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Scalar and tensor perturbation in vacuum inflation

Recently, it was proposed that a small true vacuum universe can inflate spontaneously, in principle. Furthermore, there should be matter creation in vacuum inflation due to quantum fluctuations, and the matter created will influence the inflation simultaneously. In this paper, scalar and tensor perturbations in this model are analyzed and confronted with recent observations. These perturbations are derived and expressed with Hubble flow-functions. By comparing our calculations with experimental results, we can determine all the parameters in this model. Finally, with the determined parameters, we compute the evolution of the matter density and show that the matter produced in inflation roughly fits the observations at present.


I. INTRODUCTION
It has been widely believed that there was a fast expansion period of the early universe, called inflation, which successfully solved several severe problems in the Big Bang cosmology [1]. Although the concept of inflation is successful in the standard cosmology model, the origin and the power of the inflation are still unknown. It was thought that the Higgs field might drive the inflation, but there are some problems in the Higgs inflation model [2], such as the naturalness problem [3]. Recently, it was mathematically proved that the universe can be spontaneously created from nothing, in principle [4]. For a small true vacuum universe, its quantum potential can drive its inflation. When the universe becomes large enough, its accelerating expansion ends. This vacuum inflation model does not need a matter field to drive inflation, which is different from slow-roll inflation [5]. However, vacuum inflation should be developed because there are some parameters to be determined. When the issue of matter creation during inflation is considered, the vacuum inflation derived by the quantum potential should be modified slightly. Furthermore, the quantum fluctuations of the early universe, viewed as seeds for the growth of the structure of the universe, should also be given [6]. This paper is organized as follows. Sec. II, we first briefly review the vacuum inflation deriven by the quantum potential and then investigate the modification of the inflation by the matter created in the inflation. In Sec. III, we calculate the scalar perturbations up to second-order perturbations of the Hamiltonian for the scalar field and gravity in ADM form, without considering the concrete form of the scalar fields. Then, we discuss the quantization of gravity waves and calculate the tensor perturbations. In Sec. IV, by comparing our calculations with experimental observations, we determine * Corresponding author. Electronic address: qycai@wipm.ac.cn all the necessary parameters in the model. In Sec. V, the matter density is calculated. With the previously determined parameters, we fit the value of matter density which is compatible with current observations. Finally, we give discussions and conclusions in Sec. VI.
For simplicity, we use Planck units in this paper.

II. VACUUM INFLATION AND MATTER CORRECTION
We first briefly review the vacuum inflation derived by the quantum potential and then show the modification of the vacuum inflation by the matter created in inflation.

A. Vacuum inflation
There is a long-standing debate over whether a vacuum can inflate spontaneously [7][8][9]. If the universe indeed came from a vacuum, the problem of creation in the Big Bang model can be avoided [10]. Recently, it was proved by the minisuperspace model that a small true vacuum bubble can inflate because of its quantum potential [4], which makes it possible to construct a theory in which the universe grows up spontaneously. For a true vacuum bubble, its action S gr can be written as where √ −g is the square root of the absolute value of the determinant of the metric, R is the curvature and d 4 x = dτ d 3 x. τ is the conformal time defined as τ ≡ dt/a, where a is the scale factor. Because the vacuum bubble is homogeneous and isotropic, the metric can be expressed by one parameter, the scale factor a, By substituting the metric above into Eq. (1), it is easy to obtain the Hamiltonian of this model H = −l −2 1 4 l 4 P 2 a + Ka 2 , where P a = −2l −2 a ′ and l 2 ≡ 8π/3. Here, the primer denotes the derivative with respect to the conformal time. The parameter K = 1, 0, −1 represents closed, flat, and open bubbles, respectively. We adopt K = 0 as default for all calculations and only keep the K explicitly in some formula for the purpose of comparing with classical case. In the minisuperspace model, the momentum operator is defined as where p is the factor of ordering and denotes the ambiguity of rules for quantization. Because there is no complete quantum theory for gravity, we treat p as an uncertain parameter. In principle, the cosmological wavefunction satisfies the Wheeler-DeWitt equation, Hψ(a) = 0. Decomposing the cosmological wavefunction into two real functions ψ(a) = R(a) exp(iS(a)) and using Eq. (3), we have 1 4 l 4 (∂ a S) 2 + Ka 2 + Q = 0, and which is called the quantum potential. Using the guidance relation [4] one can obtain where H is the Hubble parameter. It is often defined as d ln a/dt. The L in the guidance relation is the the Lagrangian of Eq. (1). The formula of the Hubble parameter above shows that the quantum potential Q has the capacity to drive the vacuum bubble towards accelerating expansion. The analytical formula of the wavefunction can be obtained by solving the Wheeler-DeWitt equation, ψ(a) = iC 1 a 1−p /(1 − p) − C 2 , where p = 1 and C 1 and C 2 are arbitrary complex numbers. In order to get an inflation solution, we set C 1 and C 2 as real numbers. Because only the value of |1 − p| is crucial for the inflation solution, we set p < 1 in the paper for convenience. Then, we have [4]. With the guidance relation Eq. (5), when | C1 C2(1−p) a 1−p | 2 ≪ 1, we can obtain the asymptotic form of the Hubble parameter We can see how small a should be when we determine C 1 , C 2 and p. From the results of Sec. IV, this equation is roughly right when a < 50. We can also get this conclusion from Fig. (1), where H is almost constant till a ∼ 50. Now, it is easy to obtain the time-dependent evolution for the scale factor, When p = −2, we have an exponential inflation solution. When −2 < p < −1, we get power-law inflation. Eq. (11) without matter correction. We use the parameter C1/C2 fixed in Sec. IV to get these exact results. When comparing it with asymptotic expression Eq. (6), we can easily find when Eq. (6) is approximately right. The late time of this plot means that the universe starts to exit the inflation. The quantum effect does decrease strongly, but is still much bigger than the influence of matter in that time.
In Fig. (1), we show the evolutions of the quantum potential and the Hubble parameter with the variable of the scale factor a for a specific ordering factor p = −2. The Hubble parameter is nearly constant in the early stage of the universe, which is quite similar to slow-roll inflation. After a while, the Hubble parameter decreases rapidly to zero, which means that the inflation can exit spontaneously. The quantum potential is the power of the vacuum inflation and plays the role of the scalar field in the slow-roll inflation model [11]. We also find that different values of p have different quantum potentials, which leads to different evolution of the universe.

B. Matter correction
There are quantum fluctuations such as the creation and annihilation of virtual particles even in the vacuum of space. Due to the accelerating expansion, the virtual particles may be separated before annihilation and then become real particles. Therefore, there is matter creation during vacuum inflation [12]. With the matter creation, the action of the early universe is modified by a scalar field, where V (ϕ) is the potential of the scalar field. Here, we don't need to restrict the form of the potential of the scalar field, which is different to the slow-roll inflation theory. By substituting Eq. (2) into Eq. (8), we can obtain the Hamiltonian of the system where P ϕ = a 2 ϕ ′ and P a was defined in Eq. (3). the wave function of the universe still satisfies the Wheeler-DeWitt equation, Hψ(a, ϕ) = 0, but it is difficult to obtain analytical solutions for this equation. For the vacuum inflation model, the inflation is driven by the quantum effects of the universe, particularly at the beginning of the early universe. Under this condition, the wavefunction of the universe can be approximately written as ψ(a, ϕ) = ψ a (a)ψ s (a, ϕ), where ψ a (a) is the wavefunction for the space of the universe and ψ s (a, ϕ) is the wavefunction of the scalar field. The quantity of P a ψ s (a, ϕ) is negligible compared with P a ψ a (a). In this way, Eq. (9) can be simplified as where ρ(a) can be treated as matter density, which we will explain using Eq. (11). By expressing ψ a (a) as two real functions ψ a (a) = R a (a) exp(iS a (a)) and substituting Eq. (3) into Eq. (10), we can obtain where Q is determined by Eq. (4). Using guidance relation Eq. (5) we find that the Hubble parameter takes the form Comparing Eq. (11) with the classical Friedmann equation [13] when Q → 0, we know that ρ(a) is exactly the matter density.
From the first equation of Eqs. (10), we have H ϕ = a 4 ρ(a), where H ϕ is the ϕ part of the H. The derivative of the Eq. (11) with respect to the normal time can give where L ϕ ≡ P ϕ 2 − H ϕ . Eq. (12) is the Einstein acceleration equation of this model. When Q → 0, it goes back to the classical Einstein acceleration equation.
Because vacuum inflation is essentially driven by the quantum potential, the contribution of matter to the evolution of the early universe can be neglected. With Eqs. (4) and (7), the quantum potential of the early universe can be written as where C 1 and C 2 are two parameters to be determined. As discussed above, there are energy fluctuations as the virtual particles are created and annihilated. The virtual particles may become real particles probabilistically when tunneling through the cosmological horizon [14]. If we consider Eq. (10) in de Sitter space, the radius r of the horizon is 1/H, and the temperature of radiation from the horizon is H/(2π) [12]. Using the Stefan-Boltzmann law, the creation rate for the matter density iṡ where the dot denotes derivative with respect to the normal time t, j is the energy flux from the horizon, and σ is the Stefan-Boltzmann constant. The particles created during inflation are relativistic due to the high temperature at this stage, so the mass-conservation equation givesρ = −4Hρ. Considering both creation and dilution of matter simultaneously, we havė Combining Eqs. (11) and (15), we can obtain the equation for the change of the matter density with the growth of the early universe. The evolution of the Hubble parameter can also be obtained, which is numerically shown in Fig. (2). When the universe is small, the Hubble parameter is modified slightly by the matter created during inflation, which is the previous assumption that the inflation is mainly driven by its quantum potential. When the early universe becomes large, the contribution of matter to inflation is relatively large, whereas the dynamics of the universe are classical because Q → 0 for a large enough universe [13].

III. COSMOLOGICAL PERTURBATIONS
The measurement of CMBR shows that there were quantum fluctuations of gravity during inflation, which were the seeds of the galaxies of our universe. With the results of CMBR, we can look into the details of inflation and verify the validity of the inflation model. Although all kinds of fields can generate corresponding cosmology perturbations, we only consider scalar and tensor fluctuations during inflation, because they can be determined by the measurement of CMBR. In the following, we will first follow the standard cosmological perturbation theory but express the scalar and tensor perturbations with the Hubble parameter and its derivative. Then, we will show that it is right to use these classical conclusions in this model. From the calculations, we see that the Hamiltonian of perturbations can be expressed by the Hubble parameter and its derivative. The perturbations are decided by how the universe inflated rather than the concrete form of the scalar fields. In this model, quantum potential, together with field perturbations leads to cosmological perturbations. It is different from the standard model of inflation, where the scalar field and its perturbations lead to cosmological perturbations.

A. Scalar perturbations
Scalar perturbations are gravity perturbations caused by fluctuations of scalar fields. We need the four quantities B, ψ, E and φ to describe the most general form of scalar metric perturbations where Furthermore, we use the quantity ϕ 0 + δϕ to describe perturbations of the scalar field.
The scalar perturbations can be tested by the intrinsic curvature perturbation of the comoving hypersurface R, which is a gauge invariance quantity, where H ≡ a ′ /a. Using the back ground equations to cancel the background action, the first-order perturbations and the background scalar field ϕ 0 , we can simplify the action in Eq. (8) to the second-order [15] Furthermore, we choose a simple gauge as δϕ = 0 and E = 0 to eliminate B and φ. A straightforward calculation gives the action that only contains one field, where we use both the normal time and the conformal time to simplify our calculations.Ḣ is dH/dt and ψ ′ is dψ/dτ . For convenience, we introduce two variables z = −a 2Ḣ /H 2 = a −Ḣ/H and u = −zR/(2 √ π) in our calculations. Varying Eq. (19) and using Fourier transformation in the space variables, we can obtain where u k is the image function of u. For easy comparison with experimental results, we use the Hubble flowfunctions (HFFs) [16] to simplify our calculations, where i ≥ 1. Expressing z with HFFs, we obtain z = a √ ǫ 1 and 1 z (22) Integrating the conformal time τ = 1/a dt = 1/(a 2 H) da by parts repeatedly gives where f 0 = 1. Combining Eqs. (22) and (23), we obtain where F (ǫ 1 , ǫ 2 , ...) represents the multinomial of ǫ 1 , ǫ 2 , . . . , and does not contain a constant term. In this model, HFFS and F (ǫ 1 , ǫ 2 , ...) is constant during the early stage of inflation. We will show that in Sec. IV. With Eq. (24) and constant F (ǫ 1 , ǫ 2 , ...), we can solve Eq. (20) analytically. In the limit k/(aH) → ∞, u k has a normalized solution as u k (τ ) → −e −ikτ / √ 2k. Thus, the solution of Eq. (20) is where H ν is the Hankel function of the first kind with ν ≡ 3 2 + F (ǫ 1 , ǫ 2 , ...). We can then calculate the vacuum fluctuations of the curvature perturbation R. The power spectrum P R (k) is defined in terms of the expectation value of inflation state R k R l Considering the quantization of u k , we have R k R l * = 4π u k u l * /z 2 = 4π|u k | 2 δ 3 (k − l)/z 2 . Finally, we can obtain (25) The scalar perturbations do not depend on special scalar fields. All scalar fields in the fundamental theory contribute a part of the scalar perturbations P R (k). In the standard model, we have scalar fields Φ with complex doublets. Therefore, there are two scalar field contributions to the scalar perturbations [17], and the total power spectrum P ′ R (k) = 2P R (k). We show how curvature perturbations change along with the scale factor in Fig. (3). The perturbations are close to constant in the early stage of inflation.

B. Gravitational waves
Gravitational waves are gravitational perturbations that are produced by the vacuum fluctuations of gravity. The linear tensor perturbations can be written as g µν = a 2 (τ ) [η µν + h µν ]. In the transverse traceless gauge as h 0 = h 0i = ∂ i h ij = h ii = 0, there are two independent states [18]. The action of the perturbations can be expressed as It is convenient to calculate the perturbations with rescaled variables v λ (x) = (32π) −1/2 a(τ )h λ (x), where h λ (x) and λ = +, × represent two independent states. By varying Eq. (26) and using Fourier transformation in the space variables, we can obtain where v k is the image function of v. Using of Eq. (21), we can simplify the following expression Substituting Eq. (23) into the equation above, we can obtain where G(ǫ 1 , ǫ 2 , ...) represents the multinomial of ǫ 1 , ǫ 2 , . . . and does not contain a constant term. Repeating the same procedure as the case of scalar perturbations, we find that v k (τ ) has the same solutions as that of u k (τ ) except that ν is replaced by µ, and µ is defined as µ = 3 2 + G(ǫ 1 , ǫ 2 , ...). The spectrum of gravitational waves P g (k) is defined as h k,λ h l,λ * ≡ 2π 2 P g δ 3 (k−l)/k 3 [5]. Using the relation between v λ (x) and h λ (x), we can obtain With the quantization of v k,λ , we have that v k,λ v l,λ ′ * = |v k,λ | 2 δ 3 (k − l)δ λ,λ ′ . Furthermore, we can obtain the spectrum of the gravitational waves as (31) In Fig. (3), we show the change of the power spectrum of gravitational waves with the scale factor. The power spectrum is also close to a constant in the early stage of inflation. It decreases in the same way as the Hubble parameter does when the scale factor becomes large. The perturbations is nearly constant when they leave the horizon, all the calculation and experimental observations are based on k = aH [18].

C. Perturbations and quantum potential
The calculation of Sec. III A and Sec. III B are totally classical. This model is based on quantum theory. It seems unreasonable to use Eq. (19) in this model. We need to discuss how to get a reasonable perturbation theory in this model. The derivation satisfies the following principle. First, all the variables of the Hamiltonian are operator. They may not communicate with each other. Second, the operator must satisfy the classical equations of motion. Third, when the operator acts on the wavefunction which is not the eigenstate of this operator, we need add the corresponding quantum potential in and replace the operator with its classical definition. For example, we takeĤ Hψ a (a) = Ḣ + 3 2 With Eq. (32), we can easily obtain Eq. (11) from Eq. (10). First, we discuss the gravitational waves. The wave function of the universe satisfies H(â,Ĥ,φ) + H g (â,ĥ) ψ(a, ϕ, h) = 0, where H(â,φ) is defined by Eq. (9) and H g (â,ĥ) is the Hamiltonian of Eq. (26). We can separate the wave function similar to that of the Sec. II B and obtain ψ(a, ϕ, h) = ψ b (a, ϕ)ψ h (a, ϕ, h). With that we can simplify Eq. (33) to Comparing it with Eq. (10), we can know that ρ h (a) is the energy of gravitational waves and ψ b (a, ϕ) is the background wave function of an inflation universe which contains the correction of gravitational waves. ψ b (a, ϕ) represents gravitational waves at the scale factor a. As the energy of perturbations is negligible, the ψ b (a, ϕ) approximates to the ψ(a, ϕ) of the Sec. II B. The ψ h (a, ϕ, h) goes back to the calculation of Sec. III B. Second, we discuss the scalar perturbations. The Hamiltonian is H(â,Ĥ,φ) + H s (â,Ĥ,Ĥ,φ,ψ,B,Ê,δϕ), where H s (â,Ĥ,φ,ψ,B,Ê,δϕ) is the total Hamiltonian of scalar perturbations. Under the second principle, We can use the Hamiltonian of Eq. (19) which means the wave function satisfies H(â,Ĥ,φ) + H R (â,Ĥ,Ĥ,ψ) ψ(a, ϕ, ψ) = 0, (36) where H R is the Hamiltonian of Eq. (19). We can separate the wave function similar to that of the Sec. II B and obtain ψ(a, ϕ, ψ) = ψ b (a, ϕ)ψ R (a, ϕ, ψ). With that we can simplify Eq. (36) to R (a, ϕ, ψ), compared with Eq. (10), the ρ R (a) is the energy of scalar perturbations and ψ b (a, ϕ) is the background wave function of an inflation universe which contains the correction of scalar perturbations. ψ R (a, ϕ, ψ) represents scalar perturbations at the scale factor a. Since the Hubble parameter has been determined for the given a in this model, the H andḢ in H R will not bring any vagueness. As the energy of perturbations is negligible, the ψ b (a, ϕ) here approximates to the ψ(a, ϕ) of the Sec. II B. The ψ R (a, ϕ, ψ) goes back to the calculation of Sec. III A.
With these derivation, we know it is right to use the conclusions of Sec. III A and Sec. III B. The quantum potential will have impact on the background wave function. It can give the relation between the scale factor and the Hubble parameter, which will determine the perturbations.

IV. EXPERIMENTAL CONSTRAINTS OF VACUUM INFLATION
There are two parameters in the vacuum inflation model that should be determined: C 1 /C 2 and p. Only when all parameters are determined, can this model be falsified by experimental observations. With the progress of measuring CMBR, this model can be fixed by observational results.
There are some parameters that constrain inflation in the current observations: the tensor-to-scalar ratio r ≡ P g /P ′ R < 0.11(95%CL), the spectral index of curvature perturbations n s ≡ 1 + dP ′ R /dlnk = 0.968 ± 0.006 and its scale dependence dn s /dlnk = −0.003 ± 0.007 [19]. From Eq. (25) and Eq. (31), we express these parameters up to the second-order of HFFs: where C = −2 + ln2 + γ. Because the perturbations are mainly produced in the early stage of inflation, we calculate the fluctuations when the scale factor is small.
At that time, the influence of matter is negligible. In this case, Eq. (6) gives Then, Eq. (38) can be represented by the ordering factor p as r = 8(p + 2)(p + 3), n s − 1 = −2(p + 2)(p + 3) and d n s /d ln k = 0. The measurement precision of n s is more dependable among the constraints of inflation, so we choose it to determine p, and get p + 2 = 0.016 ± 0.003. If we choose the limitation on r can also be fixed. The restriction on d n s /d ln k is satisfied regardless of the value of p. The r and n s − 1 are constant in the early stage of inflation, which is similar to slow-roll inflation.
To constrain the parameter C 1 /C 2 , we need to specify the starting point problem of scale factor. In this model, we can not give the starting point of a. Luckily enough, with Eq. (11), we can roughly determinate the ending point of the inflation by comparing the quantum potential with the contribution of matter. Once the ending point of a is determined, we can use it to fix the starting point of a with the general requirement of 60 e-folds of expansion [5]. Generally, the observational perturbations are mainly crossed the Hubble radius in the range from 50 to 60 e-foldings before the end of inflation [5].
With these general considerations, from Eq. (25) and Eq. (39), we can get The observations give that P ′ R ≈ 2 × 10 −9 [19]. Before we constrain the parameter C 1 /C 2 with this observation result, we still have an unsolved problem. We can not numerically compute the ending point of the inflation without parameter C 1 /C 2 . Since p+2 is a very small number, P ′ R is a slowly varying function of a. The estimation of the end point of inflation is pretty accurate. Using a = 1 in Eq. (6) and Eq. (41), we can get C 1 /C 2 ≈ 1.55 × 10 −6 . By numerical calculation, we can roughly get the end point of inflation around a ≈ 10 4 . This means the observational perturbations are mainly crossed the Hubble radius in the range from a = 10 −22 to a = 2 × 10 −18 . With Eq. (6), Eq. (41) and this range, we can rederive C 1 /C 2 more accurately, which ranges from C 1 /C 2 = 8.0 × 10 −7 to C 1 /C 2 = 9.2 × 10 −7 . The range is really narrow and we set for simplification. With this value of C 1 /C 2 , we can recompute the ending point of inflation. It changes within a magnitude of 10, which proves that the estimation for C 1 /C 2 is pretty good. Thus, all the parameters in the vacuum inflation model have been determined.

V. COMPARISON AND PREDICTION
With the fixed parameters C 1 , C 2 and p , we will first calculate the matter generation. Then, we can test the model by comparing the calculated matter density with current experimental observations. The evolution of the early universe is roughly determined by Eq. (11) and Eq. (15). Before we do any calculation, we must clarify the evolution process of universe in this model. As all the parameter have been determined, we can divide the evolution process into several stages. At first, the cosmos is beginning to inflate due to quantum potential. The scale factor grew very fast from about 10 −22 to about 50, which is obvious in Fig. (1). In this stage, matter is produced with little influence on the inflation. With the growth of the scale factor, the quantum potential decreases rapidly. It means that the universe is about to exit the inflation. This process lasts from about 50 to the magnitude of 10 4 , which can be seen from Fig. (1), Fig. (2) and Fig. (4). After the end of inflation, the radiation produced by inflation takes the role of the quantum potential and drives the expansion of the cosmos slowly. Combining Eqs. (15) and (11) and considering that H is very small in this stage, we have the relation The quantum potential is negligible at this stage, which is shown in Fig. (4). After the radiation-dominated stage, the matter produced during inflation replaces the radiation and continues to drive the expansion of the cosmos. Because the Hubble parameter grew rapidly during the initial stage of inflation and slowly after that stage, we divide the calculations into two parts to obtain more accurate results. First, we plot the function H(a) to find when the quantum potential becomes negligible. Fig. (4) shows that the quantum potential and matter production no longer play a part when a is very large. The relation between H and a exactly fits the picture of the radiationdominated stage. In this case, we can ignore the effects of the quantum potential.
When the quantum potential dominates the inflation, we use to evaluate the time t f , we can use a = 2 × 10 4 from Fig. (4). We use the time when a = 1 as t i . The reason why we use t i to calculate t f is that we have to calculate this period with sufficiently long step length. The scale factor a = 1 does not mark the start of the universe as we clarified before. If we choose a = 10 −22 at the start of time, t i will be a very short time. The universe is still inflating at t i according to Fig. (1). The Hubble parameter can be described by Eq. (6) at this time. Therefore, we have where t P is the Planck time. Using Eqs. (11), (13) and (15), we can obtain the evolution of the Hubble parameter along with the scale factor numerically. With the numeric solution of the Hubble parameter we can integrate Eq. (44) numerically. We calculate t f −t i ≈ 3.40×10 16 t P , which is much bigger than t i . The matter density can be calculated as ρ f ≈ 1.58 × 10 −35 m P · l −3 P . We now address the classic case. With the Friedmann equation H 2 = 8πρ/3 and the mass-conservation equatioṅ ρ = −3(w + 1)Hρ we can obtain the evolution of matter density where w represents the ratio of the pressure to the den- We can see that the evolution of energy density coincide with the picture of radiation-dominated stage after a ∼ 2 × 10 4 .
sity of matter, which we treat as constant because it changes very slowly. Before we use Eq. (46), we must be careful about the meaning of t. t = 0 does not mark the time when a = 10 −22 . We must calculate asymptomatic axis where ρ → ∞ from Eq. (46) and Fig. (4). We can get t a ≈ −0.95 × 10 16 t P by comparing ρ and t of points after a = 2 × 10 4 . The right relation can be get by replacing t with t − t a in Eq. (46). The age of the universe is estimated as t now ≈ 4.3 × 10 17 s [20]. Using revised Eq. (46), we estimate which is roughly in agreement with experimental observation, which gives ρ ≈ 1.8 × 10 −27 kg · m −3 [21]. The model can also provide some new predictions for further test. The HFFs of this model are different from other models. The high-order terms of the HFFs are zero as shown in the Eq. (39). Independent of the value of p, we have two strict relations r/(1 − n s ) = 4, n s /d ln k = 0 (48) These can be used to falsify this model in future, when more accurate observations are available.

VI. DISCUSSION AND CONCLUSION
The vacuum inflation model derived by the quantum potential is interesting because it can avoid the problem of creation [10]. In contrast to other inflation models, the early universe is a vacuum bubble that grows rapidly due to its quantum potential. Matter can be created during the inflation stage, and inflation ends when the early universe becomes large enough. In this paper, we have considered the scalar perturbations and gravitational waves for the vacuum inflation derived by the quantum potential with HFFs directly. We find that matter contributes little to inflation, but becomes significant after the inflation. All the parameters in the vacuum inflation model are determined by comparison with observations. The matter density of the current universe is calculated with the vacuum inflation model, and the result roughly fits the observations, which indicates that the vacuum inflation model may be the correct answer.

VII. ACKNOWLEDGMENTS
We thank the referees for their helpful comments and suggestions that significantly polish this work. This work is supported by the National Natural Science Foundation of China (Grant Nos. 61471356).