The TT, TB, EB and BB correlations in anisotropic inflation
The ongoing and future experiments will measure the B-mode from different sky coverage and frequency bands, with the potential to reveal non-trivial features in polarization map. In this work we study the TT, TB, EB and BB correlations associated with the B-mode polarization of CMB map in models of charged anisotropic inflation. The model contains a chaotic-type large field complex inflaton which is charged under the gauge field. We calculate the statistical anisotropies generated in the power spectra of the curvature perturbation, the tensor perturbation and their cross-correlation. It is shown that the asymmetry in tensor power spectrum is a very sensitive probe of the gauge coupling. While the level of statistical anisotropy in temperature power spectrum can be small and satisfy the observational bounds, the interactions from the gauge coupling can induce large directional dependence in tensor modes. This will leave interesting anisotropic fingerprints in various correlations involving the B-mode polarization such as the TB cross-correlation which may be detected in upcoming Planck polarization data. In addition, the TT correlation receives an anisotropic contribution from the tensor sector which naturally decays after . We expect that the mechanism of using tensor sector to induce asymmetry at low to be generic which can also be applied to address other low CMB anomalies.
The apparent detection of B-mode polarization of the Cosmic Microwave Background (CMB) on by the BICEP2 observation has stirred significant interests in primordial gravitational waves (1). If confirmed to be primordial (2); (3), this detection implies the existence of primordial gravitational waves with the scalar-to-tensor ratio . In the near future, the BICEP2 detection of will be cross-checked by many on going and forthcoming experiments. For example, Planck (4), SPTPol (5), ACTPol (6), PolarBear (7) and CLASS (8).
These forthcoming experiments can probe more detailed properties for the tensor mode. For example, when the statistical features of the primordial perturbations are not isotropic, a large number of new observables arises, including the correlation functions with different multipole moment , and TB and EB cross-correlations, which have been forbidden by the isotropic statistics of the primordial fluctuations (9); (10); (11).
Motivated by the future data, in this paper we study inflationary dynamics producing statistical anisotropies in details. We investigate the anisotropic inflation scenario with a charged scalar inflaton coupled to the gauge field. This model is studied originally in (12) at the background level and its predictions for statistical anisotropies in curvature perturbation power spectrum were studied in (13). However, the perturbations in tensor sectors and their correlation with the scalar perturbations have not been explored so far. We show that the gauge coupling induces large statistical anisotropies in tensor perturbations as compared to model of anisotropic inflation with no charge coupling (14). We show that while statistical anisotropies in temperature power spectrum can be small as required by the Planck data (15), the tensor mode can develop significant statistical anisotropies. Therefore it is important for the forthcoming experiments to look for the statistical anisotropies in the B-mode polarization even though the statistical anisotropies in temperature map is well-constrained. For this purpose, in this work we calculate the primordial correlations of the curvature perturbation , and the two tensor modes and . The CMB temperature and polarization correlations TT, TE, EE, TB, EB and BB are then calculated from these primordial perturbations, for the same and different multipole moments.
A novel result in our analysis is that the anisotropies in the tensor sector also contribute to the TT correlation on the CMB. However, the transfer function from primordial tensor to CMB temperature decays towards large . Thus the TT anisotropy coming from the primordial tensor has a decaying amplitude and is highly suppressed after . As a result, we naturally obtain anisotropies at low multipoles of TT without modifying the high multipoles. This scale-dependent anisotropies will have a better fit to the CMB anomalies.
Another motivation of the present work is that, the Planck experiment puts a limit of tensor-to-scalar ratio . This result is in tension with the BICEP2 detection. The tension is reported to be as unlikely as . Recently, it is proposed (16) that with anti-correlated curvature and tensor perturbations, the tension between Planck and BICEP2 may be reconciled. However, our detailed study show that such a mechanism does not work. As we shall show, there is indeed an anti-correlation between the curvature and tensor perturbations. However the contribution to the TT power spectrum is cancelled when summing over the angular mode .
The rest of the paper is organized as follows. In Section II we present our model of anisotropic inflation and review the background dynamics and the perturbations. In Section III we calculate the anisotropic scalar and the tensor power spectra and the scalar-tensor cross-correlations. In Section IV we present the predictions of our model for CMB. We relegate the technical details into appendices.
Ii Anisotropic Inflation
Here we review anisotropic inflation. For earlier works on various aspects of anisotropic inflation see (9); (17); (14); (15); (18); (12); (19); (20); (21); (22); (23); (24); (25); (13); (26); (27); (28); (29); (30); (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46); (47); (48); (49); (50), for a review of anisotropic inflation see (51); (52). Also see (53); (54); (55) for related works on primordial anisotropies.
As mentioned before, the model contains a gauge field , as in Maxwell theory, which is turned on at the background level. However, it is well-known that the Maxwell theory suffers from the conformal invariance on expanding backgrounds. Therefore, the background gauge field is diluted exponentially during inflation. Furthermore, the quantum excitations of the gauge field are not scale-invariant. One interesting mechanism to break the conformal invariance is to couple the gauge field to the inflaton field non-trivially.
With these general discussions in mind we present our model of anisotropic inflation. The model contains a complex inflaton field which is charged under the gauge field with the electric charge (coupling) . The action is given by
where is the reduced Planck mass.
As usual, is the gauge field strength given by
In addition, the covariant derivative associated with the gauge field is given by
where represents the electric gauge coupling. The model of anisotropic inflation based on the above action was studied at the background level in (12) and its analysis for curvature perturbations were performed in (13).
Note that in Maxwell theory . However, as explained above, we need a time-dependent gauge kinetic coupling in order to break the conformal invariance such that the background gauge field survives the exponential expansion and the gauge field excitations acquire a nearly scale-invariant power spectrum. As we shall verify below one requires in order to obtain a scale-invariant power spectrum for the gauge field quantum excitations. This choice of the gauge kinetic coupling corresponds to a constant background electric field energy density during inflation.
The complex scalar field can be decomposed into a radial part and an axial part via . As usual, we assume the model is axially symmetric in field space so the potential and are only functions of . It is convenient to go to the unitary gauge in which . In this gauge, becomes real-valued and in the analysis below we take .
We consider the coordinate system in which the background gauge field is turned on along the -direction so . With the background gauge field in the -direction the background space-time becomes anisotropic, taking the form of type I Bianchi Universe, with the metric
Note that the metric (4) still has a residue two-dimensional rotational symmetry on the plane. In this convention measures the averaged Hubble expansion while measures the level of anisotropic expansion. However, on the observational grounds, as we shall see in next Section, the level of anisotropy in curvature perturbations is not more than few percent which subsequently is translated into the conclusion that . As a result, we can treat the analysis prturbatively in term of background anisotropy.
The background fields equations are (12)
in which a dot indicates derivative with respect to and and so on.
Here we summarize the main features of the above equations. Eq. (5) is the Maxwell equation in the inflationary background with the gauge coupling appearing in the source term. Eq. (6) is the modified Klein-Gordon equation in the presence of the gauge field. As we see the scalar field dynamics are affected by the gauge field via the last two terms in the bracket. The first term in the bracket in Eq. (6) comes from the gauge kinetic term while the last term comes from the charge coupling . As we shall see, these two terms play crucial roles both at the background and the perturbation levels. The remaining equations are the Einstein equations with Eq. (9) controlling the dynamics of the anisotropy .
In general it is not easy to solve the above set of equations analytically, even in the slow-roll limit. However, as we mentioned before, the level of anisotropy in curvature perturbations is small which also yields . This means that the background expansion is nearly isotropic. Therefore, as in in conventional models of inflation, the background expansion is controlled by the potential term . However, we expect the gauge field also to contribute in the background expansion in the form of electric field energy density where . In order for the background to be nearly isotropic we require that the electric field energy density to be very small compared to . It is convenient to parameterize the gauge field (electric) energy density by the parameter via
In order for the anisotropy to be small we assume .
Alternatively, for the perturbation analysis, it is more convenient to express the background metric (4) in the following form
where and and the conformal time is defined via via .
We will work in the slow roll limit where the change in expansion rates in all directions are small. Defining the average Hubble expansion rate via , the slow-roll parameters are given by
We work in the slow-roll limit where and to leading order in slow-roll parameters and anisotropy .
Before we study the attractor solution in next subsection, let us discuss the effects of the gauge coupling . The main effect of the gauge coupling is captured by the third term in bracket in Eq. (6). From this term we see that the interaction induces a time-dependent mass for the inflaton. Since this induced mass is exponentially time-dependent, we expect it becomes important only towards the end of inflation where the exponential growth of the gauge field becomes significant. As studied in details in (12) inflation ends when the induced mass from the back-reaction becomes comparable to the bare inflaton mass . Therefore, in order for inflation to sustain long enough, the back-reaction is negligible during much of the period of inflation and it only controls the mechanism of end of inflation. As studied in (12) the end of inflation depends logarithmically on where where dots indicate the dependence on other parameters such as the inflaton value and its mass. Therefore, the larger is the gauge coupling , the shorter is the period of inflation. As explained above this is easily understood from the fact that the induced mass for the inflaton field from the Higgs mechanism, , becomes comparable to the bare inflaton mass.
In the remaining analysis of the background dynamics we neglect the effects of during inflation. In particular, one can neglect the source term in Maxwell equation (5) during much of period of inflation. In this approximation one can easily solve the Maxwell equation (5) to get
where is a constant of integration. Of course, this approximation breaks down near the end of inflation where the induced mass term from the interaction becomes important.
ii.1 The Attractor Solution
As we discussed above the anisotropy is small and the average Hubble expansion rate in Eq. (7) mainly comes from the isotropic potential term. However, the back-reactions of the gauge field on the inflaton field induce an effective mass term for the inflaton as captured by the last term in Eq. (6). Therefore, the dynamics of the inflaton field is affected by the back-reactions of the gauge field. A key observation was made in (9) where it is shown that for a general form of slow-roll potential and with an appropriate choice of the system reaches an attractor regime where . As a result, the modified Klein-Gordon equation still admits a slow-roll solution but now with a modified effective mass for the inflaton which is induced from the back-reactions of the gauge field.
In order for to reach a nearly constant value, and neglecting the contribution of in the small anisotropy limit, one has to choose such that . Now to find as a function of , our job is to solve the background expansion equation and express in terms of . In the small anisotropy limit, and for a given potential , the background isotropic expansion is given by
This indicates that if one chooses
then one obtains .
The exact form of depends on . To be specific, in this work we consider the simple chaotic potential
where is a constant.
Now we examine under what conditions the system allows for an attractor solution where the anisotropy reaches a sub-dominant but constant value. In the small anisotropy limit, the background equation is given by the potential term and we have
On the other hand, in the slow-roll regime, the scalar field equation, Eq. (6), is given by
Note that in order to get the second term in the right hand side the solution for and the form of , given respectively in Eq. (13) and Eq. (18), have been used. Eliminating from Eqs. (19) and (20) we obtain an equation for in terms of the number of e-folds as
One can easily solve this differential equation to get
where is a constant of integration. Now the important point to note is that for , the second term in the square bracket above decays exponentially during inflation and we quickly reaches the attractor regime
On the other hand, using the remaining Einstein equation, one can also verify that the slow-roll parameter is
As a result, we conclude
where we have introduced the anisotropy parameter via
Note that in order for the gauge field to survive the exponential expansion and the system reaches the attractor regime we require .
This equation implies that the back-reaction of the gauge field has reduced the effective mass of the inflaton such that .
The above equation can be easily solved to find as a function of the number of e-foldings as
where represents the value of at the end of inflation. Note that the above solution is valid until end of inflation when the effects of the gauge coupling starts to dominate. As we discussed before, there is additional induced mass from the term which terminate inflation quickly when it becomes comparable to the bare mass .
Here we present cosmological perturbations in anisotropic inflation. The perturbation analysis for various models of inflation were studied in (20); (21); (22); (23); (24); (25); (13); (26); (27); (28); (14). The general form of the metric and matter perturbations have been studied in (13), see also (22). For the perturbations in the metric we note that there are some non-dynamical degrees of freedom, namely , which should be integrated out in order to calculate the dynamical action. Integrating out these non-dynamical metric degrees of freedom lead us to extra terms in the action. Therefore, one important question is what the leading contributions in the final action are. Are they from the metric sector perturbations or from the matter sector perturbations? Fortunately, as it has been verified in (13), it turns out that the metric sector perturbations are either slow-roll suppressed or would cancel with each other. As a result, we conclude that the leading terms in the interactions come from the matter sector. In other word, we do not need to consider the perturbations from the non-dynamical degrees of freedom in the metric. To simplify the situation further, we go to flat gauge where the curvature perturbations is given by the inflaton perturbations . As a result, the metric has no scalar perturbations and we are left with the simple form of metric perturbations
Note that since we work in small anisotropy limit, we can set to leading order. The corrections in our results below from using this assumption will be suppressed by additional factors of or which are small. The perturbations represents the tensor modes subject to the transverse and traceless conditions and where the repeated indices are summed. We denote the two independent polarizations of the metric by and .
As for the perturbations in gauge field sector there is one non-dynamical degree of freedom, , which must be integrated out from the action. However, similar to the case of non-dynamical degrees of freedom from the metric perturbations, it turns out that the new terms from integrating out are also sub-leading. As a result, the leading interaction terms in the total Lagrangian come from the dynamical degrees of freedom .
In order to simplify the analysis further, we can use the remaining two-dimensional rotational symmetry on the plane to set so in Fourier space. In addition, since the gauge field has three polarizations in the unitary gauge, two transverse and one longitudinal polarizations, we can choose the following ansatz for the gauge field perturbations, i.e. ,
Where we have defined, and to be and respectively. Here refers to one transverse polarizations in the vector sector while represents the two polarizations in the scalar sector. Furthermore, we can decompose the two polarizations in into one transverse and one longitudinal polarizations as follows (13)
In this decomposition represents the transverse polarization while refers to the longitudinal polarization of the gauge field. However, as it has been demonstrated in (13), the interactions containing the longitudinal mode are exponentially suppressed during inflation and can be neglected from the analysis. Physically, this is understandable since the interactions containing the longitudinal mode originate from the “Higgs mechanism” via the interaction which are exponentially suppressed during much of the period of inflation as discussed before.
We can quantize the curvature perturbation and the gauge field perturbations as usual. For the curvature perturbation, note that we work in the flat gauge so
Expanding the quantum operator in terms of the annihilation and the creation operator and we have
where the creation and the annihilation operators satisfy the usual commutation relation .
The wave function of the curvature perturbation has the standard form of the excitations of a massless scalar field on a dS background
The power spectrum of the curvature perturbations is given by
In particular, the power spectrum for the free isotropic theory is
Similarly, the quantum excitations of the gauge field perturbations and can be expanded in terms of their annihilation and creation operators with the wave functions
Now we present our decomposition of the tensor perturbations into and polarizations following the method of (14). Decomposing into in Fourier space, the traceless and transverse conditions, , yields
with representing the two polarizations. In addition, we choose the following normalization
where represents the complex-conjugation. Note that we also have .
The quantum operators in Fourier space are represented in terms of the annihilation and creation operators by
with the commutation relations .
As we mentioned before, we chose the convention that . With this choice, the polarizations and become
We will use this expression later on when calculating the cross-correlation between the tensor mode and the curvature as well as the gauge field.
The profile of the tensor excitations has the standard form
The power spectrum of the tensor perturbations is given by
In the absence of anisotropy the power spectrum has the standard form
Therefore, defining the tensor-to scalar ratio we have for the isotropic theory.
ii.3 The Interaction Lagrangian
Having presented the background in some details, here we separate the Lagrangian into the free field part and interaction part. Here and below we call the latter the interaction Lagrangian. The starting Lagrangian from the action (1) is
Expanding the above action around the background values, neglecting the contributions of the non-dynamical field which are sub-leading as discussed before, and using the relation , the interaction Lagrangians in the Fourier space is calculated as (see Apendix for further details)
where c.c stands for complex conjugation.
The above interaction Lagrangians are needed in order to calculate the anisotropy corrections in and the cross-correlations . Note that in the free (isotropic) theory with there is no anisotropy corrections in power spectra and as expected.
Iii Anisotropic Correlations
Having calculated the interaction Lagrangians as given in Eqs. (49)-(52) now we are ready to calculate the anisotropic correlation functions by using the in-in formalism. For this purpose, we need to obtain the interaction Hamiltonian from the interaction Lagrangian. One should notice that is not necessary true with kinetically coupled interactions. So it is worth to check it in this model before proceeding with the in-in calculation of the correlation functions. We have calculated it in the Appendix B. It turns out that the above formula is true for the whole of the interactions except . So in the following we use everywhere except that in , special care is taken of.
iii.1 Anisotropies in curvature power spectrum
Here we calculate the anisotropic contributions in curvature perturbation power spectrum . We denote the change in curvature perturbation power spectrum from the anisotropic sources by . This analysis were performed in (13) and here we outline the analysis briefly.
where represents the interaction Lagrangian. The lower limit of the integral should be set corresponding to initial modes being deep inside the horizon. However, as studied in (23); (13), the interactions responsible for anisotropies operate on super-horizon scales so to a good approximation one can safely take corresponding to the time when the mode leaves the horizon. The upper limit of the above integral as usual corresponds to .
The interaction Lagrangians relevant to are and given in Eqs. (49) and (50). A look at these two equations show that is suppressed compared to by the factor . Therefore, the leading order anisotropic corrections in curvature perturbation power spectrum comes from . In addition, has two independent terms denoted by and :
Depending on whether one chooses either or in place of and in the integral in Eq. (53), there are four possible contributions in denoted by where with the assumption that and . For example, means and . With this identification we have
The details of the in-in analysis are presented in Appendix C. As a sample analysis, here we present the integral form of which is (here and hence after, the momentum conservation -function is omitted to save writing)
Expanding the integrand for small arguments and assuming and as explained above, the above integral yields
in which represents the number of e-folds when the mode has left the horizon. Taking to be the CMB scales we need in order to solve the flatness and the horizon problem.
Performing the same procedure for other integrals, we obtain (13)
So combining these four contributions, and assuming , we have
where we have defined
We have also written in terms of and by using .
Note that is a measure of the gauge field coupling . In particular, in the model of (22); (23); (14) with we have . With , and with we obtain . For larger value of we see that grows like .
The anisotropic power spectrum therefore is
Correspondingly, the total anisotropic power spectrum is
Now defining the anisotropy estimator via
where is the preferred anisotropic direction in the sky (the -direction in our example), we obtain
It is important to note that the form of we have defined here is with respect to the primordial curvature power spectrum. In the TT and other correlation functions, the anisotropy not only comes from the here, but also comes from an “effective ” contribution from the tensor sector. Such an “effective ” has a scale dependence on the TT and other correlations because the tensor mode is decaying after it returns to the horizon. We will return to this issue later. Also we see that the gauge coupling appears in via the parameter . Taking from the Planck data constraint (15) we require .
One may ask what the theoretical limits on the value of or the parameter are. First, we have to make sure that we get enough number of e-folds of inflation at the background level. As we mentioned before, the number of e-folds depends logarithmically on so as studied in (12) one can take say to get a long enough period of inflation. In addition, our assumption in parametrizing the anisotropy was that the anisotropic power spectrum is smaller than the isotropic power spectrum, i.e. so our perturbative approach using the leading order in-in formalism is valid. Therefore, demanding we need