Production of gravitational waves during preheating in the Starobinsky inflationary model
PProduction of gravitational waves during preheating in theStarobinsky inflationary model
Guoqiang Jin, Chengjie Fu, Puxun Wu and Hongwei Yu
Department of Physics and Synergetic InnovationCenter for Quantum Effects and Applications,Hunan Normal University, Changsha, Hunan 410081, China
Abstract
The production of GWs during preheating in the Starobinsky model with a nonminimally coupledauxiliary scalar field is studied through the lattice simulation in this paper. We find that the GWspectrum Ω gw grows fast with the increase of the absolute value of coupling parameter ξ . This isbecause the resonant bands become broad with the increase of | ξ | . When ξ <
0, Ω gw begins togrow once the inflation ends and grows faster than the case of ξ >
0. Ω gw reaches the maximumat ξ = −
20 ( ξ = 42 for the case ξ >
0) and then decreases with slight oscillation. Furthermorewe find that the GWs produced in the era of preheating satisfy the limits from the Planck andnext-generation CMB experiments.
PACS numbers: 98.80.Cq, 04.50.Kd, 05.70.Fh a r X i v : . [ g r- q c ] J u l . INTRODUCTION Inflation, a phase of quasi-exponential expansion in the early universe, is an elegant ideaproposed to resolve most of theoretical problems in the big bang standard cosmology [1–4]. Furthermore, the scalar quantum fluctuations during inflation provide the seed for theformation of large scale structure [5]. Inflation usually is assumed to be driven by a slow-rollsingle scalar field, which predicts a scale-invariant spectrum of scalar fluctuations on thesuper-horizon scale. This prediction is consistent with the Cosmic Microwave Background(CMB) observations [6]. The latest CMB limits the spectral index of the power spectrumto be n s = 0 . ± . f ( R ) gravity [1]. Since the tensor perturbationslead to the B-mode polarization for the CMB photon, its amplitude can be constrained bythe CMB observations. The allowed region of the ratio of tensor to scalar fluctuations r is r . < .
10 at the 95% CL [7]. Thus, the combination of n s and r can discriminate a hostof inflationary models. In this regard, it is well-known that the Starobinsky inflationarymodel is in excellent agreement with the latest CMB observations [7].After inflation, the inflaton usually will undergo a periodical oscillation around the min-imum of its potential. During this oscillation the inflation field will decay into some lightparticles to thermalize the universe, which is called reheating [10]. In the first stage ofreheating, i.e., preheating, the periodical oscillation of the inflaton may lead to an explo-sive production of the inflaton quanta or other light particles coupled to the inflation fieldthrough the parametric resonance [11–16]. When the mode momenta of the inflaton quantaor the light fields are in the resonance bands, these Fourier modes will grow exponentially.Since only a part of modes have the resonant momenta, the matter distribution has largeand time-dependent inhomogeneities in the position space, which results in that the mat-ter possesses substantial quadruple moments, and becomes an effective source of significantgravitational waves (GWs) [17]. The production of GWs during preheating has been stud-ied widely in [18–24]. Different from vacuum fluctuations of tensor perturbations during2nflation, the amplitude of the GW spectrum generated during preheating is independent ofthe energy scale of inflation which only determines the present peak frequency [25, 26]. Forlow energy scale inflationary models, the peak frequency of GWs produced after inflationmay well occur in the range which in principle can be detected by future direct detectionexperiments (like LIGO/VIRGO) [23, 27–29]. This opens a unique observational windowfor us to test inflation and the subsequent dynamics of the very early universe.It has been found that in the Starobinsky model the creation of matter can be realizedthrough the decay of scalarons [1, 30, 31]. However, the inflaton field in the Starobinskymode is absent of self-resonance, and in order to realize the preheating process an auxil-iary scalar field χ coupled non-minimally with the gravity is required [32]. Recently, therescattering between the χ particles and the inflaton condensate during preheating has beenstudied in [33] by using the lattice simulation. The result shows that the rescattering is anefficient mechanism promoting the growth of the χ field variance, and knocks copious infla-ton particles out of the inflaton condensate. Thus, both the auxiliary field and the inflatonfield become the effective gravitational wave sources. However, Ref. [33] does not investigatethe production of GWs during preheating, which motivates us to finish the present work.In this paper, we investigate the production of GWs during preheating in the Starobinskyinflation model. The energy density and spectrum of GWs will be analyzed detailedly byusing the lattice simulation. Throughout this paper, we adopt the metric signature (-, +, +,+). Latin indices run from 0 to 3, Greek letters do from 1 to 3, and the Einstein conventionis assumed for repeated indices. II. BASICS EQUATIONS
For the Starobinsky model, when an auxiliary scalar field χ coupled non-minimally withgravity is considered, the Lagrangian has the form [32] L = √− g (cid:20) κ (cid:18) R + R µ (cid:19) − ξRχ −
12 ( ∇ χ ) (cid:21) , (1)where g is the determinant of metric tensor g µν , R is the Ricci scalar, κ − = M p with M p being the reduced Planck mass, µ is a constant, which is fixed at 1 . × − M p by themagnitude of the primordial density perturbations [9, 34], and ξ is an arbitrary couplingparameter. In our following analyses, besides the weak coupling, the case of strong coupling3 ξ | (cid:29) | ξ | ( ξ < r decreases and enters the 68% region allowed by the Planck observations inthe nonminimally coupled inflation model with a self-coupling potential [35]. Usually, it isconvenient to discuss the cosmic dynamics in the Einstein frame. By taking a conformaltransformation: ˆ g µν = Ω g µν with Ω being Ω = 1 − ξκ χ + R/ (3 µ ), and introducing ascalar field φ ≡ (cid:112) / κ − ln Ω as the inflaton, one can obtain the Lagrangian in the Einsteinframe L = (cid:112) − ˆ g (cid:20) κ ˆ R −
12 ( ˆ ∇ φ ) − e − √ κφ ( ˆ ∇ χ ) − V ( φ, χ ) (cid:21) , (2)where V ( φ, χ ) = e − √ κφ (cid:20) µ κ (cid:16) e √ κφ − ξκ χ (cid:17) (cid:21) . (3)In a flat Friedmann-Robertson-Walker (FRW) background, we obtain the following fieldequations from the Lagrangian (2)¨ φ + 3 H ˙ φ + ∂V∂φ − a (cid:53) φ + κ √ e − √ κφ (cid:18) ˙ χ − a ∂ k χ∂ k χ (cid:19) = 0 , (4)¨ χ + 3 H ˙ χ + e √ κφ ∂V∂χ − a (cid:53) χ − (cid:114) κ (cid:18) ˙ φ ˙ χ − a ∂ k φ∂ k χ (cid:19) = 0 , (5)and 1 κ (cid:18) ˆ R µν −
12 ˆ g µν ˆ R (cid:19) = ∂ µ φ∂ ν φ + e − √ κφ ∂ µ χ∂ ν χ − ˆ g µν (cid:18) ∂ α φ∂ α φ + 12 e − √ κφ ∂ α χ∂ α χ + V ( φ, χ ) (cid:19) . (6)Here a dot denotes a derivative with respective to the cosmic time t , a is the scale factor,and H is the Hubble parameter. During the inflation which is driven by the inflaton field φ , the χ field has a negligible effect and thus can be neglected. After inflation, the field φ behaves as an inflaton with the quadratic potential and oscillates coherently around φ = 0.This oscillations lead to that the χ field may have a tachyonic mass since the square of itseffective mass, which has the form m χ, eff = d Vdχ = e − √ κφ (cid:104) µ ξ (cid:16) e √ κφ − ξκ χ (cid:17)(cid:105) , (7)4an be negative. The tachyonic instability causes the parametric resonance of the χ parti-cles, which excites the production of copious χ particles of small-momentum modes. Sincethere are rescatterings between the χ particles and the inflaton condensate, the produced χ particles can knock inflaton quanta out of the condensate into low-momentum modes. So,the rapid growth of the inflaton and χ fluctuations can be expected, which means that theycan become significant GW sources.The best way to obtain the production of GW during preheating in the Starobinsky modelis to perform numerical lattice simulations. Here, we use the publicly available packageHLattice [36] to do simulations. In original HLattice the symplectic integrator is used tohandle scalar fields with canonical kinetic terms. Since the scalar fields considered in thispaper have non-canonical kinetic terms, we modify the HLattice by adopting the fourth-order Runge-Kutta integrator instead of the symplectic one. The results are simulated with N = (128) points. The lattice length of side L is chosen to to satisfy LH i < π , whichmeans that all field modes are always within the horizon, where H i ( (cid:39) . × − M p ) isthe Hubble parameter when the preheating begins. Different values of LH i are chosen forthe different values of ξ . For example, when ξ = 5 we set LH i to be 5, which indicates thatthe minimum of the comoving wave number k is k min = 1 . H i . Since all field modes aresub-horizon, we will initialize the fluctuations of φ and χ fields to be those in the Bunch-Davies vacuum. In addition, we set a UV cutoff k UV on k when initializing the fluctuations,i.e., k UV = 30 k min when ξ = 5, which is larger than the maximum resonance frequency inorder to prevent it from affecting the results. Furthermore, we have checked that the resultsare independent of the values of k UV . In the original HLattice, the produced results are alsolimited to be in the regions k ∈ [ k min , k UV ]. Here, we relax this limitation to include somemomenta larger than k UV . This operation is similar to what was done in [37]. The initialconditions of the homogeneous part of φ field during preheating are determined by the timewhen the inflation ends, while for the χ field the homogeneous part is initialized to zero.We stop the simulation when the density spectrum of the inflaton quanta and the χ field donot change appreciably. 5 II. PRODUCTION OF GRAVITATIONAL WAVES
Now we study the production of GWs during preheating in the Starobinsky inflation-ary model. GW can be represented by the transverse traceless part of the spatial metricdisturbance of the FRW background ds = g µν dx µ dx ν = − dt + a ( t ) ( δ ij + h ij ) dx i dx j (8)with ∂ i h ij = h ii = 0. The perturbation h ij corresponds to two independent tensor degreesof freedom and satisfies the equation of motion¨ h ij + 3 H ˙ h ij − a ∇ h ij = 2 κ a Π T Tij , (9)where the source term Π T Tij is the transverse-traceless part of the anisotropic stress Π ij ,which is defined as Π ij ≡ T ij − (cid:104) p (cid:105) g ij (10)with T ij and p being the energy-momentum tensor and the pressure of the system, respec-tively. For the model considered in this paper the energy-momentum tensor has the form T µν = ∂ µ φ∂ ν φ + e − √ κφ ∂ µ χ∂ ν χ − g µν (cid:18) ∂ α φ∂ α φ + 12 e − √ κφ ∂ α χ∂ α χ + V ( φ, χ ) (cid:19) , (11)and the anisotropic stress can be expressed asΠ ij = ∂ i φ∂ j φ + e − √ κφ ∂ i χ∂ j χ − δ ij (cid:16) ∂ k φ∂ k φ + e − √ κφ ∂ k χ∂ k χ (cid:17) . (12)From the above expression, one can obtain the transverse-traceless part of the anisotropicstress Π T Tij = (cid:104) ∂ i φ∂ j φ + e − √ κφ ∂ i χ∂ j χ (cid:105) T T . (13)In our numerical calculation, it is more convenient to work in the Fourier space. Usingthe convention f ( k ) = 1(2 π ) / (cid:90) d x e i k · x f ( x ) , (14)6ne can obtain that the GW equation (9) in Fourier space has the form¨ h ij ( t, k ) + 3 H ˙ h ij ( t, k ) − k a h ij ( t, k ) = 2 κ a Π T Tij ( t, k ) (15)where k = | k | . After introducing the transverse-traceless projection operator [19]:Λ ij,lm ( k ) = P il ( k ) P jm ( k ) − P ij ( k ) P lm ( k ), where P ij ( k ) = δ ij − k i k j k is the spatial projectionoperators, we achieve the transverse-traceless part of Π ij in the momentum spaceΠ T Tij ( k ) = Λ ij,lm ( k )Π lm ( k ) (16)It is easy to see that k i Π T Tij ( t, k ) = Π T Tii ( t, k ) = 0 in any time t . Using this projection oper-ator, one can define a new tensor u ij which satisfies the following relation in the momentumspace h ij ( t, k ) = Λ ij,lm ( k ) u lm ( t, k ) . (17)Then the GW equation (9) can be re-written as¨ u ij + 3 ˙ aa ˙ u ij − a ∇ u ij = 2 κ a [ ∂ i φ∂ j φ + e − √ κφ ∂ i χ∂ j χ ] . (18)We assume no GWs at the beginning of preheating, so u ij and its derivative are initializedto be zero.The GW energy density is given by ρ gw = (cid:88) i,j κ (cid:104) ˙ h ij (cid:105) . (19)Here (cid:104)· · ·(cid:105) is the spatial average. According to the Parseval’s theorem, the GW energydensity can be expressed as an integral in momentum space ρ gw = 1 L κ (cid:90) d k ˙ h ij ( t, k ) ˙ h ∗ ij ( t, k )= 1 L κ (cid:90) d k Λ ij,lm ( k ) ˙ u lm ( t, k )Λ ij,rs ˙ u ∗ rs ( t, k )= 1 L κ (cid:90) d k Λ lm,rs ( k ) ˙ u lm ( t, k ) ˙ u ∗ rs ( t, k ) , (20)where L is the length of one side of the lattice. The corresponding GW spectra, normalizedto the critical energy density ρ c , can be obtained throughΩ gw ( k ) ≡ ρ c dρ gw d ln k = πk H L (cid:90) d Ω k π Λ ij,lm ( k ) ˙ u ij ( k ) ˙ u ∗ lm ( k ) . (21)7 = ϕχ - - - - k / H i k f k ξ =- ϕχ - - - - k / H i k f k FIG. 1: The evolutions of the spectra of scalar fields as a function of k/H i with time t . Thespectra from bottom to up are plotted with the time interval ∆ t = 0 . H − i . Bold lines showthe final results. Since the produced GWs during preheating are determined by the spectra of scalar fields,we first investigate the evolutions of χ and φ and show their spectra in Fig. 1 where only ξ = − χ particles are produced rapidly. When ξ = − ξ = 5, which results from that all modes of the χ field with k /a < | m χ, eff | arealready tachyonic when the preheating begins [33]. After the increase of the χ spectrum,the spectrum of the φ field begins to increase since inflaton particles are knocked out of theinflaton condensate through the rescattering between χ particles and the inflaton condensate.Finally, both the inflaton field and the auxiliary field become the effective GW sources.Fig. 2 shows the evolutions of the GW density spectra from the lattice simulation with ξ = 5 ,
10, 20 and 30. The outputs are plotted from bottom to top per same time step∆ t = 0 . H − i , where H i denotes the value of Hubble rate when inflation ends. One cansee clearly that the low frequency modes increase rapidly as parametric resonance begins,and then the growth of the higher frequency modes is very fast due to the nonlinear effect.The amplitude increases faster and faster with the increase of ξ , which results from the factthat the resonant bands become broader and broader with the increase of ξ . The blue curves,which show the final GW spectra, indicate that the maximum value of Ω gw ( k ), Ω gw , max , isnot a monotonous function of ξ , since when ξ = 10 the value is larger than the ones of ξ = 5and ξ = 30, which is different from the case of non-minimally coupled inflation model inwhich the GW spectra produced in preheating increase with the increase of the couplingparameter [29]. 8 = - - - - / H i Ω g w ( k ) ξ = - - - - - k / H i Ω g w ( k ) ξ = - - - - - k / H i Ω g w ( k ) ξ = - - - - / H i Ω g w ( k ) FIG. 2: The evolutions of the density spectra of GWs as a function of k/H i with time t for ξ > t = 0 . H − i , with Blueline corresponding to the final result. Fig. 3 give the evolutions of the GW density spectra for the case of negative ξ . We takefour different values: ξ = − − −
20 and −
30. The results are very similar with theones shown in Fig. 2. This is because the resonant bands become broad with the increaseof | ξ | . But in the case of ξ <
0, Ω gw begins to grow once the inflation ends and growsfaster than the case of ξ >
0. The reason is that when ξ <
0, all modes of the χ field with k /a < | m χ, eff | are already tachyonic when the preheating begins [33]. In addition, we findwhen ξ <
0, Ω gw , max with ξ = −
20 is larger than the one from other three cases.Although Figs. 2 and 3 do not indicate a simple relation between Ω gw , max and the couplingconstant ξ , there, however, indeed exists a subtle relation as we will now demonstrate. Forthis purpose, we plot the evolution of Ω gw , max as a function of ξ in Fig. 4. We find thatwhen ξ is very small the production of GWs is very weak, which is agreement with theresult obtained in [39] where it was found that the direct generation of GWs after the endof inflation is suppressed in the Starobinsky model. One can see that the maximum of Ω gw is about 6 . × − , which occurs at ξ (cid:39) −
20 . In the case of ξ >
0, the maximum of Ω gw =- - - - - / H i Ω g w ( k ) ξ =- - - - - - k / H i Ω g w ( k ) ξ =- - - - - - k / H i Ω g w ( k ) ξ =- - - - - / H i Ω g w ( k ) FIG. 3: The evolutions of the density spectra of GWs as a function of k/H i with time t for ξ < t = 0 . H − i , with Blueline corresponding to the final result. - -
50 0 50 10010 - - - - ξ Ω g w , m a x FIG. 4: The amplitude for the final GW density spectrum peak as a function of ξ . is about 5 . × − at ξ (cid:39)
42. It is easy to see that Ω gw , max first increases fastly with theincrease of | ξ | , and then reaches the maximum after several oscillations. Finally it decreaseswith slight oscillation with the further increase of | ξ | .Since we are interested in the energy density of GWs today, the GW spectra generated inthe preheating era need to be transformed to the present-day case. The present scale factor10ompared to that at the time when GW production stops can be expressed as a ∗ a = (cid:18) a ∗ a j (cid:19) − (1+ w ) (cid:18) ¯ g j ¯ g (cid:19) − / (cid:18) ρ r, ρ ∗ (cid:19) / , (22)where subscripts ∗ , j and 0 denote the time when GW production stops, thermal equilib-rium is established and today, respectively. Here, w is the mean equation of state of thecosmic energy from t ∗ to t j , ρ r is the radiation energy density, ¯ g is the number of effectivelyrelativistic degrees of freedom, and ρ ∗ is the total energy density of scalar fields. So thepresent-day GW physical frequency is given by f = k πa = ka ∗ ρ / ∗ (cid:18) a ∗ a j (cid:19) − (1+ w ) × (4 . × ) Hz . (23)Here, we take the reheating temperature to be about 3 . × Gev [38], which means thatone can take ¯ g j / ¯ g = 106 . / . (cid:39)
31 in analysis. The transformed function to obtaintoday’s GW amplitude is [26]Ω gw , ( f ) h = Ω gw ( f ) (cid:18) ¯ g j ¯ g (cid:19) − / Ω r, h (cid:18) a ∗ a j (cid:19) − w , (24)where Ω r, h = h ρ r, /ρ c, = 4 . × − is the abundance of radiation today and h is thepresent dimensionless Hubble constant. Thus, the present-day GW energy density has theform Ω gw , h = (cid:90) d ln f Ω gw , ( f ) h . (25)During reheating the mean equation of state w varies with time, but its form is unknown.In our analysis we assume w = 1 / t ∗ and t j for simplicity. This assumption isreasonable since in [33] it has been found that after preheating the value of w is close to 0 . N eff [41] Ω gw , h Ω γ, h = 78 (cid:18) (cid:19) / ∆ N eff , (26)where Ω γ, h = 2 . × − is the present energy density of photons, N eff is the effective extrarelativistic degrees of freedom and ∆ N eff = N eff − . -
50 0 50 10010 - - - - ξ Ω g w , h FIG. 5: The total GW energy density today as a function of ξ . | ∆ N eff | (cid:46) .
33, which requires that the GW energy density must satisfy Ω gw , h (cid:46) . × − . The next-generation CMB experiments, such as CMB-S4, will probe ∆ N eff ≤ .
03 at1 σ CL and ∆ N eff ≤ .
06 at 2 σ CL [43], which potentially constrains the energy density tobe Ω gw , h (cid:46) . − . × − (27)In Fig. 5, we show the total GW energy density today as a function of ξ . One can seethat the maximum of energy density is 1 . × − , which occurs at ξ (cid:39) −
21. Thus, theGWs produced during preheating in the Starobinsky inflationary model satisfy the Plancklimit and the next-generation CMB experiment constraints.
IV. CONCLUSION
The Starobinsky inflationary model satisfies the latest CMB observations very well. In thisinflationary model, to achieve the preheating process requires an auxiliary scalar field whichcouples non-minimally with the gravity. This auxiliary scalar field becomes an effective GWsource due to the parametric resonance. Furthermore, the rescattering between auxiliaryscalar field and inflation field can knock copious inflaton particles out of the inflaton con-densate, which results in that the inflaton field also becomes an effective GW source. In thispaper, using the lattice simulation, we study the production of GWs during preheating inthe Starobinsky model. We find that the GW spectrum Ω gw grows fast with the increase ofthe absolute value of the coupling parameter ξ . This is because the resonant bands becomebroad with the increase of | ξ | . When ξ <
0, Ω gw begins to grow once the inflation ends and12rows faster than the case of ξ >
0. Ω gw reaches the maximum at ξ (cid:39) −
20 ( ξ (cid:39)
42 for thecase ξ >
0) and then decreases with slight oscillation. By assuming that all extra radiationdensity during the CMB formation is comprised of GWs, we investigate the constraint onthe total GW energy density from observations and find that the GWs producing duringpreheating in the Starobinsky mode satisfy the limits from the Planck and next-generationCMB experiments.
Acknowledgments
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