Slow roll inflation and BB mode angular power spectrum of CMB

The BB-mode angular correlation power spectrum of CMB is obtained by considering the primordial gravitational waves in the squeezed vacuum state for various inflationary models and results are compared with the joint analysis of the BICEP2/Keck Array and Planck 353 GHz data. The present results may constrain several models of inflation.


Introduction
Cosmic inflation is the most widely known scenario proposed for resolving several problems associated with the standard model of cosmology [1,2]. A number of inflationary models have been proposed over several decades [1,2,3,4,5,6,7,8]. The recent observations on the cosmic microwave background (CMB) anisotropy data may constrain many of the inflationary models [9,10,11,12,13]. It is believed that inflation seeded the formation of the large scale structures in the universe. Inflation also predicts a nearly scale invariant spectrum for the scalar and tensor perturbations which occurred in the early universe. The tensor perturbations of cosmological origin are known as primordial gravitational waves (GWs).
It is believed that the primordial gravitational waves have left its imprint on the cosmic microwave background. Therefore the primordial GWs can be studied with the aid of CMB anisotropy and polarization. The CMB is polarized in the early universe due to the Thomson scattering. The density (scalar) fluctuations generate the E-mode polarization of the CMB, while the gravitational waves generate both E-mode and B-mode polarizations [23,24]. The primordial gravitational waves are a unique source of B-mode of CMB and its detection will help in understanding the inflation as well as the primordial gravitational waves itself.
The gravitational waves were generated during the inflation period due to the zero-point quantum oscillations [14]. An initial vacuum state (no graviton) can evolve into a multi-particle quantum state known as the squeezed vacuum state [15], which is a well-known state in the context of quantum optics [16,17,18]. Therefore the primordial gravitational waves are believed to exist in the squeezed vacuum state. [19,20,21]. The primordial gravitational waves are placed in the squeezed vacuum state and its effect on the BB mode angular correlation power spectrum of CMB is studied with WMAP data [51]. Recently, it is shown that the BB mode angular power spectrum gets enhanced at its lower multi poles by considering the primordial gravitational waves in thermal state [31]. These studies show that the primordial gravitational waves may exhibit both the squeezing as well as thermal features and hence it is worthwhile to examine their combined effects on the BB mode angular correlation power spectrum in light of the recent joint BICEP2/Keck Array and Planck data.
The aim of the present work is to study effect of primordial gravitational waves in the squeezed vacuum state on the BB mode angular correlation power spectrum of CMB with various slow roll inflationary models. Thus the obtained BB mode angular correlation power spectrum of CMB for the squeezed vacuum as well as the joint effect of squeezing and thermal cases are compared with the joint BICEP2/Keck Array and Planck data.

Tensor power spectrum in squeezed state
The perturbed metric for a flat Friedmann-Lemaître-Robertson-Walker universe can be written as where δ ij is the flat space metric and h ij is the tensor perturbation, |h ij | ≪ δ ij , ∂ i h ij = 0, δ ij h ij = 0 and dτ = dt R is the conformal time. In quantum theory, the field h ij (x, τ ) can be written in the Fourier mode as where D = √ 16πl pl is the normalization constant, l pl = √ G is the Planck length and k is the wave vector. The wave number is k = (δ ij k i k j ) 1 2 and is related to wavelength, λ = 2πR k . The two polarization states ε (p) ij , p = 1, 2 satisfy the conditions ε ij (k). These linear polarizations are respectively known as plus (+) polarization and cross (×) polarization.
The creation ( b (p) † k ) and annihilation (b The evolution of these operators are governed by the Heisenberg equations of motion, The initial vacuum state |0 is defined as Under the Bogoliubov transformation the creation and annihilation operators become where b † k (0) and b k (0) are the initial values of the operators, u k (τ ) and v k (τ ) are complex functions and they satisfy the condition The coupling of the mode functions h k (τ ) with R(τ ) gives where χ (p) k can have the following form 1 which satisfies the equation of motion where prime indicates the derivative with respect to the conformal time τ .
Two-point correlation function of the tensor perturbation is given by: where the angle bracket denotes ensemble average and P T is known as the tensor power spectrum.
Using eqs. (2) and (6), taking the contribution from each polarization to be the same, The primordial gravitational waves are created due to the zero-point quantum oscillations in the early universe [14,34]. The initial vacuum state with no graviton evolves into multi-particle quantum state through parametric amplification. Hence the primordial GWs are possible to be considered in the squeezed vacuum state [19,35].
The squeezed vacuum state is defined as [16,36] where Z(ζ) is the single mode squeezing operator and is given by where ζ = r s e iγ is a complex number, r s is the squeezing parameter and γ is the squeezing angle. The unitary transformations of the squeezing operator S on the annihilation and creation operators lead to: Using eqs. (2) and (13), the two point correlation for the tensor perturbation in the squeezed vacuum is obtained as Thus from eqs. (9) and (14), we get the tensor power spectrum in the squeezed vacuum state as In the case of quasi de Sitter universe during inflation, R(τ ) = −1 Hτ (1−ǫ) , where the slow roll parameter ǫ is related to the scalar field potential V and is given by, ǫ = If the slow roll parameter ǫ is considered as constant, then eq. (8) gets modified as The general solution for eq. (16) is given by where H (1) ϑ and H (2) ϑ are respectively the Hankel functions of the first and second kind, and C 1 and C 2 are the constants of integration. For within the horizon (k >> RH), the approximate solution is given by Using the above approximate solution, one can obtain Therefore, for long wavelength limit (k << RH), eq. (17) leads to Using eq. (19) in eq. (15), the tensor power spectrum in the superhorizon limit (k << RH) is obtained as, where 2 is the normalization constant and H k0 is the Hubble parameter at RH = k 0 during the inflation, k 0 being the pivot frequency.

Inflationary models and tensor power spectrum
In most models of inflation [3], a homogeneous scalar field, called inflaton, is considered as candidate for the inflation. The inflaton field φ is governed by the equation of motion given bÿ where dot and prime indicate derivatives with respect to time (t) and the field (φ) respectively. The Hubble parameter H is determined by the energy density of the inflaton field, thus the Friedmann equation can be written as In the slow-roll limit, the energy density of the inflaton field is dominated by its potential, (22), the Hubble parameter and the inflaton potential are related by This condition is characterized by the slow-roll parameters which are defined in terms of the inflaton potential V and its derivatives as [4] ǫ ≡ m 2 and so on. Inflation lasts as long as the slow-roll conditions are satisfied, i.e., ǫ ≪ 1 and |η| ≪ 1.
The duration of inflation is characterized by the e-fold number, N , which can be written in terms of the inflaton potential, In the slow-roll approximation, the power spectrum of the scalar perturbations (P S ) and the tensor perturbations (P T ) generated outside the horizon are respectively given in terms of the potential [30,32] where k = RH indicates that H and hence, V is evaluated at the time when the mode with wave number k crosses the horizon. For all calculations, we take the scalar power spectrum to be P S = 2.43 × 10 −9 .
The tensor-to-scalar ratio can be written in terms of the parameter ǫ as This parameter is often used to characterize the amplitude of the tensor perturbation at the CMB scale. Physically, r is a measure of the slope (of the quantum hill) down which the scalar field is rolling. Since inflation predicts a nearly scale invariant spectrum, the slope is small but not flat. Hence, r is small and can differentiate between the many inflation models.
Next, we obtain the slow roll parameter and tensor spectral index corresponding to various slow roll inflation models given below. For this purpose, we use the e-fold number N = 60, for all the inflationary models under the present work [40].

Quadratic Chaotic Inflation
The chaotic inflation model assumes that the scalar field rolls down its potential and rests at its vacuum state for a while, then after getting displaced due to some fluctuations, rolls back to its true vacuum state and the same mechanism repeats itself. For quadratic chaotic inflation, the scalar field has the potential given by [4] where m is the mass of the inflaton field and is taken to be m = 1.53 × 10 13 GeV.
Using eqs. (25) and (29), the slow roll parameter and index of the tensor power spectrum for the quadratic chaotic inflationary model are obtained as: The corresponding tensor-to-scalar ratio is obtained as, r = 0.132.

Quartic Chaotic Inflation
The quartic chaotic inflation model suggests the existence of the curvaton which generates the curvature fluctuations during inflation after the inflaton field has decayed, while it does not drive the inflation itself. The model has the potential given by [5] where normalization gives the self-coupling of the scalar field (φ) as λ = 5.94 × 10 −14 GeV, and hence The tensor-to-scalar ratio is for this model is found as r = 0.26.

New Inflation
The new inflation model [6] is based on the Coleman-Weinberg potential [41,42], In this model, the parameters are taken to be σ = 10m pl , where m pl is Planck's mass, and λ = 2.36 × 10 −14 GeV. σ is the vacuum expectation value of the scalar field at minimum and λ is the quadratic self coupling of the field. Therefore we get The tensor-to-scalar ratio in this case is obtained as r = 0.186.

Hybrid Inflation
In this model, the inflation is driven by two scalar fields, in which one of the fields (φ) is responsible for the normal slow-roll inflation while the other field (σ) triggers the end of inflation.
The potential for this model is [7,8] where the parameters are taken to be as follows: g = 8 × 10 −4 , λ = 1 are the self-coupling constants of the inflaton field and the trigger field respectively, and the masses of the fields are m = 1.5 × 10 −7 m pl , M = 1.21 × 10 16 GeV respectively.
The φ field drives the inflation, thus determining the period of inflation while the σ field determines the rate of inflation. Therefore we get: The tensor-to-scalar ratio in this case is found as r = 4.24 × 10 −3 .

BB-mode angular power spectrum of CMB
The CMB radiation can be polarized due to the gravitational waves [43] called the B-mode [44,45] of CMB.
The angular correlation power spectrum of the B-mode CMB is given by [46,47] where g(τ ) = κe −κ is the visibility function, κ is the differential optical depth for the Thomson scattering, x = k(τ H − τ ) and j l is the spherical Bessel function.
The BB mode correlation angular spectrum of CMB is obtained for the gravitational waves in the squeezed vacuum and thermal vacuum states for the quadratic chaotic, quartic chaotic, new  Figure 1: BB-mode angular spectra for various inflation models with lensing for zero squeezing effect with BICEP2/Keck and Planck joint analysis result inflation and hybrid inflationary models. The angular power spectra for the different inflation models are generated using the CAMB code with the tensor spectral index n T corresponding to each inflationary model. For all the cases, the optical depth is taken as κ = 0.08, the pivot wave number for tensor modes is taken as k 0 = 0.002 Mpc −1 and that for scalar mode it is k 0 = 0.05 Mpc −1 . The tensor to scalar ratio used for each inflationary model is taken from the previous section where their values are computed. The obtained BB mode angular power spectrum for various values of the squeezing parameter and temperature are compared with the limit of the BICEP2/Keck Array and Planck 353 GHz joint analysis data. The implemented limit (BK x BK -α BK x P)/(1 -α) at α = α f id = 0.04 is evaluated [48] from the auto-spectra and cross-spectra of the combined BICEP2/Keck 150 GHz maps and Planck 353 GHz maps to clean out the dust contribution, BK x BK indicates the BICEP2/Keck auto-spectra at 150 GHz and BK x P indicates the cross-spectra of BICEP2/Keck maps at 150 GHz and Planck maps at 353 GHz. This combination is taken after the subtraction of the dust contribution which is 0.04 times as much in the BICEP2 band as it is in the Planck 353 GHz band (see ref. [48] for details).
The BB-mode correlation angular spectra for the different inflation models for zero squeezing with lensing effect are obtained and their corresponding plots are given in fig.(1). The BB mode angular spectrum for various inflationary models for the squeezing, thermal and their combined effects with lensed and unlensed cases are also obtained and the results plots are in figs. (2)(3)(4)(5)(6)(7)(8)(9)(10). It can be seen that for various values of the squeezing parameter and squeezing angle, the quadratic chaotic, quartic chaotic and new inflation models are out of the limit of BICEP2/Keck Array and Planck 353 GHz joint data. For the thermal effect with various values of the squeezing parameter and squeezing angle, the quadratic chaotic, quartic chaotic and new inflation models are also not found within the limit of BICEP2/Keck Array and Planck 353 GHz joint data.
It can be observed that the BB mode angular correlation power spectrum corresponding to the hybrid inflation model for various values of squeezing parameter and squeezing angle is found within the limit of the BICEP2/Keck and Planck joint analysis data at l ≃ 145, even though the squeezing and thermal effects enhances the angular correlation power spectrum moderately.

Conclusion
The primordial gravitational waves are very important in cosmology. It is believed that the origin of the primordial gravitational waves are due to inflation mechanism. Thus the primordial gravitational waves are placed in a special quantum state called the squeezed vacuum state, a well known state in quantum optics. The gravitational waves have not been detected directly yet but its effect is expected to be observed through the B mode of CMB. If this is the case, then the role of squeezing effect is also expected to reflect on the BB mode angular correlation power spectrum of CMB.
The BB-mode correlation angular power spectrum for various slow-roll inflationary models are studied by considering the primordial gravitational waves in the squeezed vacuum state. The obtained angular spectrum for the squeezing, thermal as well as their combined effects are compared with the BICEP2/Keck Array and Planck 353 GHz joint analysis data. The comparative study of the BB mode angular spectrum obtained for the various inflationary models with the BICEP2/Keck Array and Planck 353 GHz joint analysis data show that the hybrid inflation model is most favorable with the primordial GWs in the squeezed vacuum state. The BB mode angular correlation power spectrum can be obtained even with higher or lower values of the squeezing parameters and thermal parameter but the results do not alter the present conclusions. Note that the BB mode angular correlation power spectrum for the vacuum case can be recovered in absence of squeezing and thermal effects.
Further, the result of the joint analysis data does not rule out the existence of gravitational waves in the squeezed vacuum or thermal squeezed vacuum states which are in agreement with previous studies. Thus, eq. (38) modifies to [40]: The power spectrum for gravitational waves in the presence of thermal effect can then be written as This can be used to compute the BB mode correlation angular power spectrum of CMB in the thermal state.