EVALUATION OF THE COSMOLOGICAL CONSTANT IN INFLATION WITH A MASSIVE NON-MINIMAL SCALAR FIELD

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1 EVALUATON OF THE OSMOLOGAL ONSTANT N NFLATON WTH A MASSVE NON-MNMAL SALAR FELD JUNG-JENG HUANG Department of Mehanial Engineering, Physis Division Ming hi University of Tehnology, Taishan, New Taipei ity 40, Taiwan huangjj@mail.mut.edu.tw n Shrödinger piture we study the possible effets of trans-anian physis on the quantum evolution of massive non-minimally oupled salar field in de Sitter spae. For the nonlinear oey-jaobson type dispersion relations with quarti or sexti orretion, we obtain the time evolution of the vauum state wave funtional during slow-roll inflation, and alulate expliitly the orresponding expetation value of vauum energy density. We find that the vauum energy density is finite. For the usual dispersion parameter hoie, the vauum energy density for quarti orretion to the dispersion relation is larger than for sexti orretion, while for some other parameter hoies, the vauum energy density for quarti orretion is smaller than for sexti orretion. We also use the bareation to onstrain the magnitude of parameters in nonlinear dispersion relation, and show how the osmologial onstant depends on the parameters and the energy sale during the inflation at the grand unifiation phase transition.. ntrodution n the standard inflationary senario, usual realiation of inflation is assoiated with a slow rolling inflaton minimally oupled to gravity []. Nevertheless, it is well nown that the extension to the non-minimal oupling with the Rii salar urvature an soften the problem related to the small value of the self-oupling in the quarti potential of haoti inflation []. Further, non-minimal oupling terms also an lead to orretions on power spetrum of primordial perturbations [], a tiny tensor-to-salar ratio [4, 5] and non-gaussianities [6]. A broad lass of models of haoti inflation in supergravity with an arbitrary inflaton potential was also proposed. n these models the inflaton field is non-minimally oupled to gravity [7, 8]. Reently, the viability of simple non-minimally oupled inflationary models is assessed through observational onstraints on the magnitude of the non-minimal oupling from the BEP experiment [9]. Moreover, the standard inflationary senario has two possible extensions. The first extension is assoiated with the ambiguity of initial quantum vauum state, and the hoie of initial vauum state affets the preditions of inflation [0, ]. The seond extension onerns with the trans-anian problem [, ] of whether the preditions of standard

2 osmology are insensitive to the effets of trans-anian physis. n fat, nonlinear dispersion relations suh as the oey-jaobson (J type were used to mimi the trans-anian effets on osmologial perturbations [-4]. These J type dispersion relations an be obtained naturally from quantum gravity models suh as Horava gravity [5, 6]. Reently, in several approahes to quantum gravity, the phenomenon of running spetral dimension of spaetime from the standard value of 4 in the infrared to a smaller value in the ultraviolet is assoiated with modified dispersion relations, whih also inlude the J type dispersion relations [7, 8]. n the previous wo [9-] we used the lattie Shrödinger piture to study the free salar field theory in de Sitter spae, derived the wave funtionals for the Bunh-Davies (BD vauum state and its exited states, and found the trans-anian effets on the quantum evolution of massless minimally oupled salar field for the J type dispersion relations with sexti orretion. n this paper we extend the study to the ase of massive non-minimally oupled salar field. The paper is organied as follows. n Setion, the theory of a generially oupled salar field in de Sitter spae is briefly reviewed in the lattie Shrödinger piture. n Setion, we onsider the massive non-minimally oupled salar field during slow-roll inflation, and use the J type dispersion relations with quarti or sexti orretion to obtain the time evolution of the vauum state wave funtional. n Setion 4, using the results of Setion, we alulate the finite vauum energy density, use the bareation to onstraint the parameters in nonlinear dispersion, and evaluate the osmologial onstant. Finally, onlusions and disussion are presented in Setion 5. Throughout this paper we will set h.. De Sitter Salar Field Theory in Shrödinger Piture n this setion, we begin by briefly reviewing the theory of a generially oupled salar field in de Sitter spae in the lattie Shrödinger piture (for the details of some derivations in this setion see []. The Lagrangian density for the salar field we onsider is µν [ g ( x ( x ( x ] L g, µ, ν ξr / V (, V ( m /, ( where is a real salar field, V ( is the potential, m is the mass of the salar quanta, R is the Rii salar urvature, ξ is the oupling parameter, and g det g µν, µ, ν 0,,,d. For a spatially flat (+d-dimensional Robertson-Waler spae-time with sale fator a (t, we have

3 i ds dt a ( t d x, i,,..., d, d L a [( ( ] / 0 a i ξr m /. ( n the (+d-dimensional de Sitter spae we have a ( t exp( ht, where h a& / a is the Hubble parameter whih is a onstant. For d, in the lattie Shrödinger piture, we obtain from ( the time-dependent funtional Shrödinger equation in momentum spae [] where H p + hp H i t, ( N / H l r H, (4 + a ωl + + ( m ξr, (5 Here ( / ε sin( lπ N of lattie, l / N / [, ] (, t t (, t, (6 l r ω, ε W / N, i.e., W is the overall omoving spatial sie, p l p + l ipl, p l is the onjugate momentum for + l l i l and the subsripts and denote the real and imaginary parts respetively. For eah real mode, we have H + a l, i, r, (7 t ( m + ξr h 4 ωl + i t. (8 Note that (8 arises from the field quantiation of the Hamiltonian (5 through the funtional Shrödinger representation p i /, where operators and P p, p satisfy the equal time ommutation relations [, p ] i, and setting + h i so that [, P ] [, p ] i. Thus (8 governs the of the Hamiltonian operator H in the time evolution of the state wave funtional { } representation. n terms of the onformal time τ defined by d dt / a τ h exp ht h a, <τ <0, (9 τ, ( the normalied wave funtionals of vauum and its exited states are

4 4 ( iθ, τ R (, τ exp,, ( n ( τ ( n ( ( n ( n τ ( n τ with the amplitude R (, and phase Θ (, R / n 0,,, (0 h / π (, τ H ( ( exp( n ( n η η, ( π n! H ν ( n ( hω τ ( H l ν π τ Θ( n, τ ( + n dτ ( ( H ν H (. ( ν Here η is defined by η ( π ν ( h / / H, H( ( η is the n th-order n ( ν Hermite polynomial, H ( ω τ is the Hanel funtion of the first ind of order ν, ν / 4 + ( m ξr / l h, and the prime in ( denotes the derivative with respet to ω l τ. The omplete wave funtionals an be written as [ n] [, t] ( n (, t, where [ n] ( n i, n j, L means that mode i is in the n i exited state, mode j is in the n exited state, et. For n 0, the ground state wave funtional orresponds j to the BD vauum. For d, we have ν 9/ 4 ( m + ξr / h, R h and the mode index l in ω arries labels ( l i, i,, whih will be suppressed below. Furthermore, from l equations (-(8 we get in the ontinuum limit ( ω i t l 9 ( m + ξr h } { + a + 4. (. Trans-anian Effets on Vauum Wave Funtional For the inflationary potential V ( m /, the bounds on ξ derived from the joint data analysis of an+wp+bao+high-l for the number of e-foldings N 60 are 4. 0 < ξ <. 0 (68%L, 5. 0 < ξ 0 (95%L [4].

5 5 For the mass in the tree-level potential, we have m.46 0 GeV [5]. Moreover, 4 the reent BEP experiment suggests that h. 0 GeV [6-8]. Therefore, from 9/ 4 ( m ξr / h, we have m + ξ R << 9 h / 4 and ν /, ν + whih will be used below. To study further the effets of trans-anian physis, we use the J type dispersion relations + s ω ( / a bs ( am, (4 where M is a utoff sale, s is an integer, and b is an arbitrary oeffiient [-4]. s.. J Type Dispersion Relations with Quarti orretion First, we use the J type dispersion relations (4 with s and b 0 to obtain the time evolution of the vauum state wave funtional. Reall that these J type dispersion relations an be obtained from theories based on quantum gravity models [5-8]. > Using τ / ah whih is the ratio of physial wave number phys / a to the inverse of Hubble radius, ( beomes 9 ( + σ h h } i { + t 4, (5 where σ b ( h / M, and the ground state solution of (5 beomes, (6 where A ( and B satisfy ( 0 τ ( 0 A (0( τ exp( B ( τ a (τ A ( 0 ( τ exp i B ( τ dτ + onst, (7 db ( τ B ( τ 9 B ( τ i + ( + σ 0 dτ τ 4τ. (8 n region where phys / a > M, i.e., > M / h, the dispersion relations an be approximated by initial BD vauum state is [, 9] ω ( / a σ, and the orresponding wave funtional for the

6 6, A ( exp( B ( a ( 0 (0 τ τ A ( 0 ( τ exp i B ( τ dτ + onst, (9 4 ( H π τ / 4 i σ, (0 H H B ( τ ( ( / 4 / 4 where the prime in (0 denotes the derivative with respet to σ /. On the other hand, in region where phys / a < M, i.e., < M / h, linear relations reover ω vauum state is [, 9] B ( τ, and the orresponding wave funtional for the non-bd, A ( exp( B ( a ( 0 (0 τ τ A ( 0 ( τ exp i B ( τ dτ + onst, ( + H ( / π τ + Re ( [ ( ] H / ( [ ( H / ] ( ( H ( H / Re + + i, ( ( + H / + Re[ / ] where the prime in ( denotes the derivative with respet to, and and satisfy. Let τ be the time when the modified dispersion relations tae the standard linear form. Then σ where τ M / b h >> for /

7 7 b ~. The onstants and an be obtained by the following mathing onditions at τ for the two wave funtionals (9 and ( (0 Z (0 Z, ( 0 whih an also be rewritten respetively as d ( d (0 Z Z d d, (4 ( B Z Re( B Z Re, (5 d Re d ( B d Re( B, (6 d Z Z by requiring B B,, and ( 0 A (0 A when. H ( /4 σ 4 / πσ + 5/8σ + 4/ / K πσ with 4 Using ( ( ( σ and >>, we have from (0, (, and (5 θ + + os( -, (7 where we hoose θ and exp( i, and θ is a relative phase parameter. Then from (7 and we have ( θ s, ot( θ, (8 where sin( θ > 0, os( θ < 0. Substituting (0 and ( into (6 and eeping terms up to order / on the right-hand side of (6, we obtain os( 8 θ + 4 sin θ (. (9

8 8 Using (8 in (9 gives Here we hoose ot( ot( θ or 4 θ 4 θ. (0 4 ot( +, so that is small for >> to avoid an unaeptably large bareation on the baground geometry. Then we have, + +, ( 4 or sin( θ, os( θ. ( 4.. J Type Dispersion Relations with Sexti orretion n this subsetion, we use the J type dispersion relations (4 with s and b 0 to obtain the time evolution of the vauum state wave funtional. For this ase, only (5, (8, and (0 are hanged into i t 4 9 ( + σ h h } > { + 4, ( db ( τ B ( τ 4 9 B ( τ i + ( + σ 0 dτ τ 4τ, (4 B 6 ( π τ H/ i σ, (5 ( ( H / H/ ( τ 4 where σ b ( h/ M, and the prime in (5 denotes the derivative with respet to ( σ /. Using H ( σ / ( 6 πσ with / 4 / / σ and τ M / b h >> for b ~, we obtain from (5, (, (5 and θ + + os( -, (6

9 9 ( θ s, ot( θ, (7 where sin( θ > 0, os( θ < 0. Substituting (5 and ( into (6 and eeping terms up to order Using (7 in (8 gives Here we hoose / on the right-hand side of (6, we find os( ot( 8 θ + 4 sin θ (. (8 ot( θ or θ θ. (9 ot( +, so that is small for >> to avoid an unaeptably large bareation on the baground geometry. Then we have, + +, (40 8 or sin( θ, os( θ. (4 4. Vauum Energy, Bareation and osmologial onstant Using the results of Setion, we proeed to alulate the finite vauum energy density and use the bareation onstraint to address the osmologial onstant problem. Note that in the slow-roll approximation, the energy density of the salar field is ρ V (, where V ( m /. Therefore the relation between the expetation value of the vauum energy density ρ and the vauum wave funtional (0 in (6 is ρ ( 0 ρ (0 m ε du (0( u, τ u m 8 d a π Re( B ( τ a m π Re( B ( τ a a d, (4 where we use a field redefinition u a /,

10 0 Re( B ( (, / ( 0 exp Re( (, τ a u u τ a B τ a (4 π a Re( B ( τ a denotes the real part of B ( τ a, and the fator / a in (4 appears through the normaliation ondition ( 0 du ( u, τ. (44 For s and 0 ( b, in region, we have H ( σ / > /4 πσ 4 / with σ. Then, using a / hτ / h and (0 in (4, we obtain α ρ m h ( s 8 d m h α, (45 π where / M / b h and α M h ( M / 9 G. 0 GeV is the / an mass are the boundaries of the interval of integration. On the other hand, in region, ( an be expressed as where ( B H / is defined as ( / ( ( H / π τ ( τ i H H, (46 / ( [ ] / ( ( ( H / / Re ( / + H + H, (47 ( with + ( H / (. From (7, (, and (, we note that H / π an be approximated by ( / H as dereases from >> to (horion exit. Then, using a / hτ / h and ( in (4, we obtain

11 + ρ m h s d m h + ln. (48 From (45 and (48 we have ρ ρ + ρ s s s m h ( α + ln, (49 For >> and α b ( M / M /, (49 beomes / > ρ s m h α. (50 From (50 we see that there is no bareation problem if the energy density due to the quantum flutuations of the inflaton field is smaller than that due to the inflaton potential, i.e., ρ s < V (. (5 n the slow-roll approximation, using h V ( M / and (50 in (5 gives the onstraint on the parameter b as b > 9 π M M m h 4. For 6 M ~ 0 GeV (the energy sale during inflation implied by the BEP experiment [6, 7], we have b >.. 0 For s and 0 ( b, in region, we have H ( σ / > / 6 / πσ with σ. Then, using a / hτ / h and (5 in (4, we obtain ρ s β m h d m h ln β, (5 where / 4 M / b h and β M h are the boundaries of the interval of / integration. On the other hand, in region, ( an be again expressed as (46 with ( H / defined by (47. From (6, (40, and (4, we also note that H ( /

12 an be approximated by ( / H as dereases from >> to. Then, using a / hτ / h and ( in (4, we obtain + ρ m h s d m h + ln. (5 From (5 and (5 we have ρ ρ + s ρ s m h s (ln β + + ln. (54 For >> and (ln β + / >> ln whih are satisfied if 4 b > 6., 0 (54 beomes ρ m h (ln β + s. (55 Moreover, we notie that there is no bareation problem if ρ V (. (56 < s Using h V ( M / and (55 in (56 gives the onstraint on the parameter / b as Mm b ( > (ln β +. For π M h b 7 > M ~ 0 GeV, we have omparing (50 with (54, we find that ρ < s ρ if the inequality s (ln β + / < α, or [( ln + / /( M / M ] < b / b β is satisfied. For example, the usual parameter hoie b ~ satisfies the inequality. On the other ~ b hand, we have ρ > s ρ if the inequality (ln β / > α s +, or [( ln + / /( M / M ] > b / b β is satisfied. For example, the parameter hoie b ~ and 6 b ~ satisfies the inequality. 0 n the ase that ρ is larger than s ρ s, using (50 in the osmologial

13 onstant Λ ρ va / M gives m M Λ / πb M whih is GeV for b ~ and GeV for b ~ 0. n the ase that ρ is larger s than ρ s, using (54 in the osmologial onstant ρ va / M Λ gives Λ πb / m M M ln β + whih is GeV for ~ b and.9 0 GeV for 6 b ~ onlusions and Disussion n the Shrödinger piture, we have onsidered the theory of a generially oupled free real salar field in de Sitter spae. To investigate the possible effets of trans-anian physis on the quantum evolution of the vauum state of salar field, we fous on the massive non-minimally oupled salar field in slow-roll inflation, and onsider the J type dispersion relations with quarti or sexti orretion. We obtain the time evolution of the vauum state wave funtional, and alulate the expetation value of the orresponding vauum energy density. We find that the vauum energy density is finite and has improved ultraviolet properties. For the usual dispersion parameter hoie, the vauum energy density for quarti orretion to the dispersion relation is larger than for sexti orretion. For some other parameter hoies, the vauum energy density for quarti orretion is smaller than for sexti orretion. We also use the bareation to onstrain the magnitude of parameters in nonlinear dispersion relation, and show how the osmologial onstant depends on the parameters and the energy sale during the inflation at the grand unifiation phase transition. From (50 and (54 we see that the value of the osmologial onstant an be redued signifiantly through inreasing the dispersion parameters in nonlinear dispersion relation and dereasing the utoff energy sale assoiated with phase transition. However, the fat that the dispersion relation of a salar field an not be modified on energy sales small than TeV maes the osmologial problem still unsolved.

14 4 Anowledgments The author thans M.-J. Wang for stimulating disussions on the osmologial onstant, and his olleagues at Ming hi University of Tehnology for useful suggestions. Referenes. A. D. Linde, Partile Physis and nflationary osmology, Harwood Aademi Publishers, hur, Switeand, R. Fair and W. G. Unruh, mprovement on osmologial haoti inflation through nonminimal oupling, Phys. Rev. D, vol. 4, 78, D.. Kaiser, Primordial spetral indies from generalied Einstein theories, Phys. Rev. D, vol. 5, 495, E. Komatsu and T. Futamase, omplete onstraints on a nonminimally oupled haoti inflationary senario from the osmi mirowave baground, Phys. Rev. D, vol. 59, 06409, J. Hwang and N. Noh, OBE onstraints on inflation models with a massive non-minimal salar field, Phys. Rev. D, vol. 60, 00, T. Qiu and K.. Yang, Non-Gaussianities of single field inflation with non-minimal oupling, Phys. Rev. D, vol. 8, 0840, R. Kallosh and A. D. Linde, New models of haoti inflation in supergravity, J. osmol. Astropart. Phys., vol. 0, 0, A. D. Linde, M. Noorbala, and A. Westphal, Observational onsequenes of haoti inflation with nonminimal oupling to gravity, J. osmol. Astropart. Phys., vol. 0, 0, D.. Edwards and A. R. Liddle, The observational position of simple non-minimally oupled inflationary senarios, arxiv: [astro-ph.o]. 0. U. H. Danielsson, Note on inflation and trans-anian physis, Phys. Rev. D, vol. 66, 05, Armendari-Pion and E. A. Lim, Vauum hoies and the preditions of inflation, J. osmol. Astropart. Phys., vol. 0, 006, 00.. J. Martin and R. Brandenberger, Trans-planian problem of inflationary osmology, Phys. Rev. D, vol. 6, 50, 00.. R. Brandenberger and J. Martin, The robustness of inflation to hanges in super-an-sale physis, Mod. Phys. Lett. A, vol. 6, 999, J. Martin and R. Brandenberger, oey-jaobson dispersion relation and trans-anian inflation, Phys. Rev. D, vol. 65, 054, P. Horava, Quantum gravity at a Lifshit point, Phys. Rev. D, vol. 79, , P. Horava, Spetral dimension of the universe in quantum gravity at a Lifshit point, Phys. Rev. Lett., vol. 0, 60, G. Amelino-amelia, M. Arano, G. Gubitosi, and J. Magueijo, Dimensional redution in the sy, Phys. Rev. D, vol. 87,5, G. Amelino-amelia, M. Arano, G. Gubitosi, and J. Magueijo, Rainbow gravity and sale-invariant flutuations, Phys. Rev. D, vol. 88, 040, 0.

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