Data from the satellites COBE (Hinshaw et al., 1996), WMAP (Bennett et al., 2003) and Planck (Akrami et al., 2019), have consistently found a lack of two-point correlations at low multipoles, or at large angular scales, compared to what is expected in the ΛCDM model. Visually, this lack of correlations is evident in the real space two-point correlation function C(θ), shown in the right panel of Figure 2: for angles larger than 60°, the two-point function is surprisingly low. The WMAP team had come up with an appropriate observable to quantify this lack of power (Spergel et al., 2003). It is defined by S 1 / 2 = ∫ − 1 1 / 2 C θ 2 d cos θ . (2.5)

Its physical meaning is obvious: it captures the total amount of correlations squared (to avoid cancellations between positive and negative values of C(θ)) in angles θ > 60°. The ΛCDM model predicts S _1/2 ≈ 42000 μK ⁴, while the Planck satellite has reported a measured value ¹ of S _1/2 = 1,209.2 μK ⁴ (Akrami et al., 2019), which corresponds to a p-value less than 1% [≤0.5% according to (SchwarzSchwarz et al., 2016)]. Put in simpler terms, this p-value tells us that if we were able to observe one thousand universes ruled out by the ΛCDM model, only about a handful will show such a low value of S _1/2.

A second important anomaly that has been also observed by multiple satellites, is the presence of a dipolar modulation of the entire CMB signal (Akrami et al., 2019). This dipole should not be confused with the multipole ℓ = 1. Rather, the anomaly makes reference to correlations between multipoles ℓ and ℓ + 1, which can be explained by a modulation of dipolar character, as we further discuss below.

Such modulation was first modeled mathematically in (Gordon et al., 2005), by adding a simple dipole to the temperature map as follows T n ^ = T 0 n ^ 1 + A 1 n ^ ⋅ d ^ , (2.6) where T 0 ( n ^ ) is the unmodulated (statistically isotropic) temperature field, A ₁ is the amplitude of the modulation, and d ^ its direction. It is easy to check that such modification affects not only the ℓ = 1 angular multipole, but actually all multipoles equally, and for this reason it is known as a scale-independent dipolar modulation. Its main effect is to create correlations between multipoles ℓ and ℓ + 1. Such correlations, as mentioned above, violate isotropy [see Appendix B of (Agullo et al., 2021b) for further details].

The Planck team has carried out a likelihood analysis of such modulation of the CMB, and arrived at constraints on the amplitude and direction of the dipolar modulation in different bins of multipoles ℓ. Surprisingly, the analysis has revealed a non-zero amplitude of the dipolar modulation only for low multipoles, in the bin ℓ ∈ [2, 64]. The amplitude reported in this bin is A ₁ ≈ 0.07 (Ade, 2016a), and the significance of the detection is greater than 3σ. This reveals, not only a significant deviation of the ΛCDM model, but also that the dipolar modulation is scale-dependent, since it only appears for low multipoles. Therefore, the simple model (Eq. 2.6) is insufficient to account for the observed modulation. Finding a mechanism to generate a scale-dependent dipolar modulation, without introducing other undesired effects, has challenged the imagination of theorists during the last decade (Dai et al., 2013).

Observations from both WMAP and Planck have found a preference for odd parity two-point correlations, as opposed to the predictions of the standard ΛCDM model, which predicts that the primordial perturbations generated in our Universe are parity neutral. The parity of the primordial perturbations can be studied by analyzing the multipoles in the range [2, 50], known as the Sachs-Wolfe plateau. This range of multipoles corresponds to long wavelength perturbations which entered the horizon in the recent past, and hence have been relatively unmodified by late time physics. The asymmetry in the parity can be quantified using the estimator R T T ℓ max = D + ℓ max D − ℓ m a x , (2.7) where D _±(ℓ _max) quantify the sum of power contained in even (+) or odd (−) multipoles, up to ℓ _max . More specifically, D _±(ℓ _max) are defined as D ± ℓ max = 1 ℓ t o t ± ∑ 2 , ℓ m a x ± ℓ ℓ + 1 2 π C ℓ (2.8) where the + or − signs on the right refer to the fact that we include only even or odd multipoles in the sum, respectively, and ℓ t o t ± refers to the total number of multipoles in the sum. Figure 3 illustrates that CMB data in the multipole range of [2, 50] shows a clear preference for odd parity compared to the parity neutral, i.e., R ^TT (ℓ _max ) = 1, prediction of the standard model. Although this anomaly, as well as the anomaly in the lensing amplitude discussed in the next subsection, are not as severe as the previous ones due to their lower statistical significance (≲ 2σ), we will later argue that they may be related to the power suppression.

The cosmic microwave background radiation undergoes lensing by the intervening distribution of matter, as it propagates from the surface of last scattering to us. An important observable in the CMB, in addition to temperature and polarization, is the lensing potential. From the CMB maps, Planck has reconstructed the lensing potential and computed its power spectrum (Aghanim, 2018b). The effect of lensing is the smoothing of CMB power spectrum at small angular scales. The amount of smoothing observed in the CMB angular power spectrum should be consistent with the smoothing derived from the power spectrum of the lensing potential. In order to check this consistency, (Aghanim, 2018a), considered a test-parameter, known as the lensing parameter A _L , that multiplies the lensing power spectrum. Theoretically, the value of lensing parameter should be A _L = 1, and in fact Planck assumes this value during the process of parameter estimation. However, if A _L is left as a free parameter, along with the six parameters of the ΛCDM model, in the Markov Chain Monte Carlo (MCMC) analysis, one finds that A _L = 1.243 ± 0.096 for PlanckTT + lowE data, which is more than 2σ away from one. If the reconstructed lensing data is also used, along with Planck EE and TE data, then the lensing parameter is consistent with 1 within 2σ.

A key feature of the anomalies discussed above, except perhaps for the lensing anomaly, is that they appear clearly associated with the largest angular scales we can observe. This suggests a common origin in primordial physics for these diverse set of anomalies. The next section introduces a proposal for a mechanism that can provide such common origin, namely the phenomenon of non-Gaussian modulation. Together with the scale dependence introduced by the quantum bounce of LQC, this mechanism constitutes a promising candidate for the origin of the anomalies we have just described.

We are interested in computing the two-point correlation function of the curvature perturbation R K ⃗ for a mode k ⃗ that is observable in the CMB, in the presence of a longer wavelength mode R k ⃗ , when both modes are correlated.

A convenient and general way to model the effects of non-Gaussian correlations, is to write the curvature perturbations at a given time t in terms of a Gaussian field R G as follows (Schmidt and Kamionkowski, 2010) R K ⃗ ( t ) = R K ⃗ G ( t ) + 1 2 ∫ d 3 q ( 2 π ) 3 f NL ( q ⃗ , K ⃗ − q ⃗ ) R q ⃗ G ( t ) R k ⃗ − q ⃗ G ( t ) . (3.1)

The convolution in the integral is the Fourier transform of a quadratic combination of R G in position space, and is the source of the non-Gaussian character of R k ⃗ ( t ) , and the function f NL ( K ⃗ 1 , K ⃗ 2 ) contains the information about the strength and details of the non-Gaussianity. The goal of this equation is simply to parameterize the non-Gaussianity in a simple and tractable way, while the form of the function f NL ( K ⃗ 1 , K ⃗ 2 ) is expected to come from a concrete microscopic model of the early Universe.

Statistical isotropy and homogeneity implies that the function f NL ( K ⃗ 1 , K ⃗ 2 ) depends only on the modulus of the two wavenumbers involved, k 1 ≡ | k ⃗ 1 | and k 2 ≡ | K ⃗ 2 | , and on the (cosine of the) angle between them, μ: f NL ( K ⃗ 1 , K ⃗ 2 ) = f NL ( k 1 , k 2 , μ ) . From it, the three-point correlation function is given by ⟨ R K ⃗ 1 R K ⃗ 2 R K ⃗ 3 ⟩ = ( 2 π ) 3 δ ( K ⃗ 1 + K ⃗ 2 + K ⃗ 3 ) B R ( K ⃗ 1 , K ⃗ 2 , K ⃗ 3 ) , where the bispectrum B R ( K ⃗ 1 , K ⃗ 2 , K ⃗ 3 ) is B R ( K ⃗ 1 , K ⃗ 2 , K ⃗ 3 ) = f NL ( K ⃗ 1 , K ⃗ 2 ) [ P R ( K ⃗ 1 ) P R ( K ⃗ 2 ) + P R ( K ⃗ 2 ) P R ( K ⃗ 3 ) + P R ( K ⃗ 3 ) P R ( K ⃗ 1 ) ] , (3.2) and P R ( K ⃗ ) is the power spectrum of R G , defined as ⟨ R K ⃗ 1 G R K ⃗ 2 G * ⟩ = ( 2 π ) 3 δ ( K ⃗ 1 − K ⃗ 2 ) P R ( K ⃗ 1 ) . (3.3)

The dimensionless power spectrum is defined as P R ( K ⃗ ) = k 3 P R ( K ⃗ ) / 2 π 2 .

Our goal is to compute the two-point function of R K ⃗ in the presence of the spectator mode R K ⃗ . Using (Eq. 3.1), one obtains ⟨ R K ⃗ 1 R K ⃗ 2 ∗ ⟩ | R q ⃗ = ⟨ R K ⃗ 1 G R K ⃗ 2 G ∗ ⟩ + 1 2 ∫ d 3 q ′ ( 2 π ) 3 f NL ( q ′ ⃗ , K ⃗ 2 − q ′ ⃗ ) ⟨ R q ′ ⃗ G R K ⃗ 1 − q ′ ⃗ G R K ⃗ 2 G * ⟩ + 1 2 ∫ d 3 q ′ ( 2 π ) 3 f NL ( q ′ ⃗ , K ⃗ 2 − q ′ ⃗ ) ⟨ R k ⃗ 1 G R G q ′ ⃗ ∗ R G K ⃗ 2 − q ′ ⃗ ∗ ⟩ + O ( f NL 2 ) . (3.4)

In order to evaluate the impact of the spectator modes R K ⃗ G , it must be taken out of the statistical average ⟨ R K ⃗ 1 R K ⃗ 2 ∗ ⟩ | R q ⃗ = ( 2 π ) 3 δ ( K ⃗ 1 − K ⃗ 2 ) P R ( K ⃗ 1 ) + f NL ( K ⃗ 1 , − K ⃗ 2 ) 1 2 P R ( K ⃗ 1 ) + P R ( K ⃗ 2 ) R q ⃗ + ⋯ . (3.5) where the trailing dots indicate terms that are higher order in non-Gaussianity, and will be subdominant.

It is interesting to note the following facts about the above expression. First of all, non-Gaussianity leads to a modulation of the primordial power spectrum, and the strength of modulation depends on both the size and shape of f NL ( K ⃗ 1 , K ⃗ 2 ) , as well as the size of the spectator mode R q ⃗ . Secondly, statistical isotropy and homogeneity constrain the wavenumber of the spectator mode to be q ⃗ = K ⃗ 1 − K ⃗ 2 . In other words, this is the only mode that can affect the two-point correlation function between K ⃗ 1 and K ⃗ 2 . Additionally, the effect of the modulation is to introduce “non-diagonal” elements in the two-point function, i.e., terms not proportional to δ ( K ⃗ 1 − K ⃗ 2 ) . But recall that such non-diagonal terms break homogeneity and isotropy. It is not surprising that we see deviations from these fundamental symmetries, since we are not averaging over the spectator mode: such average would make those terms disappear, since ⟨ R q ⃗ G ⟩ = 0 . But, as it happens for the magnitude of the temperature anisotropies, the quantity that is more interesting for observations is the typical value of such term, and not only its statistical average.

The primordial perturbations R K ⃗ are related to the CMB multipole coefficients a _ℓm through the relation a ℓ m = 4 π ∫ d 3 k ( 2 π ) 3 ( − i ) ℓ Δ ℓ ( k ) Y ℓ m ∗ ( k ^ ) R K ⃗ , (3.6) where Δ _ℓ (k) are the CMB temperature transfer functions, which encode the post-inflationary evolution of the perturbations from the re-entry of perturbations into the horizon during late radiation domination till today. From this equation, one can compute the covariance matrix ⟨ a ℓ m a ℓ ′ m ′ ∗ ⟩ = ( 4 π ) 2 ∫ d 3 k 1 ( 2 π ) 3 ∫ d 3 k 2 ( 2 π ) 3 ( − i ) ℓ − ℓ ′ Δ ℓ ( k 1 ) Δ ℓ ′ ( k 2 ) Y ℓ m * ( k ^ 1 ) Y ℓ ′ m ′ ( k ^ 2 ) ⟨ R K ⃗ 1 R K ⃗ 2 ∗ ⟩ | R q ⃗ , (3.7) which is obtained from the two-point functions of the curvature perturbations given in (Eq. 3.5). Upon expanding f _NL in terms of Legendre polynomials, f NL ( k 1 , q , μ ) = ∑ L G L ( k 1 , q ) 2 L + 1 2 P L ( μ ) , and using the multipole expansion R q ⃗ G = ∑ L ′ M ′ R L ′ M ′ G ( q ) Y L ′ M ′ ( q ^ ) , one can write (Eq. 3.7) as (Agullo et al., 2021b) ⟨ a ℓ m a ℓ ′ m ′ ∗ ⟩ = C ℓ δ ℓ ℓ ′ δ m m ′ + ( − 1 ) m ′ ∑ L M A ℓ ℓ ′ L M C ℓ m ℓ ′ − m ′ L M . (3.8)

The above expression consists of two terms. The first term is the usual temperature power spectrum that is diagonal in ℓ and m. The second term arises from the non-Gaussian modulation and, as before, introduces non-diagonal terms. C ℓ m ℓ ′ m ′ L M are Clebsch-Gordan coefficients, and the information about the primordial non-Gaussianity is encoded in the coefficients A ℓ ℓ ′ L M = 4 ( 2 π ) 3 ∫ d k 1 k 1 2 d q q 2 ( − i ) ℓ − ℓ ′ Δ ℓ ( k 1 ) Δ ℓ ′ ( k 1 ) P R ( k 1 ) G L ( k 1 , q ) R L M G ( q ) × C ℓ 0 ℓ ′ 0 L 0 ( 2 ℓ + 1 ) ( 2 ℓ ′ + 1 ) 4 π ( 2 L + 1 ) . (3.9)

These coefficients are known as bipolar spherical harmonic (BipoSH) coefficients (Hajian and Souradeep, 2003; Joshi et al., 2010). As we shall see, the BipoSH coefficients provide a convenient way to organize the effects of the non-Gaussian modulation.

The Clebsch-Gordan coefficients present in the above expressions enforce certain properties on the BipoSH coefficients. In particular, Clebsch-Gordan coefficients C ℓ 1 , m 1 , ℓ 2 , m 2 L M are nonzero only if ℓ ₁ + ℓ ₂ ≥ L ≥ |ℓ ₁ − ℓ ₂| and if M = m ₁ + m ₂. This, together with properties of the Clebsch-Gordan coefficient C ℓ 0 ℓ ′ 0 L 0 , implies that, if

i. L = 0, then ℓ ₁ = ℓ ₂

Two remarks are in order now.

i. The strength of non-Gaussian modulation is dictated by the size of f _NL(k ₁, q, μ). But it is the dependence of f _NL on μ, the cosine of the angle between K ⃗ 1 and q ⃗ , what determines the relative size of the BipoSH coefficients for different L’s, i.e., the “shape” of the modulation. On the other hand, the dependence of f _NL on the moduli k ₁ and q determines the ℓ-dependence of the modulation. The two multipoles should not be confused: the L-dependence dictates the shape of the modulation, while the ℓ-dependence controls the variation of the amplitude of the modulation at different angular scales in the CMB. The non-Gaussianity generated in slow-roll inflation is small and nearly scale-invariant. Hence, the strength of modulation generated is also quite small. Since the anomalies observed in the CMB are scale dependent, we need a scenario with a strongly scale-dependent and large non-Gaussianity. Such scale dependence is also needed to explain why we have not observed non-Gaussian correlations directly in the CMB, since a strong scale dependence can make these correlations large only when at least one super-horizon mode is involved. In such situation, we could only observe the indirect effects that the non-Gaussian correlations induce in the CMB.

ii. A ℓ ℓ ′ L M given in (Eq. 3.9) depend on the mode R q ⃗ G . Since, R q ⃗ G is a random variable, we cannot predict the exact value of A ℓ ℓ ′ L M . We can only compute the standard deviation of the BipoSH coefficients, i.e. 〈 | A ℓ ℓ ′ L M | 2 〉 = 1 2 π ∫ d q q 2 P R ( q ) | C ℓ ℓ ′ L ( q ) | 2 1 / 2 × C ℓ 0 ℓ ′ 0 L 0 ( 2 ℓ + 1 ) ( 2 ℓ ′ + 1 ) 4 π ( 2 L + 1 ) , (3.10) where C ℓ ℓ ′ L ( q ) ≡ 2 π ∫ d k 1 k 1 2 ( i ) ℓ − ℓ ′ Δ ℓ ( k 1 ) Δ ℓ ′ ( k 1 ) P R ( k 1 ) G L ( k 1 , q ) . (3.11)

These are the typical values that the BipoSH coefficients are expected to take in the sky. If these values are large, the effects they entail should be expected in the CMB or, more precisely, they would have a large p-value and should not be considered anomalous.

Consider a spatially flat Friedmann-Lemaitre-Robertson-Walker spacetime. We shall describe the perturbations following the dressed metric approach. This approach has been discussed in (Agullo et al., 2012; Agullo et al., 2013a; Agullo et al., 2013b; Agullo and Morris, 2015; Agullo et al., 2018) [for a recent review, see (Agullo et al., 2017a)] and we refer the reader to these references for details omitted here. For the purpose of this article, it suffices to say that we consider perturbations as test fields propagating on the background described by the effective equations of LQC (Taveras, 2008; Ashtekar and Singh, 2011; Agullo et al., 2017a). The essential features of perturbations generated in LQC can be summarized using Figure 4. The left panel of this figure plots a | R / 6 | as a function of time, where a refers to the scale factor and R is the Ricci scalar. In making this plot, we have worked with a scalar field governed by a quadratic potential, and minimally coupled to gravity. Similar results are obtained for other potentials (Bonga and Gupt, 2016a; Bonga and Gupt, 2016b; Zhu et al., 2017). Different solutions to the effective equations of LQC with a scalar field as the dominant source are parameterized by the value of the scalar field at the bounce. As we further discuss below, different choices of this quantity translate to different amounts of cosmic expansion from the bounce to the end of inflation. The Ricci scalar attains its largest value at the bounce, and its maximum value sets a characteristic scale in LQC denoted by k L Q C ≡ a ( t B ) R ( t B ) / 6 ≈ a ( t B ) κ ρ B , where t _B indicates the time of the bounce and ρ _B is the energy density of the scalar field at the time of the bounce. As the inset in the plot shows, inflation occurs at late time, when a | R / 6 | grows exponentially fast. Regarding scalar perturbations, they are in an adiabatic regime before the bounce, and we choose them to start in an adiabatic vacuum at those early times [see e.g., (Agullo, 2015; Agullo and Morris, 2015; de Blas and Olmedo, 2016; Ashtekar and Gupt, 2017; Elizaga Navascués et al., 2019; Navascués et al., 2020; Martín-Benito et al., 2021) for other choices of initial state]. As perturbations evolve across the bounce, modes with wavenumbers k ≲ k _LQC are excited. These excitations get further amplified as they cross the curvature scale during inflation. Wavenumbers that are ultraviolet compared to k _LQC , k > k _LQC , are not excited during the bounce, and remain in the adiabatic vacuum at the onset of inflation. Hence, only for those modes one recovers the familiar Bunch-Davies vacuum at the onset of inflation, while more infrared modes keep memory of the bounce. Consequently, as shown in Figure 4, the power spectrum of curvature perturbations shows a strong scale dependence at infrared scales, while approaches the more familiar scale-invariant shape for large k’s. In particular, we see that the power spectrum for infrared modes k ≲ k _LQC is enhanced and oscillatory. In the extreme infrared limit, modes are neither excited during bounce nor during inflation, and this leads to a power spectrum which scales as k ². The scale at which these effects appear in the CMB depends on the physical size of the mode k _LQC today, compared to the Hubble scale [recall that the physical wavenumber scales with time as k _LQC /a(t)]. This depends on the expansion accumulated—i.e., the number of e-folds N—from the time of the bounce until the end of inflation. This is a free parameter in LQC. In this article, we investigate whether there is a value of N for which this model can explain the origin of the anomalies in the CMB.

The dressed metric approach was extended beyond linear perturbation theory in (Agullo, 2018), and we provide here a short summary. Primordial curvature perturbations whose wavenumbers are comparable to or smaller than k _LQC not only get excited, as described above, but also become non-Gaussian as they cross the bounce. The non-Gaussianity thus generated is further enhanced as the perturbations cross the horizon during inflation. Equal-time three-point functions are computed using time dependent perturbation theory, generalizing the pioneering calculations in (Maldacena, 2003) to bouncing geometries: 〈 0 | R ^ K ⃗ 1 t e R ^ K ⃗ 2 t e R ^ K ⃗ 3 t e | 0 〉 = i ∫ t i t e d t ′ 〈 0 | R ^ K ⃗ 1 t R ^ K ⃗ 2 t R ^ K ⃗ 3 t , H ^ i n t t ′ | 0 〉 , (4.1) where H ^ i n t is the interaction Hamiltonian [whose lengthy expression can be found in (Agullo et al., 2018)], t _i refers to the time at which initial conditions are imposed and t _e is the time at which the correlation is evaluated. Usually, t _e is chosen at the end of inflation, after all the three modes have crossed the Hubble radius. With the knowledge of the background dynamics and the initial conditions, we can exactly evaluate the three-point function and hence obtain the function f NL ( K ⃗ 1 , K ⃗ 2 ) which characterizes the non-Gaussianity. Our exact computations reveal that the non-Gaussianity generated in LQC is strongly scale-dependent, large and oscillatory, similar to the power spectrum. As for the power spectrum, the non-Gaussianity quickly approaches the inflationary result for wave numbers k > k _LQC (Agullo, 2018; Sreenath et al., 2019), and in particular they become negligibly small when the moduli of the three wave numbers K ⃗ 1 , K ⃗ 2 and K ⃗ 1 − K ⃗ 2 are larger than k _LQC , in such a way that they are too small to be observed directly in the CMB. However, the non-Gaussianity becomes large when at least one of the modes involved is infrared, k < k _LQC , or equivalently, when one of the modes has wavelength larger than the Hubble radius today. These are the correlations which can account for the CMB anomalies, as we argue in the next section.

The strong oscillatory character of the non-Gaussianity generated in LQC, makes it computationally difficult to obtain an exact evaluation of (Eq. 3.10). For this reason, in this work, rather than working with the exact numerically-evaluated non-Gaussianity, we shall work with an analytical approximation derived in (Agullo, 2018) f NL k 1 , k 2 , k 3 ≃ f NL e − α k 1 + k 2 + k 3 / k L Q C , (4.2) where α = 0.647, f NL ≈ 2750 , and k 3 = k 1 1 + k 2 2 k 1 2 + 2 μ k 2 k 1 . The value of α is determined from the behavior of the scale factor around the time of the bounce, while the amplitude f NL is determined from numerical simulations (Agullo, 2018). As showed in (Agullo, 2018), this expression provides a good approximation for the non-Gaussianity generated in LQC, and is significantly easier to manipulate. This approximation, however, neglects the oscillatory nature of f _NL(k ₁, k ₂, μ) with k ₁ and k ₂. The oscillations will generically reduce the size of the effects we describe below. Therefore, the numbers obtained in the next section should be understood as an upper bound for the predictions of LQC, rather than an exact result. This is the main technical limitation of our analysis, and it arises from the highly oscillatory nature of the perturbations.

We first consider the monopolar term (L = 0). The properties of the Clebsch-Gordan coefficients for L = 0 impose the constraints ℓ = ℓ′ and m = − m′. Therefore, the monopolar modulation introduces an isotropic shift in the value of C _ℓ , although the shift can be different for different values of ℓ. More concretely, the modulated power spectrum C ℓ m o d is given by C ℓ m o d = C ℓ 1 − − 1 ℓ C ℓ A ℓ ℓ 00 2 ℓ + 1 . (5.1)

Note that A ℓ ℓ 00 can be either positive or negative, leading to an enhancement or suppression of C ℓ m o d with respect to C _ℓ . As explained before, we cannot predict the exact value of A ℓ ℓ 00 . The interesting quantity is rather the root-mean-square value of the modulation: σ 0 2 ℓ = 1 C ℓ 2 〈 | A ℓ ℓ 00 | 2 〉 2 ℓ + 1 = 1 C ℓ 2 1 8 π 2 ∫ d q q 2 P R q | C ℓ ℓ 0 q | 2 , (5.2) where C ℓ ℓ 0 ( q ) was defined in (Eq. 3.11). This quantity determines the typical size and scale dependence of the monopolar modulation expected in the CMB. A large value of σ ₀ would make deviations from the unmodulated power spectrum, C _ℓ more likely to be observed in the CMB. The result of our calculations, using the power spectrum and the form of f _NL(k ₁, q, μ) described in the previous section, is plotted in Figure 5.

We will assume that the probability distribution for the modulation is well approximated by a Gaussian, and hence completely characterized by σ ₀(ℓ). This is a reasonable approximation, since the deviations are expected to be of second order in non-Gaussianity, and therefore very small. With this probability distribution for the monopolar modulation, we can now investigate the connection with the power suppression observed in the CMB. In particular, we want to answer the following question: what is the p-value given the observed value of S _1/2? We obtain that the probability to find S _1/2 ≤ 1,209.2 once the non-Gaussian modulation is taken into account is approximately 16%. This is equivalent to saying that the observed suppression is around one standard deviation from the mean. Figure 6 shows the form of the T-T power spectrum for a simulation for which the monopolar modulation produces S _1/2 in agreement with observation, along with the 1σ confidence contour arising from cosmic variance. For comparison, we provide the corresponding quantities arising from the standard model, as well as data from Planck (Aghanim, 2019).

These results show that, in presence of the LQC bounce occurring before inflation, a power suppression as the one we observe in the CMB should not be considered anomalous. It is important to emphasize the precise sense in which the suppression is explained: not because the theory predicts that we should observe a suppression in the CMB, but rather because the probability of observing such a suppression is much larger than in the standard ΛCDM model with Bunch-Davies initial conditions. In this sense, the resolution of the anomaly has precisely the same character as its origin: probabilistic.

An important check is to confirm that the non-Gaussian effects are not large enough to jeopardize the validity of the perturbative expansion on which the calculations rest. This question was explored in detail in Ref. (Agullo, 2018), confirming that, in LQC, perturbation theory does not break down when non-Gaussianity is included. Regarding the non-Gaussian modulation discussed in this paper, we find that the correction to the unmodulated angular power spectrum is not small, and it is in fact a significant fraction of the final result, particularly for the smallest multipoles. The relative contribution is, however, smaller than one in all our calculations. In quantitative terms, the relative contribution of the non-Gaussian modulation is of order f NL P R , which is smaller than one for f NL ∼ 1 0 3 . More importantly, higher order corrections introduce additional powers of the power spectrum P R ≪ 1 . So the next-to-leading-order correction to the non-Gaussian modulation is of order f NL ( P R ) 3 / 2 , which is negligible due to the smallness of P R . Therefore, our results are robust under the addition of higher perturbative corrections.

Next, we discuss the effects of the L = 1, dipolar modulation, induced by the BipoSH coefficients, A ℓ ℓ + 1 1 M , and compare the results with those reported by Planck. As discussed in section IIB, the Planck team quantifies the dipolar modulation in terms of a scale-dependent amplitude A ₁(ℓ) (Ade, 2016a), which can be related with the BipoSH coefficients A ℓ ℓ + 1 1 M as follows. First, define from A ℓ ℓ + 1 1 M the multipole coefficients m _1M by A ℓ ℓ + 1 1 M ≡ m 1 M G ℓ ℓ + 1 1 , w h e r e G ℓ ℓ + 1 1 ≡ C ℓ + C ℓ + 1 2 ℓ + 1 2 ℓ + 3 4 π 3 C ℓ , 0 , ℓ + 1,0 10 (5.3) is called a form factor. The m _1M (ℓ) defined above can take three values corresponding to M = −1, 0, +1, and in general, they depend on ℓ. From them, the amplitude of the dipolar modulation is defined as A 1 ℓ ≡ 3 2 1 3 π | m 1 − 1 | 2 + | m 1 0 | 2 + | m 1 1 | 2 . (5.4)

Hence, from the value of the root-mean-square of A ℓ ℓ + 1 1 M we can obtain the root-mean-square of A ₁(ℓ). It is given by the expression A 1 ℓ = 3 2 1 π 1 C ℓ m o d + C ℓ + 1 m o d 1 2 π ∫ d q q 2 P R q | C ℓ ℓ + 1 1 q | 2 , (5.5) where we have used the modulated (i.e., suppressed) C ℓ m o d since, as emphasized in (Ade, 2016a), the dipole amplitude must be evaluated relative to the observed angular power spectrum. Hence, the fact that the observed C ℓ m o d are smaller than the ones predicted by ΛCDM, increases the amplitude of the observed dipole. In this sense, the power suppression and the dipolar modulation are not completely independent. However, the amplitude of the dipole is ultimately dictated from the angular μ-dependence of the primordial non-Gaussianity f _NL(k ₁, k ₂, μ).

The result for A ₁(ℓ) is plotted in Figure 7. We find that the dipolar modulation is strongly scale-dependent, as a consequence of the scale-dependent nature of the non-Gaussianity. Although Planck observations for A ₁(ℓ) are limited, in the sense that only its mean value in the range ℓ ∈ [2, 64] is reported, the order of magnitude and scale dependence agree with our results.

We have also checked that higher order multipolar modulations, L = 2, 4, … have amplitudes significantly smaller than the dipolar one (Agullo et al., 2018), and therefore additional modulations are not expected in the CMB according to LQC, in agreement with observations. Hence, interestingly, the form of f _NL (k ₁, k ₂, μ) derived from LQC produces a hierarchy in the amplitude of the modulations which is dominated by a monopole, and a smaller dipole.

In this subsection we briefly discuss the results for the parity and lensing anomalies. As discussed in previous sections, the statistical evidence for these two features is weaker than the power suppression and the dipolar anomaly. Nevertheless, it is interesting to see what the predictions of LQC are.

We find that the monopolar modulation induces also a preference for odd parity multipoles ℓ, in agreement with observations. After inspection, this fact is not surprising, and it is a consequence of the simple fact that, in a power suppressed angular power spectrum, the sum of ℓ ( ℓ + 1 ) 2 π C ℓ m o d starting from ℓ = 2 is larger for odd multipoles, precisely because the sum starts at an even multipole—it would have been otherwise if the sum starts at ℓ = 1. Therefore, we find that in LQC there is a preference for odd-parity multipoles ℓ, as measured by R ^TT (ℓ _max), and the result is a consequence of the power suppression. We report our result in Figure 8. For comparison, we provide the corresponding values obtained in the ΛCDM model and the observations made by Planck (Aghanim, 2019). Although the result for R ^TT (ℓ _max) from LQC is closer to the data, the value of R ^TT (ℓ _max) observed by Planck is smaller than what we find in LQC, but the significance of the deviation is modest. In the absence of a better estimator for the parity anomaly, it is not possible for us to make a more precise comparison.

Yet another effect of the power suppression caused by the monopolar modulation is the alleviation of the lensing tension. The relation between a power suppression and the lensing anomaly was discussed in (Ashtekar et al., 2020), also in the context of LQC, and our analysis confirms the relation. The value of A _L is obtained from data by performing MCMC simulations involving the standard six free parameters, together with the lensing amplitude A _L . We repeat the analysis with the modified probability distribution obtained from LQC, using TT + lowE data (Aghanim, 2019), and find that the marginalized mean value of the lensing parameter is A _L = 1.20 ± 0.092. This value is 3.5% smaller than the result obtained from ΛCDM. This is a modest change. However, as shown in Figure 9, the joint probability distribution of τ-A _L , with τ the optical depth, shows that the value of A _L = 1 is within 2 standard deviations, and it is in this sense that the anomaly is alleviated. It should also be noted, however, that the marginalized mean value of the χ ² statistic, which is a measure of the difference between the predictions of the model and the data, scaled suitably by the expected error (Barlow, 1989), is larger for the modulated model by Δχ ² = 5.29. This lower value of the lensing parameter A _L can be explained due to the slightly larger value of τ. This is because a larger value of τ implies a slightly larger value of the scalar amplitude A _s , which in turn leads to a smaller value of A _L (Ashtekar et al., 2020).