# Evaluation of the Cosmological Constant in  Inflation with a Massive Nonminimal Scalar Field

Authors: Jung-Jeng Huang
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

In Schrödinger picture we study the possible effects of trans-Planckian physics on the quantum evolution of massive nonminimally coupled scalar field in de Sitter space. For the nonlinear Corley-Jacobson type dispersion relations with quartic or sextic correction, we obtain the time evolution of the vacuum state wave functional during slow-roll inflation and calculate explicitly the corresponding expectation value of vacuum energy density. We find that the vacuum energy density is finite. For the usual dispersion parameter choice, the vacuum energy density for quartic correction to the dispersion relation is larger than for sextic correction, while for some other parameter choices, the vacuum energy density for quartic correction is smaller than for sextic correction. We also use the backreaction to constrain the magnitude of parameters in nonlinear dispersion relation and show how the cosmological constant depends on the parameters and the energy scale during the inflation at the grand unification phase transition.


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## Body

## 1. Introduction

In the standard inflationary scenario, usual realization of inflation is associated with a slow rolling inflation minimally coupled to gravity [1]. Nevertheless, it is well known that the extension to the nonminimal coupling with the Ricci scalar curvature can soften the problem related to the small value of the self-coupling in the quartic potential of chaotic inflation [2]. Further, nonminimal coupling terms also can lead to corrections on power spectrum of primordial perturbations [3], a tiny tensor-to-scalar ratio [4, 5], and non-Gaussianities [6]. A broad class of models of chaotic inflation in supergravity with an arbitrary inflation potential was also proposed. In these models the inflation field is nonminimally coupled to gravity [7, 8]. Recently, the viability of simple nonminimally coupled inflationary models is assessed through observational constraints on the magnitude of the nonminimal coupling from the BICEP2 experiment [9].Moreover, the standard inflationary scenario has two possible extensions. The first extension is associated with the ambiguity of initial quantum vacuum state, and the choice of initial vacuum state affects the predictions of inflation [10, 11]. The second extension concerns with the trans-Planckian problem [12, 13] of whether the predictions of standard cosmology are insensitive to the effects of trans-Planckian physics. In fact, nonlinear dispersion relations such as the Corley-Jacobson (CJ) type were used to mimic the trans-Planckian effects on cosmological perturbations [12–14]. These CJ type dispersion relations can be obtained naturally from quantum gravity models such as Horava gravity [15, 16]. Recently, in several approaches to quantum gravity, the phenomenon of running spectral dimension of spacetime from the standard value of 4 in the infrared to a smaller value in the ultraviolet is associated with modified dispersion relations, which also include the CJ type dispersion relations [17, 18].In the previous work [19–23] we used the lattice Schrödinger picture to study the free scalar field theory in de Sitter space, derived the wave functionals for the Bunch-Davies (BD) vacuum state and its excited states, and found the trans-Planckian effects on the quantum evolution of massless minimally coupled scalar field for the CJ type dispersion relations with sextic correction. In this paper we extend the study to the case of massive nonminimally coupled scalar field.The paper is organized as follows. In Section2, the theory of a generically coupled scalar field in de Sitter space is briefly reviewed in the lattice Schrödinger picture. In Section 3, we consider the massive nonminimally coupled scalar field during slow-roll inflation and use the CJ type dispersion relations with quartic or sextic correction to obtain the time evolution of the vacuum state wave functional. In Section 4, using the results of Section 3, we calculate the finite vacuum energy density, use the backreaction to constraint the parameters in nonlinear dispersion, and evaluate the cosmological constant. Finally, conclusions and discussion are presented in Section 5. Throughout this paper we will set ħ=c=1.


## 2. De Sitter Scalar Field Theory in Schrödinger Picture

In this section, we begin by briefly reviewing the theory of a generically coupled scalar field in de Sitter space in the lattice Schrödinger picture (for the details of some derivations in this section see [23]). The Lagrangian density for the scalar field we consider is
(1)L=g1/212gμν(x)ϕx,μϕx,ν-ξRϕ22-V(ϕ),V(ϕ)=m2ϕ22,
where ϕ is a real scalar field, V(ϕ) is the potential, m is the mass of the scalar quanta, R is the Ricci scalar curvature, ξ is the coupling parameter, and g = det⁡gμν, μ,ν=0,1,…,d. For a spatially flat (1+d)-dimensional Robertson-Walker spacetime with scale factor a(t), we have
(2)ds2=dt2-a2td2xi,i=1,2,…,d,L=ad12∂0ϕ2-a-2∂iϕ2-ξRϕ22-m2ϕ22.
In the (1+d)-dimensional de Sitter space we have a(t)=exp⁡(ht), where h≡a˙/a is the Hubble parameter which is a constant.Ford=1, in the lattice Schrödinger picture, we obtain from (2) the time-dependent functional Schrödinger equation in momentum space [23]
(3)Hψ=i∂∂tψ,
where
(4)H=2∑l=1N/2∑r=12Hrl,(5)Hrl=12prl2+12hprlϕrl+12a-2ωl2ϕrl2+12m2+ξRϕrl2,(6)ψϕrl,t=∏l=1N/2∏r=12ψrl(ϕrl,t)≡∏rlψrlϕrl,t.
Here ωl≡2/εsinlπ/N, ε=W/N, that is, W is the overall comoving spatial size of lattice, ϕl=ϕ1l+iϕ2l, pl=p1l+ip2l, pl is the conjugate momentum for ϕl, and the subscripts 1 and 2 denote the real and imaginary parts, respectively.For each real modeϕrl, we have
(7)Hrlψrl=i∂∂tψrl,r=1,2(8)-12∂2ψrl∂ϕrl2+12a-2ωl2+m2+ξR-14h2ϕrl2ψrl=i∂ψrl∂t.
Note that (8) arises from the field quantization of the Hamiltonian (5) through the functional Schrödinger representation ϕ-rl→ϕrl, p-rl→-i∂/∂ϕrl, where operators ϕ-rl and p-rl satisfy the equal time commutation relations [ϕ-rl,p-rl]=i, and setting P-rl=p-rl+(1/2)hϕ-rl=-i∂/∂ϕrl so that ϕ-rl,P-rl=ϕ-rl,p-rl=i. Thus (8) governs the time evolution of the state wave functional ψrl of the Hamiltonian operator Hrl in the {ϕrl} representation. In terms of the conformal time τ defined by
(9)dτ=dta,τ=-h-1exp⁡-ht=-h-1a-1,-∞<τ<0,
the normalized wave functionals of vacuum and its excited states are
(10)ψrlnrlϕrl,τ=Rnrlϕrl,τexp⁡iΘnrlϕrl,τ,nrl=0,1,2,…
with the amplitude R(nrl)(ϕrl,τ) and phase Θ(nrl)(ϕrl,τ)(11)R(nrl)(ϕrl,τ)=2h/ππ2nrlnrl!Hν11/2Hnrlηrlexp⁡-12ηrl2,(12)Θnrlϕrl,τ=-hωlτ2Hν(1)′Hν(1)ϕrl2-12+nrl∫2/πτHν(1)2dτ.
Here ηrl is defined by ηrl≡2h/π/Hν(1)ϕrl, H(nrl)(ηrl) is the nrlth-order Hermite polynomial, Hν(1)(ωlτ) is the Hankel function of the first kind of order ν, ν2=1/4-m2+ξR/h2, and the prime in (12) denotes the derivative with respect to ωlτ. The complete wave functionals can be written as ψ[n]ϕrl,t=∏rlψ(nrl)(ϕrl,t), where [n]≡(ni,nj,…) means that mode i is in the ni excited state, mode j is in the nj excited state, and so forth. For nrl=0, the ground state wave functional corresponds to the BD vacuum.Ford=3, we have ν2=9/4-m2+ξR/h2, R=12h2, and the mode index l in ωl carries labels (li,i=1,2,3) which will be suppressed below. Furthermore, from (3)–(8) we get in the continuum limit (ωl→k)
(13)i∂ψ∂t=∑rk-12∂2∂ϕrk2+12a-2k2+m2+ξR-94h2ϕrk2ψ.


## 3. Trans-Planckian Effects on Vacuum Wave Functional

For the inflationary potentialV(ϕ)=m2ϕ2/2, the bounds on ξ derived from the joint data analysis of Planck + WP + BAO + high-l for the number of e-foldings N=60 are -4.2×10-3<ξ<-1.1×10-3 (68% CL), -5.1×10-3<ξ≤0 (95% CL) [24].For the mass in the tree-level potential, we havem=1.46×1013GeV [25]. Moreover, the recent BICEP2 experiment suggests that h≅1.13×1014GeV [26–28]. Therefore, from ν2=9/4-m2+ξR/h2, we have m2+ξR≪9h2/4 and ν≅3/2, which will be used below. To study further the effects of trans-Planckian physics, we use the CJ type dispersion relation
(14)ω2ka=k21+bskaM2s,
where M is a cutoff scale, s is an integer, and bs is an arbitrary coefficient [12–14].


### 3.1. CJ Type Dispersion Relations with Quartic Correction

First, we use the CJ type dispersion relation (14) with s=1 and b1>0 to obtain the time evolution of the vacuum state wave functional. Recall that these CJ type dispersion relations can be obtained from theories based on quantum gravity models [15–18].Usingz=kτ=k/ah which is the ratio of physical wave number kphys≡k/a to the inverse of Hubble radius, (13) becomes
(15)i∂ψ∂t=∑rk-12∂2∂ϕrk2+12z21+σ2z2h2-94h2ϕrk2ψ,
where σ2≡b1(h/M)2, and the ground state solution of (15) becomes
(16)ψ(0)=∏rkAk0τexp⁡-12Bkτa-1ϕrk2,
where Ak(0)(τ) and Bk(τ) satisfy
(17)Ak(0)(τ)=exp⁡-i12∫Bk(τ)dτ+const,(18)Bk2(τ)-idBk(τ)dτ+Bk(τ)τ-k21+σ2z2-94τ2=0.
In region I where kphys≡k/a>M, that is, z>M/h, the dispersion relations can be approximated by ω2(k/a)≈k2σ2z2, and the corresponding wave functional for the initial BD vacuum state is [23, 29]
(19)ψ(0)I=∏rkAk0Iτexp⁡-12BkIτa-1ϕrkI2,Ak(0)I(τ)=exp⁡-i12∫BkI(τ)dτ+const,(20)BkI(τ)=4/πτH3/4(1)2-ik2H3/4(1)2′H3/4(1)2σz,
where the prime in (20) denotes the derivative with respect to σz2/2.On the other hand, in region II wherekphys≡k/a<M, that is, z<M/h, linear relations recover ω2≅k2, and the corresponding wave functional for the non-BD vacuum state is [23, 29]
(21)ψ(0)II=∏rkAk0IIτexp⁡-12BkIIτa-1ϕrkII2,Ak(0)II(τ)=exp⁡-i12∫BkII(τ)dτ+const,(22)BkII(τ)=+2ReC1IIC2II*H3/212-12/πτ·C1II2+C2II2H3/212+2ReC1IIC2II*H3/212-1-ik2+2ReC1IIC2II*H3/212-1C1II2+C2II2H3/212+2ReC1IIC2II*H3/212′·C1II2+C2II2H3/212+2ReC1IIC2II*H3/212-1,
where the prime in (22) denotes the derivative with respect to z and C1II and C2II satisfy C1II2-C2II2=1. Let τc be the time when the modified dispersion relations take the standard linear form. Then σ2zc2=1 where zc=kτc=M/b11/2h≫1 for b1~1. The constants C1II and C2II can be obtained by the following matching conditions at τc for the two wave functionals (19) and (21):
(23)ψ0IZC=ψ0IIZC,(24)dψ0IdzZC=dψ0IIdzZC,
which can also be rewritten, respectively, as
(25)ReBkIZC=ReBkIIZC,(26)dReBkIdzZC=dReBkIIdzZC,
by requiring BkI=BkII, ϕrkI=ϕrkII, and Ak(0)I=Ak(0)II when z=zc.UsingH3/4(1)σz2/22=4/πσz21+5/8σ2z4+⋯≈4/πσz2 with σ=zc-1 and zc≫1, we have from (20), (22), and (25)(27)1=C1II2+C2II2+2C1IIC2IIcos⁡(2zc-θ),
where we choose C1II=C1II and C2II=C2IIexp⁡(iθ), and θ is a relative phase parameter. Then from (27) and C1II2-C2II2=1 we have
(28)C1II=csc2zc-θ,C2II=-cot(2zc-θ),
where sin(2zc-θ)>0, cos⁡(2zc-θ)<0. Substituting (20) and (22) into (26) and keeping terms up to order 1/zc on the right-hand side of (26), we obtain
(29)1zc=C1IIC2IIcos⁡(2zc-θ)8zc+4C1IIC2IIsin2zc-θ.
Using (28) in (29) gives
(30)cot(2zc-θ)=-14zcorcot(2zc-θ)=-zc2+14zc.
Here we choose cot(2zc-θ)=-(1/4zc), so that C2II is small for zc≫1 to avoid an unacceptably large backreaction on the background geometry. Then we have
(31)C2II≅14zc,C1II=1+C2II2≅1+132zc2≅1
or
(32)sin(2zc-θ)≅1,cos⁡(2zc-θ)≅-14zc.


### 3.2. CJ Type Dispersion Relations with Sextic Correction

In this subsection, we use the CJ type dispersion relation (14) with s=2 and b2>0 to obtain the time evolution of the vacuum state wave functional. For this case, only (15), (18), and (20) are changed into
(33)i∂ψ∂t=∑rk-12∂2∂ϕrk2+12z21+σ2z4h2-94h2ϕrk2ψ,(34)Bk2(τ)-idBk(τ)dτ+Bk(τ)τ-k2(1+σ2z4)-94τ2=0,(35)BkI(τ)=6/πτH1/2(1)2-ik2H1/2(1)2′H1/2(1)2σz2,
where σ2≡b2(h/M)4, and the prime in (35) denotes the derivative with respect to σz3/3. Using H1/2(1)σz3/32=6/πσz3 with σ=z-c-2 and z-c=kτ-c=M/b21/4h≫1 for b2~1, we obtain from (35), (22), (25), and C1II2-C2II2=1(36)1=C1II2+C2II2+2C1IIC2IIcos⁡(2z-c-θ),(37)C1II=csc2z-c-θ,C2II=-cot(2z-c-θ),
where sin(2z-c-θ)>0, cos⁡(2z-c-θ)<0. Substituting (35) and (22) into (26) and keeping terms up to order 1/z-c on the right-hand side of (26), we find
(38)2z-c=C1IIC2IIcos⁡(2z-c-θ)8z-c+4C1IIC2IIsin2z-c-θ.
Using (37) in (38) gives
(39)cot2z-c-θ=-12z-corcot(2z-c-θ)=-z-c2+12z-c.
Here we choose cot(2z-c-θ)=-(1/2z-c), so that C2II is small for z-c≫1 to avoid an unacceptably large backreaction on the background geometry. Then we have
(40)C2II≅12z-c,C1II=1+C2II2≅1+18z-c2≅1
or
(41)sin(2z-c-θ)≅1,cos⁡(2z-c-θ)≅-12z-c.


## 3.1. CJ Type Dispersion Relations with Quartic Correction

First, we use the CJ type dispersion relation (14) with s=1 and b1>0 to obtain the time evolution of the vacuum state wave functional. Recall that these CJ type dispersion relations can be obtained from theories based on quantum gravity models [15–18].Usingz=kτ=k/ah which is the ratio of physical wave number kphys≡k/a to the inverse of Hubble radius, (13) becomes
(15)i∂ψ∂t=∑rk-12∂2∂ϕrk2+12z21+σ2z2h2-94h2ϕrk2ψ,
where σ2≡b1(h/M)2, and the ground state solution of (15) becomes
(16)ψ(0)=∏rkAk0τexp⁡-12Bkτa-1ϕrk2,
where Ak(0)(τ) and Bk(τ) satisfy
(17)Ak(0)(τ)=exp⁡-i12∫Bk(τ)dτ+const,(18)Bk2(τ)-idBk(τ)dτ+Bk(τ)τ-k21+σ2z2-94τ2=0.
In region I where kphys≡k/a>M, that is, z>M/h, the dispersion relations can be approximated by ω2(k/a)≈k2σ2z2, and the corresponding wave functional for the initial BD vacuum state is [23, 29]
(19)ψ(0)I=∏rkAk0Iτexp⁡-12BkIτa-1ϕrkI2,Ak(0)I(τ)=exp⁡-i12∫BkI(τ)dτ+const,(20)BkI(τ)=4/πτH3/4(1)2-ik2H3/4(1)2′H3/4(1)2σz,
where the prime in (20) denotes the derivative with respect to σz2/2.On the other hand, in region II wherekphys≡k/a<M, that is, z<M/h, linear relations recover ω2≅k2, and the corresponding wave functional for the non-BD vacuum state is [23, 29]
(21)ψ(0)II=∏rkAk0IIτexp⁡-12BkIIτa-1ϕrkII2,Ak(0)II(τ)=exp⁡-i12∫BkII(τ)dτ+const,(22)BkII(τ)=+2ReC1IIC2II*H3/212-12/πτ·C1II2+C2II2H3/212+2ReC1IIC2II*H3/212-1-ik2+2ReC1IIC2II*H3/212-1C1II2+C2II2H3/212+2ReC1IIC2II*H3/212′·C1II2+C2II2H3/212+2ReC1IIC2II*H3/212-1,
where the prime in (22) denotes the derivative with respect to z and C1II and C2II satisfy C1II2-C2II2=1. Let τc be the time when the modified dispersion relations take the standard linear form. Then σ2zc2=1 where zc=kτc=M/b11/2h≫1 for b1~1. The constants C1II and C2II can be obtained by the following matching conditions at τc for the two wave functionals (19) and (21):
(23)ψ0IZC=ψ0IIZC,(24)dψ0IdzZC=dψ0IIdzZC,
which can also be rewritten, respectively, as
(25)ReBkIZC=ReBkIIZC,(26)dReBkIdzZC=dReBkIIdzZC,
by requiring BkI=BkII, ϕrkI=ϕrkII, and Ak(0)I=Ak(0)II when z=zc.UsingH3/4(1)σz2/22=4/πσz21+5/8σ2z4+⋯≈4/πσz2 with σ=zc-1 and zc≫1, we have from (20), (22), and (25)(27)1=C1II2+C2II2+2C1IIC2IIcos⁡(2zc-θ),
where we choose C1II=C1II and C2II=C2IIexp⁡(iθ), and θ is a relative phase parameter. Then from (27) and C1II2-C2II2=1 we have
(28)C1II=csc2zc-θ,C2II=-cot(2zc-θ),
where sin(2zc-θ)>0, cos⁡(2zc-θ)<0. Substituting (20) and (22) into (26) and keeping terms up to order 1/zc on the right-hand side of (26), we obtain
(29)1zc=C1IIC2IIcos⁡(2zc-θ)8zc+4C1IIC2IIsin2zc-θ.
Using (28) in (29) gives
(30)cot(2zc-θ)=-14zcorcot(2zc-θ)=-zc2+14zc.
Here we choose cot(2zc-θ)=-(1/4zc), so that C2II is small for zc≫1 to avoid an unacceptably large backreaction on the background geometry. Then we have
(31)C2II≅14zc,C1II=1+C2II2≅1+132zc2≅1
or
(32)sin(2zc-θ)≅1,cos⁡(2zc-θ)≅-14zc.


## 3.2. CJ Type Dispersion Relations with Sextic Correction

In this subsection, we use the CJ type dispersion relation (14) with s=2 and b2>0 to obtain the time evolution of the vacuum state wave functional. For this case, only (15), (18), and (20) are changed into
(33)i∂ψ∂t=∑rk-12∂2∂ϕrk2+12z21+σ2z4h2-94h2ϕrk2ψ,(34)Bk2(τ)-idBk(τ)dτ+Bk(τ)τ-k2(1+σ2z4)-94τ2=0,(35)BkI(τ)=6/πτH1/2(1)2-ik2H1/2(1)2′H1/2(1)2σz2,
where σ2≡b2(h/M)4, and the prime in (35) denotes the derivative with respect to σz3/3. Using H1/2(1)σz3/32=6/πσz3 with σ=z-c-2 and z-c=kτ-c=M/b21/4h≫1 for b2~1, we obtain from (35), (22), (25), and C1II2-C2II2=1(36)1=C1II2+C2II2+2C1IIC2IIcos⁡(2z-c-θ),(37)C1II=csc2z-c-θ,C2II=-cot(2z-c-θ),
where sin(2z-c-θ)>0, cos⁡(2z-c-θ)<0. Substituting (35) and (22) into (26) and keeping terms up to order 1/z-c on the right-hand side of (26), we find
(38)2z-c=C1IIC2IIcos⁡(2z-c-θ)8z-c+4C1IIC2IIsin2z-c-θ.
Using (37) in (38) gives
(39)cot2z-c-θ=-12z-corcot(2z-c-θ)=-z-c2+12z-c.
Here we choose cot(2z-c-θ)=-(1/2z-c), so that C2II is small for z-c≫1 to avoid an unacceptably large backreaction on the background geometry. Then we have
(40)C2II≅12z-c,C1II=1+C2II2≅1+18z-c2≅1
or
(41)sin(2z-c-θ)≅1,cos⁡(2z-c-θ)≅-12z-c.


## 4. Vacuum Energy, Backreaction, and Cosmological Constant

Using the results of Section3, we proceed to calculate the finite vacuum energy density and use the backreaction constraint to address the cosmological constant problem. Note that, in the slow-roll approximation, the energy density of the scalar field is ρϕ≅V(ϕ), where V(ϕ)=m2ϕ2/2. Therefore the relation between the expectation value of the vacuum energy density ρϕ and the vacuum wave functional ψ(0) in (16) is
(42)ρϕ=ψ(0)ρϕψ(0)=m22ε-3∑rk∫-∞∞durkψrk(0)(urk,τ)2urk2=m2218π3∫d3k121Re(Bk(τ)a-1)a-3=m2212π2∫k2121ReBkτa-1a-3dk,
where we use a field redefinition urk≡a-3/2ϕrk,
(43)ψrk(0)(urk,τ)2=a3/2ReBkτa-1πexp⁡-ReBkτa-1urk2a-3,Re(Bk(τ)a-1) denotes the real part of Bk(τ)a-1, and the factor a3/2 in (43) appears through the normalization condition
(44)∫-∞∞durkψrk(0)(urk,τ)2=1.Fors=1 and b1>0, in region I, we have H3/4(1)σz2/22≈4/πσz2 with σ=zc-1. Then, using a=1/hτ=k/hz and (20) in (42), we obtain
(45)ρϕs=1I=18π2m2h2zc∫zcαzcdz=18π2m2h2zc2(α-1),
where zc=M/b11/2h and αzc=MPl/h (MPl=G-1/2=1.22×1019GeV is the Planck mass) are the boundaries of the interval of integration. On the other hand, in region II, (22) can be expressed as
(46)BkII(τ)=2/πτH3/2(1)md2-ik2H3/2(1)md2′H3/2(1)md2,
where H3/2(1)md is defined as
(47)H3/2(1)md≡C1II2+C2II2H3/2(1)2+2ReC1IIC2II*H3/2(1)21/2,
with H3/2(1)(z)2=(2/πz)1+1/z2. From (27), (31), and (32), we note that H3/2(1)md can be approximated by H3/2(1) as z decreases from z=zc≫1 to z=1 (horizon exit). Then, using a=1/hτ=k/hz and (22) in (42), we obtain
(48)ρϕs=1II=18π2m2h2∫1zcz2+1zdz=18π2m2h212zc2+ln⁡zc-12.
From (45) and (48) we have
(49)ρϕs=1=ρϕs=1I+ρϕs=1II=18π2m2h2zc2α-12+ln⁡zc-12.
For zc≫1 and α=b11/2(MPl/M)>3/2, (49) becomes
(50)ρϕs=1≅18π2m2h2zc2α.
From (50) we see that there is no backreaction problem if the energy density due to the quantum fluctuations of the inflation field is smaller than that due to the inflation potential; that is,
(51)ρϕs=1<V(ϕ).
In the slow-roll approximation, using V(ϕ)≅3MPl2h2/8π and (50) in (51) gives the constraint on the parameter b1 as b1>(1/9π2)M/MPl2m/h4. For M~1016GeV (the energy scale during inflation implied by the BICEP2 experiment [26, 27]), we have b1>2.1×10-12.Fors=2 and b2>0, in region I, we have H1/2(1)σz3/32=6/πσz3 with σ=z-c-2. Then, using a=1/hτ=k/hz and (35) in (42), we obtain
(52)ρϕs=2I=18π2m2h2z-c2∫z-cβz-c1zdz=18π2m2h2z-c2ln⁡β,
where z-c=M/b21/4h and βz-c=MPl/h are the boundaries of the interval of integration. On the other hand, in region II, (22) can be again expressed as (46) with H3/2(1)md defined by (47). From (36), (40), and (41), we also note that H3/2(1)md can be approximated by H3/2(1) as z decreases from z=z-c≫1 to z=1. Then, using a=1/hτ=k/hz and (22) in (42), we obtain
(53)ρϕs=2II=18π2m2h2∫1z-cz2+1zdz=18π2m2h212z-c2+ln⁡z-c-12.
From (52) and (53) we have
(54)ρϕs=2=ρϕs=2I+ρϕs=2II=18π2m2h2z-c2ln⁡β+12+ln⁡z-c-12.
For z-c≫1 and z-c2(ln⁡β+1/2)≫ln⁡z-c which are satisfied if b2>6.3×10-14, (54) becomes
(55)ρϕs=2≅18π2m2h2z-c2ln⁡β+12.
Moreover, we notice that there is no backreaction problem if
(56)ρs=2<V(ϕ).
Using V(ϕ)≅3MPl2h2/8π and (55) in (56) gives the constraint on the parameter b2 as b21/2>(1/3π)(Mm/MPlh)2(ln⁡β+1/2). For M~1016GeV, we have b2>2.0×10-7.Comparing (50) with (54), we find that ρϕs=2<ρϕs=1 if the inequality z-c2(ln⁡β+1/2)<zc2α or (ln⁡β+1/2)/(MPl/M)2<b2/b1 is satisfied. For example, the usual parameter choice b1~b2~1 satisfies the inequality. On the other hand, we have ρϕs=2>ρϕs=1 if the inequality z-c2(ln⁡β+1/2)>zc2α or (ln⁡β+1/2)/(MPl/M)2>b2/b1 is satisfied. For example, the parameter choices b1~1 and b2~10-6 satisfy the inequality.In the case thatρϕs=1 is larger than ρϕs=2, using (50) in the cosmological constant Λ=8πρvac/MPl2 gives Λ=(1/πb11/2)m2M/MPl which is 5.6×1022GeV2 for b1~1 and 1.8×1028GeV2 for b1~10-11. In the case that ρϕs=2 is larger than ρϕs=1, using (54) in the cosmological constant Λ=8πρvac/MPl2 gives Λ=(1/πb21/2)m2M2/MPl2ln⁡β+1/2 which is 3.5×1020GeV2 for b2~1 and 1.9×1023GeV2 for b2~10-6.


## 5. Conclusions and Discussion

In the Schrödinger picture, we have considered the theory of a generically coupled free real scalar field in de Sitter space. To investigate the possible effects of trans-Planckian physics on the quantum evolution of the vacuum state of scalar field, we focus on the massive nonminimally coupled scalar field in slow-roll inflation and consider the CJ type dispersion relations with quartic or sextic correction.We obtain the time evolution of the vacuum state wave functional and calculate the expectation value of the corresponding vacuum energy density. We find that the vacuum energy density is finite and has improved ultraviolet properties. For the usual dispersion parameter choice, the vacuum energy density for quartic correction to the dispersion relation is larger than for sextic correction. For some other parameter choices, the vacuum energy density for quartic correction is smaller than for sextic correction.We also use the backreaction to constrain the magnitude of parameters in nonlinear dispersion relation and show how the cosmological constant depends on the parameters and the energy scale during the inflation at the grand unification phase transition.From (50) and (54) we see that the value of the cosmological constant can be reduced significantly through increasing the dispersion parameters in nonlinear dispersion relation and decreasing the cutoff energy scale associated with phase transition. However, the fact that the dispersion relation of a scalar field can not be modified on energy scales smaller than 1 TeV makes the cosmological problem still unsolved.



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