A Truncated Primordial Power Spectrum and Its Impact on B-Mode Polarization

The absence of large-angle correlations in the temperature of the cosmic microwave background (CMB), confirmed by three independent satellite missions, creates significant tension with the standard model of cosmology. Previous work has shown, however, that a truncation, $k_{min}$, of the primordial power spectrum comprehensively resolves the anomaly and the missing power at $\ell\lesssim 5$ (the low multipoles). Since this cutoff is consistent with the hypothesized delay of inflation well beyond the Planck time, we are strongly motivated to consider its possible impact on other observational signatures. In this Letter, we analyze and predict its influence on the most revealing probe awaiting measurement by upcoming missions -- the B-mode polarization of the CMB, whose accurate determination should greatly impact the inflationary picture. We highlight the quantitative power of this discriminant by specifically considering the LiteBIRD mission, predicting the effect of $k_{min}$ on both the angular power spectrum and the angular correlation function of the B-mode, for a range of tensor-to-scalar ratios, $r$. While its impact on the latter appears to be negligible, $k_{\rm min}$ should have a very pronounced effect on the former. We show that for $r=0.036$, $k_{min}$'s impact on $C_{\ell}^{BB}$ at low $\ell$'s should be easily detectable by LiteBIRD, but will be largely hidden by the total uncertainty of the measurement if $r\lesssim 0.02$.


Introduction
Inflationary cosmology [1,2,3,4], offering insights into the early Universe's rapid expansion, may overcome several difficulties with the standard model, including the cosmic microwave background's (CMB) temperature horizon problem and the scarcity of magnetic monopoles.Many details regarding inflation, however, including the specifics of the driving field and the onset of the rapid expansion, remain elusive.
These uncertainties have been compounded since the Planck-2013 data release [5], which shifted the focus from simple, chaotic inflation to models featuring plateau-like potentials, removing the possibility of matching the Universe's energy density to the Planck scale at the Planck time, t Pl [6,7].As we now understand, however, a possibly related anomaly emerges from the observed lack of large-angle correlations in the CMB, confirmed by three independent satellite missions: the Cosmic Background Explorer (COBE) [8], the Wilkinson Microwave Anisotropy Probe (WMAP) [9] and the Planck mission [5,10].This issue is not effectively resolved by cosmic variance, which assigns an unacceptably low probability to the observed Universe [11].
Instead, a comprehensive analysis of the Planck data suggests that the absence of large-angle correlations in the CMB can be explained in terms of a non-zero minimum wavenumber, k min , in the primordial power spectrum, P(k) [12].Previous work has shown that such a truncation would naturally arise if the de Sitter expansion began at a time t init ≫ t Pl [13].
We thus appear to be witnessing an interesting confluence of modern inflationary theories and recent observations both pointing to a delayed onset of de Sitter expansion, well past t Pl , producing a non-zero minimum wavenumber, k min , in P(k).
In this Letter, we focus on the broader impact of this cutoff on the observational signatures created by infla-tion.Starting with the role of k min in mitigating the TT discrepancies, we extend the analysis to gauge its impact on upcoming observations of the CMB polarization.If the latter confirm its reality, the existence of such a cutoff would represent one of the most important, if yet unexplored, features of the primordial power spectrum.For example, a nonzero value of k min affects the angular correlation function of the B-mode polarization, whose low-ℓ portion is predominantly generated by tensor fluctuations.
We utilize the CAMB code [14] with the latest Planck results [10] to compute the angular power spectrum and corresponding angular correlation function of the polarization signal for a range of k min values.To provide a quantitative measure of how this cutoff should impact future observations, we evaluate the expected observational signal relative to the sensitivity of LiteBIRD [15].In doing so, we highlight the significance of the tensorto-scalar ratio r in modulating the impact of k min within the conventional inflationary paradigm.

Background
The primordial power spectrum was generated when the quantum fluctuations in the inflaton field crossed the Hubble horizon.During inflation, the Hubble radius, R h ≡ c/H(t), remained nearly constant, while the mode wavelengths, λ k = 2πa(t)/k, grew at a rate proportional to the expansion factor a(t). Modes with different k's crossed R h at different times [16], so a de Sitter expansion beginning at a particular time, t init , would have produced a largest mode corresponding to k min [13].
The angular power spectra for the temperature and polarization in the CMB largely depend on P(k) [17,18,19]: In these equations, T , E, and B refer to the temperature fluctuations, the E-mode polarization, and B-mode polarization, respectively.The quantities (S ) and (T ) denote scalar and tensor modes, and P ϕ and P h represent the primordial scalar and tensor power spectra, while ∆ is the transfer function.
The angular correlation function, C(θ), may be written in terms of the angular power spectrum [17,18,19,20]: where P ℓ (cos θ) are the Legendre polynomials.We shall here introduce the quantities B and Ê to represent the angular correlation function of the polarization, which we prefer over the more commonly used E and B modes.The latter are non-local quantities requiring an all-sky integral for extraction.In contrast, B and Ê are designed to be computable from the Q and U modes (both observable quantities) by applying local spin-raising and spin-lowering operators [17,18]: A non-zero k min affects all the C l values and C(θ) at every angle.We shall reproduce this result below and demonstrate the dramatic improvement of the model fit compared to the analogous situation with k min = 0. Given that k min is primarily a large-scale feature, however, its impact is mainly felt at large angles and low ℓ's, where it suppresses the values of C ℓ .
The outcome of calculating C(θ) is quite different for the temperature and polarization cases.As demonstrated below, the suppression of C ℓ for the temperature at low ℓ's induces a notable shift in the C T T (θ) curve.But the situation changes significantly for the polarization.Due to the additional (ℓ+2)!(ℓ−2)!factor, C EE,BB (θ) is dominated by the large-ℓ terms [20], thereby weakening the impact of k min .
The effect of k min on C T T (θ) was one of the principal reasons for its introduction.Previous work [12] demonstrated that the Planck data decidedly rule out a zero k min at a confidence level exceeding 8σ.The value of the cutoff is instead measured to be for a last scattering surface at z = 1080.We shall adopt this cutoff throughout this work.Its viability for resolving the large-angle anomalies in the CMB was reinforced by a second study [21] focused on the TT angular power spectrum.This subsequent work demonstrated a clear improvement of the model fit at ℓ ≤ 5, with an optimized cutoff in this case at k min = (2.04 +1.4  −0.79 ) × 10 −4 Mpc −1 , consistent with the previous value to within 1σ.
To be clear, these two earlier papers [12,21] analyzed two separate features of the CMB anisotropies.The first was solely focused on the angular correlation function, while the second addressed the low power in the lowℓ multipole components.The strong evidence (≳ 8σ) against a zero value for k min was based purely on the impact of such a cutoff on what is essentially the standard model.The inferred k min optimized the fit to the angular correlation function.Of course, this outcome does not necessarily imply that the standard model is ruled out at this confidence level.It merely states that k min = 0 is ruled out, whether the underlying model is ΛCDM or something else.In this regard, we point out that the 'new' chaotic inflationary paradigm itself requires a delayed initiation to inflation, which necessarily implies that k min 0, unlike the original (or classical) version of the model in which the fluctuations were believed to have originated below the Planck scale, thereby implying that k min is effectively zero [22].
Given the impact of such a strong result, we here summarize the procedure followed in [12] to arrive at this 8σ confidence level.The points in the observed angular correlation function C(θ) are highly correlated.This needs to be taken into account in the statistical analysis, as described in that earlier publication.The optimized value of k min is obtained by examining the distribution of mock CMBR catalogs (using standard cosmology with the angular correlation function C[θ]).Its uncertainty is derived from the r.m.s.value within which one finds 68 % of the possible outcomes.This error is actually larger than the value one would infer from a simple χ 2 fitting, which would ignore the correlations.In other words, the estimate of 8σ is atually the most conservative limit one may get when analyzing the angular correlation function on its own.Our second publication [21] complemented this earlier work by considering the low power anomaly associated with the low-ℓ multipoles.Its purpose was to see whether or not the k min inferred from C(θ) also solves this independent problem with the traditional CMB analysis.The simple answer is 'yes,' but the confidence level associated with a non-zero k min based solely on the low power argument is weaker: ∼ 2.6σ.Nevertheless, these two analyses together point to a compelling scenario, in which a non-zero k min can easily solve both anomalies with the value quoted in Equation 8.
Crucially, this second study also confirmed that the inclusion of k min as one of the free parameters in the standard model creates no noticeable change to the other variables, whose previous optimization produced the well-known, impressive fit to the angular power spectrum at ℓ ≫ 10.For simplicity, we shall therefore simply adopt the Planck-concordance values of all the cosmological parameters except for k min .

Methodology
In spite of these compelling results with the TT anisotropies, however, there are two principal reasons for focusing on the CMB polarization going forward.First, forthcoming observations are likely to provide measurements of the polarization precise enough to distinguish between scenarios with and without k min .Second, B-mode polarization is a tell-tale signature of tensor modes, which are masked by scalar-mode signals in the TT and EE anisotropies.This information is critical for improving our understanding of inflation and the early Universe.Findings related to the TT anisotropies merely confirm the existence of k min for the primordial scalar-mode power spectrum.But if k min emerges due to the delayed commencement of inflation, it should also be present in the primordial tensor spectrum.This can only be identified through the analysis of B-mode polarization.All of our calculations are carried out using the CAMB code (version 1.4.0) with the most recent Planck results as its default parameters [14].The cutoff is introduced to the default primordial power spectrum as follows in terms of the amplitude, A s , of the primordial power spectrum, the spectral index n s of the scalar fluctuations, and the pivot scale k pivot = 0.05 Mpc −1 .The rest of the parameters used in these computations are derived from the most recent Planck optimizations, as discussed above.
The angular power spectrum C T T ℓ is shown in Figure 1, and the corresponding TT angular correlation function is shown in Figure 2. To achieve a realistic representation of the Planck measured angular correlation function along with its 1σ range, we generated a sample of one thousand mock realizations of the angular correlation function.More specifically, we used the measured C T T ℓ values and their uncertainties, as published in [10], to generate a thousand sets of mock C T T ℓ 's.To address the non-Gaussianity of the C T T ℓ errors, we assumed that the upper and lower errors in C T T ℓ follow (possibly different) half-Gaussian distributions.We then randomized the measured C T T ℓ values within these distributions.For each set of mock C T T ℓ , we calculated the corresponding angular correlation function, thus obtaining a thousand realizations of the mock C(θ).From this sample, we then estimated the non-Gaussian errors using the same approach, i.e., allowing the upper and lower error distributions to be half-Gaussians.The disparity between the predicted and measured angular correlation functions is sometimes quantified in terms of the cosmic variance associated with the theoretical calculation.The results in Figure 2 are thus reproduced in Figure 3, where the shaded region now  corresponds to this cosmic variance in the concordance model (red curve), estimated using the Planck official simulations (FFP10 CMB realizations).
These results confirm those obtained earlier for the TT distributions [12,21] based on a different methodology and the older (i.e., Planck-2014) data.As demonstrated in these previous works, the optimized value of k min significantly improves the fits in both Figures 1 and  2 (and 3) at all angles.

B-mode Polarization
The BB angular power spectrum C BB ℓ is calculated assuming a primordial tensor distribution in terms of the tensor-to-scalar ratio r and the spectral index n T of the tensor fluctuations, which we assume to be the conventional n T = −r/8.The range of values discussed here is motivated (i) by the current upper bound r < 0.036 provided by Planck and BICEP2/Keck [23], and (ii) by the expected sensitivity of LiteBIRD, which should allow a measurement to levels less than r = 0.004.It is important to note that B-mode polarization arises, not just from these tensor modes in the primordial power spectrum, but also from weak lensing effects that convert some of the E-mode polarization into B-mode.CAMB correctly takes all such effects into consideration [24].The angular power spectrum C BB ℓ is shown in Figure 4.The BB spectra reveal noticeable differences for ℓ ≤ 4, analogously to the situation with TT.Moreover, the difference increases somewhat with increasing r.Increasing r mainly affects the low-ℓ C ℓ 's because of the two sources of B-mode polarization in the CMB introduced above: primordial gravitational waves (i.e., tensor modes) and weak gravitational lensing.The influence of these two contributors to C BB ℓ varies in significance at different ℓ's.Tensor modes mainly affect the low-ℓ region because they induce B-mode polarization primarily on large scales.They generated the B-mode polarization pattern at photon decoupling (t ∼ 380, 000 years), which remained unchanged afterwards.In contrast, weak gravitational lensing contributes relatively more to C BB ℓ at higher ℓ's.The lensing effect warps the null geodesics arriving from the early Universe, converting some of the E-mode polarization into B-mode.This effect is more pronounced on small scales (high ℓ's), given that lensing depends on the distribution of matter on large scales altering the photon paths on smaller scales.Thus, when lensing effects are present, tensor modes mainly influence the low ℓ region of C BB ℓ .The portion of Figure 4 most relevant to this analysis (ℓ ≤ 4) is magnified and displayed in the four panels of Figure 5.The difference induced by a nonzero cutoff to P (T ) (k) is here compared to the expected total uncertainty in the LiteBIRD observations, believed to be the most sensitive upcoming B-mode measurements.These simulations begin with the theoretical curves for C BB ℓ , with and without the cutoff.Then, assuming Gaussian errors for these future measurements (The "total Lite-BIRD errors" we used here were published by the Lite-BIRD team [15].For the range of ℓ we are interested in (ℓ ≤ 10), the errors are presented as Gaussian), we carry out a Monte Carlo simulation randomizing the results based on the total (predicted) LiteBIRD uncertainty [15] to produce a mock sample of possible C BB ℓ distributions as a function of r .We calculate the ±1σ and ±2σ errors assuming a Gaussian distribution of the C BB ℓ values in the mock sample.
The expected total uncertainty used here is taken directly from the LiteBIRD simulations, which includes cosmic variance and the noisy foreground residuals.The contributing instrumental systematic uncertainty is considered by the LiteBIRD team when creating mock input foreground maps for the simulations.
The panels in this figure show the results for three representative values of r (the tensor-to-scalar ratio): 0.036 is the current upper limit of r consistent with the latest Planck data release [10], which produces the largest difference between the two curves (with and without k min ); 0.02 is the smallest r for which the difference between the two curves (at ℓ = 2) is still larger than the expected LiteBIRD total uncertainty; 0.00461 coincides with the value utilized by the LiteBIRD team when presenting their projected error bars [15].As noted earlier, a larger r accentuates the disparity between the cases with and without k min .
We can clearly see that the shift introduced by k min is relatively insignificant compared to the forecast Lite-BIRD total uncertainty when r is very small (r ∼ 0.005).If r is not much smaller than its current upper limit, however, the impact of k min should be measurable, especailly at ℓ = 2.The cutoff in P (T ) (k) produces an observable signature well above cosmic variance and Lite-BIRD's expected measurement error.
To complete the summary of results from this analysis, we also show in Figure 6 the angular correlation function of the B-mode polarization for r = 0.036.This calculation includes only the range ℓ ≤ 10 since only the low-ℓ C ℓ 's may be differentiated on the basis of their k min value.The ±1σ uncertainty region corresponding to LiteBIRD's expected measurement accuracy for C ℓ [15] is estimated with a ten-thousand step Monte Carlo simulation.The two curves are exceedingly close.Lite-BIRD may not be able to differentiate between them.As noted earlier, this outcome is due to the extra factor (ℓ+2)! (ℓ−2)!appearing in Equation ( 5), which amplifies the dependence of the angular correlation function on the higher ℓ's.Thus, in spite of our calculation restricting the range of multipoles to ℓ ≤ 10, the impact of k min is largely concealed by the dominance of the ℓ > 5 terms.
Figure 5: BB angular power spectrum for ℓ ≤ 4. In all four panels, the red and blue lines show the results with and without k min .The shaded regions show the expected total LiteBIRD errors, including foreground residuals.Top-left is for r = 0.00461, with the forecast ±1σ region; Top-right is for r = 0.02, with the expected ±1σ region; Bottom-left is for r = 0.036, with the forecast ±1σ region; Bottom-right is for r = 0.036, with the expected ±2σ region.

Conclusion
The hypothesized delayed initiation of inflation would produce a rather rigid cut-off, k min , to both the scalar and tensor primordial power spectra.In this Letter, we have sought to find an unambiguous observational signature of this crucial inflationary feature, which is manifested in both the TT and BB CMB anisotropies.It is already known that k min significantly impacts the TT angular correlation function.But whereas the current Planck data support the existence of such a cutoff in the scalar power spectrum, only high-precision observations of the B-mode polarization by forthcoming missions can fully reveal a truncation in both the scalar and tensor modes, thereby providing clear, untainted evidence of a delayed de Sitter expansion in the early Universe.
We have shown that a cutoff k min should be clearly measurable by LiteBIRD if the tensor-to-scalar ratio, r, is not much smaller than ∼ 0.02.At the current upper limit, r = 0.036, the differences in C BB ℓ for ℓ < 4 with and without the cutoff easily exceed LiteBIRD's total uncertainty.
The influence of k min on the B-mode angular correlation function is less pronounced, due to its greater dependence on the high-ℓ terms in the angular power spectrum.In this context, the outcomes for B-mode and TT are distincly different.An anomaly in the B B angular correlation function is thus unlikely, again contrasting with the existing situation with TT.
The impact of a measured nonzero cutoff k min in the B-mode angular power spectrum cannot be overstated.Together with the existing constraints inferred from TT, this would provide compelling evidence of a delayed de Sitter expansion in the early Universe, as required by the new, chaotic inflationary paradigm.

Figure 1 :
Figure 1: Calculated TT angular power spectrum with (red) and without (blue) the cutoff k min , in comparison with the Planck data.

Figure 2 :
Figure 2: TT angular correlation function with (green) and without (red) k min , in comparison with the angular correlation function calculated from the Planck data (blue).The shaded region represents the (non-Gaussian) 1σ uncertainty obtained from a sample of one thousand mock correlation functions generated via Monte Carlo randomization.

Figure 3 :
Figure3: Same as Fig.2, except that here the shaded region corresponds to the 1σ cosmic variance, calculated utilizing Planck's official set of simulations (FFP10) for the concordance model (red curve).

Figure 4 :
Figure 4: BB angular power spectrum for different values of r: 0.00461 , 0.02 and 0.036 , with (solid) and without (dashed) k min .

K 2 ]Figure 6 :
Figure 6: B B angular correlation function for ℓ ≤ 10.The plot compares the results with and without k min .The shaded area represents the ±1σ error region anticipated by LiteBIRD.