpl√Nt) ≃H0(1−C1e−2.952H0t−C2e0.03514H0t). (3.17)

is depicted in Fig. 3.1(b) for and , which indicates that is close to in the region (3.15).

### 3.2 End of the inflation

In the previous subsection 3.1, we have found the inflationary solution has a sufficiently large e-folding number, and after the exponential expansion, the Hubble parameter starts to decrease.

In this subsection we show that in the above solution the Hubble parameter becomes zero for large , which indicates that the inflation will end automatically. As we have seen in the previous subsection the eq. (3.5) becomes invalid sometime after starts to decrease. However the eq. (3.5) becomes valid again if is sufficiently small. Therefore if we assume that continues to decrease, we can use eq. (3.5) to describe how the inflation stops. In this region term on the right hand side of (3.5) is much smaller than term on the left hand side because they balanced before becomes small:

 GNH4≪H2.

 H2=PH2˙H+QH¨H+R˙H2+S(−6H2˙H−2H¨H+˙H2)log√2Hmμ2, (3.18)

where the constants and are given by

 P=8πN3m2pl×12{164π2763−632π2(log4π−γ+λ)+0.01333}, Q=8πN3m2pl×12{164π2203−232π2(log4π−γ+λ)+0.0005572}, R=8πN3m2pl×12{−164π2296+132π2(log4π−γ+λ)+0.02139}, S=−8πN3m2pl164π2. (3.19)

Here and are given by (3.12) and (3.13).

The solution of (3.18) rapidly decreases as increases and then, with small oscillation, as for various choices of with . For example when we impose an appropriate initial condition so as to satisfy and and perform the numerical calculation for two cases and [GeV]. The results are depicted in Fig. 3.2. Here Fig. 3.2(a) and Fig. 3.2(b) shows the results for and respectively.

## 4 Conclusion and outlook

In the present paper we investigate the effect of the backreaction to the expansion of the space-time due to the matter fields existing in the universe. On arbitrary Friedmann-Robertson-Walker space-time, we quantize, as an example of the matter field, the scalar field and then we calculate the vacuum expectation value of the energy density. Since we are not able to obtain the exact solution of the scalar field, we make use of the Eikonal approximation to construct the solution. We then plug it in the energy density of the matter field. Combining it with the Friedmann equation we obtain a self-consistent equation that describes the time evolution of the space-time.

Thus we can determine the behavior of the early universe including the backreaction of the matter fields. As a result we find the following two eras in the early universe:

1. We obtain the inflationary solution with the e-folding number if we make a fine tuning of few orders of magnitude for the initial condition. This initial condition may be determined by a scenario of the birth of the universe in string theory and/or quantum gravity.

2. After the era (1) the Hubble parameter will continue to decrease and then with small oscillation goes to zero asymptotically. Then finally the inflation ends.

In this manner by imposing some reasonable conditions we can make a inflation scenario without inflaton.

The problems next to be discussed are why the temperature fluctuation in the observed CMB is of the order and very small non-Gaussianity is realized (cf. [3], (95% CL)). In our previous papers [8], by using the de Sitter background without incorporating the backreaction, we estimated the CMB temperature fluctuation and non-Gaussianity using our theory of many scalar fields, and obtained that if then . The difference between the present and previous papers ia in the fact that in the present paper we incorporated the effect of backreaction, which may not affect largely to the fluctuation, because in both cases the fluctuation is produced during the exponentially expanding era. Furthermore in the previous papers [8] we had the problem that the spectral index exceeded 1 and contradicted the observed value. However as is mentioned in Section 1 in the present paper the condition for translated into and is satisfied in the whole range of the inflation era (see Fig 4.1).

Furthermore we consider a new mechanism to produce the temperature fluctuation in CMB that is different from our previous papers [8]. In [8], we pointed out that the fluctuation of the energy density in the universe creates the gravitational potential through the Einstein equation and turns to be the temperature fluctuation of CMB. A new mechanism we would like to discuss here is that the fluctuation of the time at which the inflation ends generates the density perturbation, as in the ordinary scenario with the inflaton, . can be roughly estimated as follows. First we express in terms of the initial values and . We define as the time when the third term of the eq. (3.14) becomes of order :

 C2e0.1012H0tend≃0.1. (4.1)

On the other hand from (3.14) we obtain

 H(0)=H0(1−C1−C2), ˙H(0)=H0(2.869H0C1−0.1012H0C2). (4.2)

By combining (4.1) and (4.2), we obtain

 tend≃10.1012H0log0.1C2≃10H0log0.11−H(0)H0−˙H(0)3H20