Non-Bunch-Davies Initial State Reconciles Chaotic Models with BICEP and Planck

The BICEP2 experiment has announced a signal for primordial gravity waves with tensor-to-scalar ratio $r=0.2^{+0.07}_{-0.05}$ [arXiv:1403.3985]. There are two ways to reconcile this result with the latest Planck experiment [arXiv:1303.5082]. One is by assuming that there is a considerable tilt of $r$, $\mathcal{T}_r$, with a positive sign, $\mathcal{T}_r=d\ln r/d\ln k\gtrsim 0.57^{+0.29}_{-0.27}$ corresponding to a blue tilt for the tensor modes of order $n_T\simeq0.53 ^{+0.29}_{-0.27}$, assuming the Planck experiment best-fit value for tilt of scalar power spectrum $n_S$. The other possibility is to assume that there is a negative running in the scalar spectral index, $dn_S/d\ln k\simeq -0.02$ which pushes up the upper bound on $r$ from $0.11$ up to $0.26$ in the Planck analysis assuming the existence of a tensor spectrum. Simple slow-roll models fail to provide such large values for $\mathcal{T}_r$ or negative runnings in $n_S$ [arXiv:1403.3985]. In this note we show that a non-Bunch-Davies initial state for perturbations can provide a match between large field chaotic models (like $m^2\phi^2$) with the latest Planck result [arXiv:1306.4914] and BICEP2 results by accommodating either the blue tilt of $r$ or the negative large running of $n_S$.

Early Universe cosmology has become a very active area of research in the last decade or so, as there is a wealth of precise cosmic microwave background (CMB) measurements pouring in. In particular, since last year two major collaborations Planck [2] and BICEP [1] have announced their results. The CMB measurements analyzed with other cosmological data favor the simple ΛCDM model for late time cosmology and inflationary paradigm for early stages of Universe evolution. According to the Planck collaboration data [2] the power spectrum of CMB temperature fluctuations (or as it is known, the power spectrum of curvature perturbations) P S is measured to be about 2.195 × 10 −9 . The spectrum is almost flat, with a few-percent tilt toward larger scales (i.e., red spectrum) and is almost Gaussian.
Planck took cosmologists by surprise as it not only did not observe non-Gaussianity, which could have been used to considerably constrain inflationary models, but also put a strong upper bound on the amplitude of primordial gravity waves during inflation. These gravity waves are tensor mode fluctuations which are produced during inflation. The power spectrum of gravity waves P T is usually reported through the tensor-to-scalar ratio r = P T /P S which Planck reported to be bounded at 2σ level as r < 0.12. This bound corresponds to the pivot scale k * = 0.002M pc −1 . The tilt in the power spectrum of curvature perturbations is customarily denoted by n S −1, n S −1 = dln P S /dln k, where k is inverse of the scale. Planck constrained n S − 1 = −0.0397 ± 0.0146 at 2σ level. Planck's measurement already disfavored many single field models, especially those with convex potential [2].
CMB besides having one-in-10 5 part temperature fluctuations is partially polarized and the parity odd polarization, the B-mode, is usually attributed to primordial gravity waves, tensor modes [4]. BICEP2 collaboration has recently announced observation of B-mode polarization [1]. BICEP results took cosmologists by an even greater surprise, when measured r = 0.2 +0.07 −0.05 . This was not an outright inconsistency between the two collaborations though, because BICEP focused on smaller scales than Planck announced bound; BICEP data is for ℓ ∼ 80. BICEP result was challenging in view of Planck results, as the measured value is already in the region which was excluded by Planck, unless the power spectrum of gravity waves considerably grows as we move to smaller scales, i.e. a blue, with relatively large tilt, for power spectrum of tensor modes. This is one possibility to reconcile BI-CEP data with Planck results [1]. Nonetheless, this potential way for Planck-BICEP reconciliation seems very hard to achieve in the context of slow-roll inflationary models composed of scalar fields minimally coupled to Einstein gravity. To see this, we need to go through the equations more closely.
The controversy is best formulated in terms of the tilt of tensor-to-scalar ratio T r , where n T is the tilt of power spectrum of tensor modes and n S − 1 is the tilt of the power spectrum of curvature perturbations. Planck requires n S − 1 to be negative and of order −0.04. Standard, textbook analysis for slow-roll inflationary models leads to the "consistency relation" n T = −r/8 [5], which is a red-tilt for gravity waves [6,7]. Therefore, n T , too, is negative and of order O(−0.01) for such inflationary models. On the other hand, BICEP-Planck reconciliation requires This clearly shows the tension between standard slowroll models, and in particular the consistency relation with Planck+BICEP data: Slow-roll inflationary models cannot easily and readily accommodate the respectively large value of tensor-to-scalar spectral tilt T r and the blue tensor spectrum required by recent observations (please see [8] for another attempt to make r run). As discussed, in particular noting (1), to remedy Planck-BICEP tension we need to relax the consistency relation n T = −r/8. The possibility which we will entertain here is based on the fact that in deriving standard cosmic perturbation theory results, besides the action of the model (which establishes the background inflationary dynamics and provides the equation of motion for cosmic perturbation fields), we also need to specify the initial quantum state over which these (quantum) cosmic perturbations have been produced. The standard initial state used is the Bunch-Davis (BD) vacuum state [9], stating that perturbation modes with physical momenta much larger than the Hubble scale during inflation H, effectively propagate in a vacuum state associated with flat space, the standard quantum field theory vacuum state.
Considering non-Bunch-Davis (non-BD) initial state for cosmic perturbations during inflation provides the setup to relax the consistency relation [10] (see [11] for some earlier works on the non-BD inflationary cosmology.) In fact, in our previous paper [3] we discussed such a setup and already used it in resolving the tension between Planck data and large-field chaotic inflationary models, including the simplest inflationary model with m 2 φ 2 potential for the inflaton field φ. Large-field models generically predict large value for tensor-to-scalar ration r, with r ∼ 0.05 − 0.2 [12]. So, they are potentially very good candidates for accommodating BICEP too. As we will discuss here, non-BD initial state can equip the large-field models with the tilt of r, T r , (equivalent with blue tensor spectrum, n T > 0) needed for BICEP-Planck reconciliation; the chaotic model m 2 φ 2 [12] with non-BD initial state nicely fits with all available cosmological data.
The rest of this Letter is organized as follows. We first briefly review the setup presented in [3] to fix our notations. We then show that a mild tilt in the non-BD initial state will accommodate BICEP as well as Planck data. In the end we make some concluding remarks.
Power spectra and non-BD initial state. Here we consider a simple single-field inflationary model described by the action where M pl = (8πG N ) −1/2 = 2.43 × 10 18 GeV is the reduced Planck mass. We take our model to be a chaotic inflation large-field model [6], motivated by the recent observation of tensor modes [1], e.g. V (φ) = 1 2 m 2 φ 2 . The details of cosmic perturbation theory analysis for this model in standard Bunch-Davis vacuum may be found in standard textbooks, e.g. [6], and the modifications due to non-BD initial state in [3,11]. For completeness we have gathered a summary of this analysis in the appendix. The power spectra and tensor-to-scalar ratio, r, with where α's and β's parameterize non-BD initial state for scalar and tensor modes and the spectral tilts are then 1 The Lyth bound [13] and the consistency relation will also be modified due to the non-BD effects to [3] where ∆φ is the inflaton field displacement during inflation. The modification in the consistency relation is, as discussed, what can resolve the mismatch of slow-roll models with BICEP+Planck data. 2 Parameterizing the initial states. Noting the normalization conditions (25) and (29) and that only the phase difference between α k and β k matters, the non-BD initial states may be parameterized as [3] We consider a crude model in which [14], 1 It is instructive to note and recall expressions for the tilts of power spectra and scalar-to-tensor ratio r for λφ n chaotic models in the BD vacuum. For these models η = 2(n − 1)ǫ/n, and (Tr) BD = + 4 n ǫ , (n S − 1) BD = − 2(n + 2) n ǫ .
Noting that r ∝ ǫ ∝ (n S − 1), one can relate the tilt of r to the running of the spectral tilt ξ. Explicitly, (or any smooth function in which |β k | 2 falls off as k −(4+δ) ). Here M is a super-Hubble energy scale associated with the new physics which leads to the non-BD initial state. In this scenario, all the k modes are pumped to an excited state as their physical momentum reaches the cutoff k a(τ ) = M . The choice in (11) indicates that M is the (cutoff) scale at which the mode gets excited from Bunch-Davies vacuum.
The physically allowed region in the four parameter space of initial states is subject to the following constraints: (1) Absence of backreaction of initial states on the inflationary background; (2) Planck normalization for P S ; (3) value of spectral tilt n S −1 as observed by Planck; (4) fitting the value of r and the corresponding tilt T r , as required by BICEP+Planck, i.e. we take r P lanck ≤ 0.12 (at k * = 0.002M pc −1 and r BICEP ≃ 0.2 (at ℓ ∼ 80). In our analysis we focus on large-field single-field models. The first three conditions were also considered in [3] while the fourth one is new.
Absence of backreaction of initial excited state on the background slow-roll inflation trajectory implies that the energy stored in the initial non-BD state for both scalar and tensor sectors should not exceed the change in the energy density in one e-fold. This condition is fulfilled if [3] sinh The above indicates that the upper bound on the deviation from BD initial state measured by χ S is inversely proportional to the scale of new physics M . Hence, larger values of M require smaller χ S . The COBE normalization implies Assuming n S takes its best fit value of Planck, n S −1 ≃ −0.04, and that ǫ ∼ 0.01, then d ln γ S /d ln k 10 −2 .
The above conditions are achieved if we take χ T and χ S to take typical values [3], i.e. sinh χ S ≃ e χ S /2 , sinh χ T ≃ e χ T /2 and hence Moreover to be able to rely the effective field theory methods, we are typically interested in larger values of M which is possible if ϕ S is close to maximal; M ≃ 20H happens when ϕ S ∼ π/2 [3].
To reconcile BICEP+Placnk we want n S − 1 ∼ −0.04 and T r ≥ +0.16 and the Planck bound on r requires γ < 3/4. Therefore, We need not impose any condition on dϕ S d ln k , as ∂ ln γ S /∂ϕ S = 0 at ϕ S = π 2 . Above we have also assumed that tan ϕ T ≫ e −2χ T . If χ T 1, in principle very small values for ϕ T could be achieved.
One interesting example that is compatible with the above constraints is when χ T = χ S . This would correspond to the case where the numbers of particles in the tensor and scalar excited states are equal. Change in n S − 1 from its Bunch-Davies value could be set to zero, if dχ S /d ln k = 0. Since χ S = χ T , one has to assume that χ T is scale independent too. A positive tensor spectral index would come totally from the scale-dependence of ϕ T . The amount of suppression of r 0.002 , will be equal to sin 2 ϕ T . As for χ T f ew very small ϕ T 's are approachable, this would correspond to the case where the tensor and scalar modes phases are complementary angles (roughly ϕ S + ϕ T = π/2) when they cross the scale of new physics M . For example, for χ T 2 would be enough to solve the tension between Planck and BICEP2 data. Another interesting option is when r for our chaotic model is reduced just enough to reconcile with the Planck data. For m 2 φ 2 , this would mean a factor of 3/4 of its Bunch-Davies value. The corresponding ϕ T for such a scenario is The variation of ϕ T with scale has to be dϕ T d ln k 0.14 (17) For such a scenario, around k ≃ 50 M pc −1 , the tilt of the tensor spectrum becomes zero and the spectrum becomes red again for larger k's. The other possibility to obtain positive T r and hence a blue tensor spectrum is to allow for running of χ T . If this is the sheer cause of a blue gravitational spectrum a value of is required to solve the discrepancy between BICEP and Planck data. Depending on the value of ϕ T , one has to ensure the required suppression through the γ factor.

CONCLUDING REMARKS
As discussed in [3] non-Bunch-Davis initial condition for inflationary perturbations provide, with a typical value of the χ S parameter (χ S 1) with the non-BD phase ϕ S close to maximum, ϕ S ∼ π/2, can reconcile the m 2 φ 2 chaotic model with Planck data, r 0.002 < 0.12, if the scale of new physics which sources the non-BD initial state M , is around 20H. Observation of B-modes by the BICEP experiment at ℓ ≃ 80 can be matched with the bound from Planck data, if the gravity wave spectrum has a blue tilt of order 0.12.
Slow-roll inflation with BD initial condition cannot provide such a blue tilt. One can obtain such a blue spectrum if the tensor Bogoliubov coefficient has a small phase (ϕ T 0.1) with small k-dependence, ∂ϕ T /∂ ln k ∼ 0.01. Simple chaotic models, in particular m 2 φ 2 model, have been of interest because they are endowed with simplicity and beauty. As our analysis indicates they can be compatible with both Planck and BICEP results, if perturbations start in a non-BD initial state at the beginning of inflation. Noting that B-modes are coming from purely tensor perturbations of the metric [15] and that initial non-BD for perturbations is provided from a high energy pre-inflationary physics, such resolutions may open up a window to the realm of quantum gravity, a territory which is untouchable by collider experiments.