Lensing anomaly and oscillations in the primordial power spectrum
LLensing anomaly and oscillations in the primordial power spectrum
Guillem Dom`enech ∗ and Marc Kamionkowski † Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at Heidelberg,Philosophenweg 16, 69120 Heidelberg, Germany Department of Physics & Astronomy, Johns Hopkins University,3400 N. Charles St., Baltimore, MD 21218, USA
The latest analysis of the cosmic microwave background by the Planck team finds more smooth-ing of the acoustic peaks in the temperature power spectrum than predicted by ΛCDM. Here weinvestigate whether this additional smoothing can be mimicked by an oscillatory feature, generatedduring inflation, that is similar to the acoustic peaks but out of phase. We consider oscillationsgenerated by oscillating modulations of the background—e.g., due to heavy fields or modulatedpotentials—and by sharp features. We show that it is difficult to induce oscillations that are linear(or almost linear) in k by oscillatory modulations of the background. We find, however, that a sharpbumpy feature in the sound speed of perturbations is able to produce the desired oscillations. Thescenario can be tested by combining CMB and BAO data. I. INTRODUCTION
As cosmic microwave background (CMB) photons travel towards us, their trajectories are deflected bythe gravitational potentials generated by the matter distribution. This weak lensing of the CMB has animpact on the CMB temperature power spectrum [1, 2]. The lensing magnifies the angular size of theprimordial fluctuations in some places on the sky and de-magnifies others. The observed peak structure inthe temperature power spectrum, when measured over the entire sky, are therefore blurred [3]: the acousticpeaks are reduced slightly, and the troughs between them filled in.Interestingly, when the theoretical prediction for this smoothing is compared with the Planck data, it isfound that the lensing smoothing is larger than expected by roughly 10% [4]. The so-called A L anomaly is persistent and recently slightly more statistically significant, with a value A L = 1 . ± .
072 (68%confidence), that constitutes a 2 σ tension with ΛCDM cosmology [4]. Moreover, the residuals between thesignal and the theoretical prediction yield an oscillatory pattern whose frequency is roughly linear in themultipole number (cid:96) and similar in shape to the acoustic peaks.If the tension persists with higher statistical significance, it might be explained by some new physics thatmimicks the smoothing effect of lensing. One possibility discussed by the Planck collaboration [4] is thatthere might be a component of cold dark matter isocurvature (CDI) perturbation with a blue tilt. Since theacoustic peaks of the CDI will have the opposite phase, this will effectively smooth out the photon acousticpeaks. A similar mechanism was also studied in Refs. [6, 7] where the isocurvature perturbations of darkmatter and baryons compensate each other. However, these models are tightly constraint by their effects onthe trispectrum [7]. Another possibility is that there are oscillations in the primordial power spectrum whichhave the same frequency but opposite phase with the acoustic peaks [4]. However, in an analysis where theoscillatory feature in the power spectrum has a k independent amplitude and a frequency linear in k , nocorrelation between the amplitude of the oscillations and A L is found [8]. Moreover, one reaches similarconclusions when reconstructing the power spectrum from raw data [9, 10]. In particular, it was alreadypointed out in Ref. [9] that there is a degeneracy between the effects of lensing and oscillating features in ∗ [email protected] † [email protected] A L parametrizes a rescaling of the lensing power spectrum such that A L = 1 for ΛCDM [5]. a r X i v : . [ a s t r o - ph . C O ] D ec the primordial power spectrum. On the theoretical side, though, it is not clear whether physical modelsthat might induce wiggles in the primordial power spectrum are required to do so with a scale-independentamplitude, nor with a precisely linear dependence on k . Previous fits to the residuals of the temperaturepower spectrum were pursued in Refs. [11–15] but a mimicking of the lensing effect from an inflationarymodel was not studied. Note that the reconstruction of the primordial power spectrum taking into accountlensing also yields an oscillating feature [10]. However, it is not clear what kind of inflationary model wouldgive rise to such specific wiggle [10].Here we explore inflationary models that might give rise to oscillatory features in the primordial powerspectrum that might account for the A L anomaly. Oscillatory features generated during inflation usuallyhave an oscillation frequency which has a logarithmic or linear dependence on k (see Ref. [16] for a review).On one hand, the logarithmic dependence could be either because there is an oscillating modulation in theLagrangian that depends linearly on the inflaton which is slowly rolling [17, 18] or because an extra massivefield is oscillating around its minimum in which case it oscillates with a constant frequency and linearly inthe cosmic time [19–21]. Also, successive turns in the multi-field inflationary trajectory yield a logarithmicdependence in k [22]. On the other hand, a frequency linear in k is typical from sharp transitions, e.g. steps inthe potential or sudden turns in the field space, with a damped amplitude depending on the sharpness of thetransition [11, 22–35]. Note that sharp transitions can be understood in terms of Bogoliubov transformations[28], since the negative frequency mode has been excited by the turn or step. It is also interesting to note thatboth logarithmic and linear k dependences of the frequency may be related to a trans-planckian modulation[36]. There is another interesting case where the frequency goes as a power-law of k [37, 38]; this mayoccur, e.g., when there is a background oscillation with a time dependent frequency. See Refs. [19, 39] for apower-law dependence in k in alternatives scenarios to inflation.This paper is organized as follows. In Sec. II we review the A L anomaly and the requirement for theoscillatory patterns in the power spectrum to mimick the lensing smoothing. In Sec. III, we study whichfeatures could potentially yield such oscillatory patterns and we conclude in Sec. IV. We discuss possiblemodels in App. A and present details of the calculations in Apps. B and C. II. A L ANOMALY
The effect of weak lensing onto the CMB power spectrum is to smooth out the acoustic peaks by blurringthe acoustic-peak structure in (cid:96) space (see Fig. 1). Using the modelled unlensed CMB power spectrum andthe lensing potential, one can estimate the magnitude of the smoothing effect of lensing A L [2, 5]. However,if nothing more than a power-law inflationary spectrum of adiabatic perturbations and ΛCDM are assumed,then the observed power spectrum has been lensed 10% more than expected [4].This tension could conceivably by explained by an oscillatory modulation of the inflationary power spec-trum which is out of phase with the acoustic peaks. To illustrate this, we introduce the following fitting tothe inflationary power spectrum [4, 19, 37, 38]:∆ P R P R , = A (cid:18) kk ∗ (cid:19) n A sin (cid:20) ω (cid:18) kk ∗ (cid:19) n o + ϕ (cid:21) (2.1)where the constants A , k ∗ , ω , ϕ , n A and n o respectively are the amplitude, the pivot scale, the frequency,the phase and the power indexes of the k in amplitude and frequency of the oscillation. These constants areultimately related to parameters of a theoretical model. For example, the fitting form Eq. (2.1) appears in The lensing power spectrum can also be reconstructed from the CMB temperature and polarization power spectra data alone[40] and the result is compatible with A L = 1 (see Fig. 3 of [4]). This further motivates us to look for an extra effect in theprimordial power spectrum which mimicks the smoothing effect of lensing. ‘ D TT ‘ [ µ K ] unlensedlensed − . − . − . − . − . . . . log[ k/k pivot ] − . − . − . − . l og [ P ( k ) ] P ( k ) P ( k ) + ∆ P ( k ) FIG. 1:
On the left, CMB temperature power spectrum. We respectively show the unlensed and lensed powerspectrum in blue and green. See how the acoustic peaks are smoothed out. On the right, the oscillatory featurelinear in k in the primordial power spectrum vs the usual power-law spectrum. sudden transitions [11, 22–27, 29–35], oscillating heavy fields [19, 37, 38] and trans-planckian modulations[36] during inflation. Now, we translate it to the multipole number for a rough comparison as∆ C (cid:96) C (cid:96), = A (cid:18) (cid:96)(cid:96) ∗ (cid:19) n A sin (cid:20) ω (cid:18) (cid:96)(cid:96) ∗ (cid:19) n o + ϕ (cid:21) , (2.2)where C (cid:96), is the unlensed power spectrum, we used the relation (cid:96) ∼ kD A ( D A ≈ (cid:96) ∗ ∼
814 whichcorresponds to k ∗ ∼ . − . To provide a rough fit to the acoustic peaks, we first focus on thefrequency of the oscillations and normalize the amplitude to unity. Then we will fit the frequency andphase by eye as we are only interested in the general behaviour. A best fit using CMB data will be studiedelsewhere. Since the maxima and minima do not exactly match a sinusoidal function linear in (cid:96) ( n o = 1), weexplore two more possibilities: the power-law index of the frequency n o is either n o > n o < n o will be important in Sec. III when we discuss the possible models as not all models areable to reproduce an exact linear behavior, that is n o = 1. We limit ourselves to the case of n o = 1 (constantfrequency) or very close to it. We leave for future studies different values of n o in which the oscillation mayonly fit few consecutive peaks.We computed the effect of the feature (2.1) in the primordial power spectrum onto the lensed CMBtemperature power spectrum using CLASS [41, 42]. We chose H = 100 h km / s / Mpc, h = 0 . T CMB =2 . b h = 0 .
02, Ω
CDM h = 0 . N eff = 3 . K = 0, Ω Λ = 0 .
69 and A L = 1. For the mainpower spectrum we took a power-law spectrum P R , ( k ) = A s ( k/k pivot ) n s − with A s = 2 . · − , n s = 0 . k pivot = 0 . − . For the oscillating feature we use the template (2.1) with the values presented inTable I. In order to numerically implement a scale dependent amplitude with CLASS we have introducedan artificial cut-off at k c in the power spectrum, otherwise the spectrum eventually blows up for n A (cid:54) = 0, sothat our power-spectrum reads P ( k ) = P R , ( k ) + ∆ P R ( k ) × (1 + tanh [ β log( k/k c )]) / , (2.3)and we respectively used k c = 0 .
001 Mpc − ( (cid:96) ∼
14) and β = 10 for n A < k c = 1Mpc − and β = − n A >
0. For the oscillations in k , we further chose n o = 1, ω = 16 . A = 0 .
01, except for the
500 1000 1500 2000 2500 ‘ A m p li t ud e n o = 1 n o = 0 . n o = 1 . C ‘ max/min position Frequency ( ω ) Power-law index ( n o ) Phase ( ϕ ) . . . π . . . . . . π FIG. 2 & TABLE I:
Left: Oscillatory modulation of the power spectrum with a constant amplitude. In blue wesee the fit for n o = 1, in light blue the one for n o = 1 . n o = 0 .
9. The red crosses are the positionsof the maxima and minima of the acoustic peaks with the amplitude normalized to an arbitrary constant. See howeven though n o = 1 offers a fairly good fit, the values of n o = 1 . n o = 0 . (cid:96) ∗ = 814. Wenote that the value of ω = 16 . n o = 1 at k = 0 . − agrees with the value ω = 14 . k = 0 .
05 Mpc − . Right: Table containing the fit by eye values for template (2.1) used in Figure 2 on the left, thephase already being corrected by π/ illustrative case where we considered A = 0 .
1. It should be noted that this artificial cut-off introduced inthis section will not be necessary when we study concrete inflationary models. The results can be seen inFig. 3. On the left, we plotted an illustrative case with A = 0 . n A = {− , , } . As one can see the residuals for n A = 1 arethe ones that best resemble the residuals from the Planck 2018 analysis [4], in particular for (cid:96) > n A > III. FEATURES DURING INFLATION
In this section, we review the computation of the primordial power spectrum when there is a feature,e.g., sharp transition or oscillations, in the background evolution. In fact, such features are rather commonin extensions of the simplest models of inflation. For example, it could be due to wiggles in the inflatonpotential, inspired from axion monodromy in string theory [18, 44], or sudden turns in the trajectory infield space in multi-field inflation [22], which would excite the heavy modes. These features will affect thebackground dynamics during inflation and the effects will be imprinted in modifications of the slow-rollparameters and/or the sound speed of perturbations. Now, for simplicity, we take an effective single fieldapproach [33, 35, 45–47] and we study the resulting oscillatory modulation of the power spectrum. Our ‘ D TT ‘ [ µ K ] × no osc.osc. 500 1000 1500 2000 ‘ − − ‘ ∆ D TT ‘ [ µ K ] × n A = 0 n A = − n A = 1 FIG. 3:
CMB lensed power spectrum and residuals for ω = 16 . n o = 1 and k ∗ = 0 . − ( (cid:96) ∗ = 814). On theleft we compare the lensed power spectrum with and without the oscillations, respectively in green and red. Wechose n A = 0 and A = 0 . n A = {− , , } with A = 0 .
01 so that all of them have the same amplitude at (cid:96) ∗ . For easiercomparison, we have included in lower opacity Fig. 24 from Ref. [4] as a background. In that image, we see in blackthe residuals of the ΛCDM model and the gray lines indicate the 1, 2, 3 σ countours. The green shadows refer todifferent values of Ω m h which are not relevant for the present discussion. The red-orangish dotted line is theremaining residuals if there were 10% more lensing. Compare our results (blue, green and purple) with the Planckresiduals. Note how in the range (cid:96) ∼ − n A >
0, since the peaks at large (cid:96) do not decay that fast. starting point is then the Mukhanov-Sasaki equation for the canonically normalized curvature perturbation[48, 49]: u (cid:48)(cid:48) k + (cid:18) c s k − z (cid:48)(cid:48) z (cid:19) u k = 0 , (3.1)where z ≡ a (cid:15)M /c s , a is the scale factor, (cid:15) ≡ − ˙ H/H , H ≡ ˙ a/a , c s is the sound speed of propagation,˙ ≡ d/dt where t is the cosmic time, (cid:48) ≡ d/dτ where dτ = dt/a is the conformal time and u ≡ z R with R being the comoving curvature perturbation. We then consider the effect of a deviation in a de-Sitterinflationary background by introducing [11] v k ≡ (cid:112) kc s u k and f ≡ πzc / s ξ (3.2)where dξ = c s dτ . With these redefinitions, Eq. (3.1) becomes [11] dv k dξ + (cid:18) k − ξ (cid:19) v k = 1 ξ f (cid:18) d fd ln ξ − dfd ln ξ (cid:19) v k . (3.3)Treating the right hand side as a perturbation one can solve the differential equation by the Green’s functionmethod at leading order in f by [11, 25, 35, 50, 51]∆ P R ( k ) P R , = (cid:90) ∞−∞ d ln ξ W ( kξ ) (cid:18) d ln fd ln ξ − d ln fd ln ξ (cid:19) (3.4)where we already used the dS approximation, i.e. v k = (cid:16) − ikξ (cid:17) e − ikξ , we defined P R ( k ) ≡ k π | v k | where P R , ( k ) ≡ π H (cid:15)c s M , (3.5)and W ( kξ ) ≡ (cid:20) sin 2 kξ ( kξ ) − kξ ( kξ ) − sin 2 kξkξ (cid:21) . (3.6)This will be our starting point in the following discussions. Any feature during inflation will be contained inthe function f ( ξ ). Thus, once we know the type of feature we can compute its effect in the power spectrumby using Eq. (3.4) through modifications of the slow-roll parameters or the propagation speed. Interestingly,one can invert this relation and find the feature given a power-spectrum modulation as in Ref. [50] (alsosee Ref. [52] for a more recent approach). At this point, we could find the change in the background thatwould lead to the desired feature in the power spectrum. However, we will be more interested in the physicalmodel behind. We will analytically compute three different regimes: ( i ) fast oscillating features, ( ii ) slowoscillating features and ( iii ) sharp features. Here we do not seek to join any of these three regimes, ratherwe are interested to see if the desired oscillations in the power spectrum fall in any of these three categories. A. Fast Oscillating feature
We begin to review the effects of an oscillating modulation of the background where its frequency is higherthan the expansion rate. This could be either induced by an oscillatory modulation of the inflaton’s potentialor by the oscillations of an extra massive field. For the moment, we will assume that the oscillations vary inamplitude and frequency and that c s = 1. Thus, in practice we have that∆ P R ( k ) P R , = 23 (cid:90) ∞−∞ d ln τ W ( kτ ) ∆ ( τ ) H (3.7)where we assumed that the frequency of the oscillation, say Ω, in the function f is ˙Ω /H = δ Ω Ω (cid:29) τ ) H ≡ d ln fd ln τ = − C ( τ ) cos [Ω( τ ) + ϕ ] , (3.8)with ϕ being an arbitrary phase. For simplicity, we will further assume that C ( τ ) = C r (cid:18) aa r (cid:19) δ C and Ω( τ ) = Ω r (cid:18) aa r (cid:19) δ Ω , (3.9)where a r is the scale factor at onset of the resonance, C r , Ω r , δ C , δ Ω are constants and we require ˙Ω /H = δ Ω Ω (cid:29)
1. Then, we can use the saddle point approximation for subhorizon scales ( kτ (cid:29)
1) in Eq. (3.7) at k = δ Ω aH Ω to find that the correction to the power-spectrum is given by∆ P R ( k ) P R , ≈ √ π C r ( | δ Ω | Ω r ) − / (cid:112) | − δ Ω | (cid:18) kk r (cid:19) δC − δ Ω / δ Ω sin Ω r (1 + δ Ω ) (cid:18) kk r (cid:19) δ Ω1+ δ Ω + ˜ ϕ (3.10)where k r = δ Ω Ω r a r H r and ˜ ϕ = ϕ ∓ π where − is for δ Ω > δ Ω < − . − . − . − . − . − . − . log( k [Mpc − ]) − ∆ P R / P R , × − δ Ω Ω ∼ δ Ω Ω (cid:28) δ Ω Ω (cid:29) n o = 1 . n o = 0 . n o = 1 . ‘ − − ‘ ∆ D TT ‘ [ µ K ] × δ Ω Ω ∼ δ Ω Ω (cid:29) n o = 0 . n o = 1 . FIG. 4:
Left: Power spectrum modulation for the fast ( δ Ω Ω (cid:29)
1) and slow ( δ Ω Ω (cid:28)
1) oscillating regimesseparated by an intermediate regime where numerical computations are needed (from δ Ω Ω ∼ / ∼ δ Ω = 9 (green line), δ Ω = −
11 (dashed blue line) both cases with δ C = 3 δ Ω / n A = 0 and n o = 0 . n o = 1 . δ Ω = 1 and δ B = 0 in Eq. (3.18) which corresponds to n A = 0 and n o = 1 in Eq. (2.1). Inall cases we used Tab. I for the values of the frequency and phase. Right: Residuals of the lensed power spectrumfor the same cases than in the left figure. For comparison, we included Fig. (24) of Planck analysis [4]. In black wehave the residuals of the ΛCDM model, gray lines are the 1, 2, 3 σ countours and the green shadows refer todifferent values of Ω m h . The red-orangish dotted line is the remaining residuals if there were 10% more lensing. In order to compare with the data, we rewrite the parameters in Eq. (3.10) in terms of the templateEq. (2.1). Comparing them at a scale k = k ∗ we find that for the frequency we haveΩ r = ω | − n o | (cid:18) k r k ∗ (cid:19) n o , δ Ω = n o − n o , (3.11)and for the amplitude C r = A √ π ( ωn o ) / (cid:18) k r k ∗ (cid:19) n o / (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) − n o − n o (cid:12)(cid:12)(cid:12)(cid:12) , δ C = n A + 3 n o / − n o . (3.12)First of all, we see that an exact linear dependence in k for the frequency, as needed to fit all the acousticpeaks, is not possible: as n o → δ Ω → ±∞ . This is because one would need that the resonancewith every mode function, which has a frequency of 2 kξ , occurred at the same time for all the modes, i.e. at ξ = ξ r . Nevertheless, the best one can do without considering a sharp feature is that the oscillating sourceterm oscillates so fast that the resonances occur almost at the same time. For this reason, we may considerthat n o = { . , . } and the fit is still reasonable, see Fig. 4. However, while they might be give feasibleresiduals to mimick the smoothing effect of lensing, they are unsatisfactory from a theoretical point of view.First of all, it must be seen whether the assumption δ Ω Ω (cid:29) δ Ω Ω r = n o ω (cid:18) k r k ∗ (cid:19) n o ≈ . × − n o ω . (cid:18) k r − k ∗ (cid:19) n o , (3.13)where we used the fact that n o ≈
1. We see that, using the values of the frequencies in Tab. I, the condition δ Ω Ω (cid:29) k ∼ . (cid:96) ∼
49) and so this kind of resonance could explain the anomalydown to small (cid:96) . However, to be able to predict what occurs for (cid:96) <
49, as it enters the observational window,we need a full detailed model. For instance, it would difficult to imagine a model where the modulationsuddenly started at k ∼ . n o = { . , . } in Eq. (2.1) which correspond to δ Ω = { , − } in Eq. (3.10). This implies that thevalue of the frequency respectively increases or decreases 4 orders of magnitude in 1 e-fold. In this respect,it is difficult to conceive what kind of model could sustain such growth or decay for more than the 3 e-foldsrequired. For example, in models where the feature is generated by an extra oscillating massive field like inRefs. [19, 37, 38], the frequency of the oscillation is associated to the mass of the field. In this case, eitherthe energy density of the field increases too fast with the mass ( δ Ω = 9) or it decreases so fast ( δ Ω = − A L anomaly.Nevertheless, it must be seen whether relaxing the assumption that the frequency be almost constant helpsfinding a viable model which resembles the lensing effect in a narrower multipole number range. B. Slow Oscillating feature
In this subsection, we study the opposite case where the oscillating modulation is slowly varying. Thus,this feature will simply act as a modulation of the background and we can estimate its effects onto the powerspectrum using the δN formalism [54–56]. We have δN = ∂N∂φ δφ + ... (3.14)where ∂N∂φ = − H ˙ φ is to be evaluated at horizon crossing. In general one has that (cid:104) δφδφ (cid:105) = H/ (2 πc s ) and sothe power spectrum is given by P R = (cid:18) ∂N∂φ (cid:19) (cid:104) δφδφ (cid:105) = 18 π H (cid:15)c s M . (3.15)The effect of any slow varying modulation of the background can be computed in this way, evaluated athorizon crossing k = aH . Now, let us assume that the modulation of the background results in a modulationof the slow-roll parameter given by (cid:15) = (cid:15) (1 − B ( τ ) sin [Ω( τ ) + ϕ ]) . (3.16)Assuming for simplicity that B ( τ ) = B r (cid:18) aa r (cid:19) δ B and Ω( τ ) = Ω r (cid:18) aa r (cid:19) δ Ω , (3.17) Since we are interested in the super-horizon limit, it could also be estimated from Eq. (3.1) or (3.3) with the approximationthat z is almost constant and then matching the solution at horizon crossing [53]. This means that (3.4) is unnecessary forthe slow oscillating feature; ∆ P R is defined as difference between P R , with and without oscillation. where B r , Ω r , δ B and δ Ω are constants, together with the requirement that δ Ω Ω (cid:28)
1, we find that themodulation of the power-spectrum reads∆ P R ( k ) P R , ≈ B r (cid:18) kk r (cid:19) δ B sin (cid:34) Ω r (cid:18) kk r (cid:19) δ Ω + ϕ (cid:35) . (3.18)Comparing this result with the template (2.1) we findΩ r = ω k r k ∗ , δ Ω = n o , B r = A and δ B = n A . (3.19)This time, we have that for δ Ω = n o = 1 at k r = 10 − k ∗ , the oscillations are slowly varying, i.e. δ Ω Ω r ≈ . × − . However, since the frequency is increasing with time as a power-law of a , the approximation of δ Ω Ω (cid:28) (cid:96) ∼
49. This is clearly not enough to explain the lensing anomaly at (cid:96) ∼ − δ Ω Ω (cid:28) δ Ω Ω (cid:29) C. Sharp feature
When one considers a sharp feature, the exact shape of the modulation is very model dependent [22,23, 27, 29–33, 35, 47, 51]. Nevertheless, let us consider the simplest example where the sharp feature is adiscontinuity, e.g. a step in the slope of the potential, which will result in a Dirac delta δ ( ξ − ξ f ) in Eq. (3.4),as the slow roll parameter (cid:15) in f ( ξ ) is proportional to the first derivative of the potential. In that case, thefrequency of the resulting oscillation will be proportional to 2 kξ f [25, 50, 51]. A quick exercise tells us thatif we require ω = 16 . k ∗ ξ r ≈ .
3, that is the transition happened 2 e-folds beforeour pivot scale at around k f ≈ . − or (cid:96) r ≈
98. Furthermore, the phase of the oscillation dependson whether the step around ξ = ξ f is odd (e.g. a hyperbolic tangent) [23, 29, 35, 51] or even (e.g. a gaussianbump) [11]. To understand that it is useful to integrate by parts Eq. (3.4) arriving at∆ P R ( k ) P R , = − (cid:90) ∞−∞ d ln ξ (cid:18) W ( kξ ) + 13 dW ( kξ ) d ln ξ (cid:19) d ln fd ln ξ . (3.20)If the step is sharp enough only the neighborhood of the transition will contribute to the integral. If the stepis odd around ξ = ξ f , its derivative is even and so the even function cos(2 kξ f ) survives asymptotically in k .Instead, if the step is even around ξ = ξ f , then its derivative is odd and the odd function sin(2 kξ f ) remains.We present now a concrete example. The simplest case is a bump in the sound speed at τ = τ f with height B and sharpness β s given by [11] c s = 1 + B e − β s log [ τ/τ f ] . (3.21)Note that if B > Although it does notpose any causality problems [57], it may obstruct the UV completion of a quantum Lorentz-invariant theory[58]. Nevertheless, the superluminality could be compensated by introducing a c s, as a common factor inEq. (3.21) with (1 + B ) c s, ≤
1. Now, integrating Eq. (3.20) under the assumption that the step is sharp( β s (cid:29)
1) yields [11] (see also App. B)∆ P R ( k ) P R , = B √ π kτ f β s e − k τ fβ s (cid:26) sin(2 kτ f ) + cos(2 kτ f ) kτ f −
12 sin(2 kτ f )( kτ f ) (cid:27) , (3.22) In Ref. [11] it is assumed that
B <
B > .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 . k [Mpc − ] − . − . . . . ∆ P R / P R , × −
500 1000 1500 2000 ‘ − − ‘ ∆ D TT ‘ [ µ K ] × FIG. 5:
Power spectrum modulation (3.22) (left) and residuals of the lensed power spectrum (right) for a bump(3.21) in c s with height B = 0 .
015 and sharpness β s = 25. Note that on the right the residuals are similar to thosein the Planck analysis [4] Fig. (24), which we included for easier comparison. Again, we have in black the residualsof the ΛCDM model, the gray lines are the 1, 2, 3 σ countours and the green shadows refer to different values ofΩ m h . The red-orangish dotted line is the remaining residuals if there were 10% more lensing. Compare our result(blue) with the Planck residuals (black and red-orangish). Note also that although the frequency of our resultingoscillations have a similar pattern, the amplitude of the Planck residuals are shifted upwards compared to our result. where we neglected terms O ( β − s ) and note that ∆ P R → k → kτ f = β s / √ B ≈ . c s [59], is barely affected as the sound speed only changesby 10%. Furthermore, the adiabatic condition s ≡ ˙ c s /Hc s is always satisfied and its maximum value is s = Bβ s / √ e ≈ .
16 (for B = 0 .
015 and β s = 25). We have plotted the oscillatory feature in the primordialpower spectrum and the residuals of the lensed power spectrum in Fig. 5. Note how the frequency of theresulting oscillatory pattern follows that of Planck [4] in the range (cid:96) ∼ − Kτ f = √ β s with amplitude f eq NL ≈ − . Bβ s . (3.23)A similar calculation for the trispectrum evaluated at its peak ( Kτ f = √ β s ) in the equilateral configuration(see App. C) yields g eq NL ≈ − . Bβ s . (3.24)The Planck results on non-gaussianities [60] yield that f eq NL = − ±
43. Thus, if B ∼ − we see that weneed β s <
60 to fall within the bounds. Regarding local (squeezed shape) non-gaussianity its magnitude is atleast suppressed by 1 /β s with respect to f eq NL [61, 62] and therefore we easily fall within Planck constraints,i.e. f loc NL = 0 . ± . Note that these constraints are looser if one allows for a scale dependence in the For instance, using the results from [61] we have that f loc NL = − dd ln k ∆ P R P R , ≈ − β s ∆ P R P R , and plugging in the same numbers,i.e. B ∼ − and β s <
60 we have that f loc NL (cid:46) − . g NL < − andfor B ∼ − it is sufficient that β s < c s with B = 0 . β s = 25 reproduces quite well the residuals in the Planck analysis [4] and is well within the bispectrumand trispectrum bounds. The development of a specific model that can produce such a bump in c s is leftfor future work, although it seems possible to build such a model using a spectator scalar field [63]. Beforeending this section, it is worth saying that no trans-planckian modulation [36] could mimick the smoothingeffect of lensing. This is due to the fact that the value of the frequency for trans-planckian modulationsdepends on the initial conditions but the frequency required to explain the anomaly correspond to a scalein the last few e-folds of inflation. IV. CONCLUSIONS
The latest analysis of the cosmic microwave background by the Planck team [4] suggests that at the 2 σ confidence level there is 10% more lensing than predicted by ΛCDM. If not a statistical fluke, one suggestedexplanation for the extra lensing is that there is new physics that mimicks the smoothing effect of lensing[8]. Here we studied what could have generated these oscillations in the power spectrum during inflation.We first considered an effective single field approach, where the effects of a sharp transition or an oscillatorymodulation in the background can be studied phenomenologically [33, 35, 46, 47]. In this way, we divided theanalysis between rapid/slow (compared to the expansion rate) oscillatory modulations and sharp transitions.We have found that for rapid oscillatory modulations [19, 37, 38], it is not possible to obtain an exact linear k dependence in the frequency of the power-spectrum’s oscillations since the modulation should oscillateinfinitely fast or be a sharp feature. Nevertheless, an almost linear dependence can be obtained for very fastoscillatory modulations. Unfortunately, when compared with the data one needs a frequency which is slowlyvarying for large scales ( (cid:96) <
50) and rapidly varying for (cid:96) >
50. We also showed that if the oscillations werecaused by an oscillating heavy field, then the mass of the field would have been smaller than Hubble at somepoint in the range of interest. Thus, this sort of feature cannot explain an oscillation over the whole rangeof (cid:96) covered by the Planck data. We discussed that the possibility of starting the oscillation at (cid:96) >
50 isnot feasible since it would be accompanied by a sharp feature which is normally larger than the oscillatoryfeature.On the other hand, we have analyzed the case of slowly oscillating modulations of the background andwe have found that it is possible to find a model where the frequency of the oscillatory feature is linear in k . In this case, there is no resonance occurring and so the frequency must evolve inversely proportional tothe conformal time so that at horizon crossing ( − kτ ≈
1) yields a linear dependence in k . However, whencompared with the data and in agreement with the results of fast oscillatory modulations, this feature couldonly explain a linear oscillation for (cid:96) <
50 which is not of interest for our work.Motivated by our results, we have studied sharp transitions within an effective single field theory for sharpfeatures [33, 35, 46, 47]. When the feature is sharp all the modes are excited at the same time (say τ = τ f )and so the resulting oscillatory feature has a frequency of 2 kτ f . If that is the case, we needed that the sharpfeature occurred at scales inside the observational window, around (cid:96) ∼
98. Although sharp features arevery model dependent [22, 23, 27, 29–33, 35, 47, 51], we see that in general terms when the sharp feature iseven [11], e.g. a bump, the oscillations with the right frequency are out of phase with the acoustic peaks.We have presented an example capable of reproducing the desired oscillatory modulation of the primordialpower spectrum times a damping function (3.22). This example consists of a bump in the sound speed givenby Eq. (3.21). Moreover, we have shown that this model can satisfy the bounds to the bispectrum andtrispectrum.We thus conclude, on one hand, that the A L anomaly in the CMB temperature power spectrum couldpotentially be explained by a bump in the sound speed of scalar perturbations, although a detailed comparisonwith the data would be needed. We presented the residuals in Fig. 5 and they are similar in frequency to the2results presented in Ref. [4]. In the future, measurements of baryon acoustic oscillations might be employed incombination with the CMB [64] to test this explanation. In the standard scenario, the Fourier wavenumbersfor the peaks in the late-time matter power spectrum are shifted relative to those for peaks in the radiationdensity at CMB decoupling [65], a result of the fact that the late-time growing mode maps at early timesto a combination of the growing and decaying modes. The relative phases of the acoustic and primordialoscillations will thus be different in the baryon acoustic oscillations (BAO) than they are in the CMB. Itwill be interesting do this analysis with high precision polarization data such as CMB-S4. Furthermore,another probe of this model would be to look for correlated features in the primordial spectra [61, 66–72].On the other hand, we have shown also that it is difficult that oscillating features in the power spectrumwhich are linear in k (or almost linear) are generated during inflation from an oscillatory modulation of thebackground and that could explain the A L anomaly at the same time. However, it has to be seen if there isany possibility for general multi-field inflationary trajectories as in Ref. [22]. ACKNOWLEDGMENTS
We would like to thank J-O. Gong, E. Kovetz, J. Mu˜noz, R. Saito, P. Shi, J. Takeda and T. Tenkanen forvery useful discussions and comments on the draft. This work was partially supported by DFG CollaborativeResearch center SFB 1225 (ISOQUANT)(G.D.). G.D. acknowledges support from the Balzan Center forCosmological Studies Program during his stay at the Johns Hopkins University. G.D. also thanks the JohnsHopkins University cosmology and gravity groups for their hospitality.
Appendix A: Model building
In this section we give the phenomenological parameters in Sec. III in terms of particular models. We willfirst consider a two-field model in which the heavy field is excited and oscillates around the minimum of itspotential. This could be a particular realization of the case studied in Sec. III A as the heavier the field thefaster the oscillations. In the second example, we will consider that the inflaton’s potential has an oscillatorymodulation superimposed. This could be an example for either fast or slow oscillations depending on themodel parameters and could be used in Secs. III A and III B.
1. Non-standard clock signal
Here we review the model studied in [38] which is a generalized version of the standard clock modelproposed in Ref. [19]. The action is given by S = (cid:90) d x √− g (cid:40) M R − (cid:16) σ Λ (cid:17) g µν ∂ µ φ∂ ν φ − f ( φ ) g µν ∂ µ σ∂ ν σ − V ( φ ) − m ( φ ) σ . (cid:41) . (A1)Assuming that the σ field is massive, does not spoil slow-roll inflation and does not backreact on theequations of motion for φ we have that σ oscillates around the minimum of the effective potential given bythe centrifugal force by∆ σ = ∆ σ r (cid:18) aa r (cid:19) − / f r f (cid:114) m eff , r m eff (cid:110) cos (cid:18)(cid:90) tt r m eff dt (cid:19) + O (1 /µ ) (cid:111) (A2)where m = m /f − ¨ f /f − H ˙ f /f and we will assume that the time derivatives of f are negligible in frontof m . All these conditions can be satisfied, at least momentarily, if the energy fraction of the massive field3is smaller than the slow-roll parameter (cid:15) ≡ − ˙ H/H . Then, the leading interacting term is given by∆( τ ) H = ¨ σH Λ (A3)which yields C r = µ r ∆ σ r Λ , Ω r = µ r δ Ω , (A4)where µ r ≡ m eff ,r H r , δ C = −
32 + 32 δ m − δ f and δ Ω = δ m − δ f . (A5)We have defined δ f ≡ d ln fdN and δ m ≡ d ln mdN , (A6)where dN = Hdt .For the values used in the main text (see Tab. I) we have that the effective dimensionless mass and theamplitude of the field oscillation at the pivot scale respectively are µ ∗ = ωn o ∼
15 and ∆ σ ∗ Λ ∼ µ − ∗ ∼ − . (A7)Thus, at the pivot scale the values are reasonable. However, we also require | δ Ω | ∼
10 and this implies δ m , δ f ∼
10. This model has a growth (or decay) of the mass and/or the coefficient of the kinetic term of 4order of magnitude per e-fold. We conclude that either there is much fine-tuning in the potential or kineticterm or the extra field will backreact in few e-folds.
2. Oscillating potential
Let us consider that the inflaton potential has an oscillating modulation of the form V = V ( φ ) (1 + W ( φ ) sin [Ω( φ ) + ϕ ]) . (A8)In order not to spoil slow-roll, i.e. η ≡ ˙ (cid:15)H(cid:15) (cid:28) W ˙Ω <
1. This can be tuned by an appropriateform of W ( φ ). Comparing with the results of Sec. III we find δ(cid:15)(cid:15) ≈ W H cos [Ω( τ ) + ϕ ] (A9)and so ∆( τ ) H = ¨ δ(cid:15) H (cid:15) (A10)which yields C r = W r δ Ω r and δ C = δ W + 3 δ Ω . (A11)We have defined δ W ≡ d ln WdN and δ Ω ≡ d ln Ω dN . (A12)4 Appendix B: Estimation of the power spectrum
Here we give a brief review of the estimation for the power spectrum. The starting point is Eq.(3.4), whichusing the fact that f ∝ z c / s ∝ c − / s and that s ≡ ˙ c s /Hc s reads∆ P R ( k ) P R , ≈ (cid:90) ∞−∞ d ln ξ (cid:18) sin (2 kξ ) kξ − cos (2 kξ ) (cid:19) s ( ξ ) . (B1)Now, using that c s = 1 + BF (ln( τ /τ f )) where F ≡ e − β s ln ( τ/τ f ) , B (cid:28) , β s (cid:29) , (B2)and that s = Bβ s dFdx , where x ≡ − β s ln ( τ /τ f ), we can write at leading order in x/β s ∆ P R ( k ) P R , ≈ B (cid:20)(cid:90) ∞−∞ dx dFdx (cid:18) xβ s (cid:19) e ikc s τ (cid:21) − B (cid:20)(cid:90) ∞−∞ dx dFdx e ikc s τ (cid:21) ≈ B √ π kτ f β s e − k τ fβ s (cid:26) sin(2 kτ f ) + cos(2 kτ f ) kτ f −
12 sin(2 kτ f )( kτ f ) (cid:27) , (B3)where used that ξ ≈ τ and we expanded the mode functions ase ikc s τ ≈ e ikτ f e − ikτ f x/β s , (B4)since τ ≈ τ f (cid:16) − xβ s (cid:17) . Appendix C: Estimation of the bispectrum and trispectrum
Here we briefly derive the estimate for the magnitude of the bispectrum and trispectrum. We work in theeffective field theory of inflation approach [11, 35, 73] and expand the action up to fourth order. By pickingup the terms that only involve the speed of sound c s and its derivatives at leading order in slow roll we find S = (cid:90) dtd xa M (cid:15)H (cid:110) (1 − c − s ) ˙ R + 2 Hsc − s ˙ R R − a − (1 − c − s ) ˙ R ( ∂ R ) (cid:111) (C1)and S = (cid:90) dtd xa M (cid:15) H (cid:40) − (1 − c − s ) (cid:104) ˙ R − a − ( ∂ R ) (cid:105) − Hsc − s a − ( ∂ R ) ˙ R R + (cid:0) H s − H ˙ s (cid:1) c − s ˙ R R (cid:41) . (C2)We will use the approximation for sharp features which consists of expanding around the transition time[11, 32, 35, 74].
1. Bispectrum
For the bispectrum we use the in-in formalism [75, 76], (cid:104)R k R k R k (cid:105) = − i Re (cid:20)(cid:90) dτ d x (cid:104)R k R k R k H I, (cid:105) (cid:21) (C3)5where H I, = −L . As usual we use the de-Sitter mode function: R k = H (cid:112) (cid:15)c s, k (1 + ikc s, τ ) e − ikc s τ . (C4)Now, picking up the highest contribution in terms of β s and evaluating the integral near the sharp featureand in the equilateral configuration ( k = k = k = k and k · k = − k /
2) we have (cid:104)R k R k R k (cid:105) eq ≈ Bβ s Im (cid:34) (cid:90) dx e iKc s τ (cid:16) F (cid:18) kτβ s (cid:19) − F β − s (cid:18) kτβ s (cid:19) (1 − ikτ ) − dFdx β − s (cid:18) kτβ s (cid:19) (1 − ikτ ) (cid:17)(cid:35) × P R , M k (2 π ) δ ( k + k + k ) (C5) ≈ − √ π Bβ s (cid:18) Kτ f β s (cid:19) e − K τ f β s sin ( Kτ f ) × P R , M k (2 π ) δ ( k + k + k ) (C6)where K = k + k + k and P R , = H π (cid:15)c s M . With this result we find that f eq NL ≈ − √ π Bβ s (cid:18) Kτ f β s (cid:19) e − K τ f β s sin ( Kτ f ) , (C7)where we used that (cid:104)R k R k R k (cid:105) = 310 f NL ( k , k , k ) k + k + k k k k P R , M (2 π ) δ ( k + k + k ) . (C8)
2. Trispectrum
Again, for the trispectrum we will use the in-in formalism. However, this time we have the possibility ofa scalar exchange [76–78], i.e. (cid:104)R k R k R k R k (cid:105) SE = (cid:90) d xdτ (cid:48) (cid:90) τ −∞ dτ (cid:48)(cid:48) (cid:104) H I, ( τ (cid:48) ) R k R k R k R k H I, ( τ (cid:48)(cid:48) ) (cid:105)− (cid:90) d xdτ (cid:48) (cid:90) τ (cid:48) −∞ dτ (cid:48)(cid:48) (cid:104) H I, ( τ (cid:48)(cid:48) ) H I, ( τ (cid:48) ) R k R k R k R k (cid:105)− (cid:90) d xdτ (cid:48) (cid:90) τ (cid:48) −∞ dτ (cid:48)(cid:48) (cid:104)R k R k R k R k H I, ( τ (cid:48) ) H I, ( τ (cid:48)(cid:48) ) (cid:105) (C9)and a contact interaction [78], that is (cid:104)R k R k R k R k (cid:105) CI = − i Re (cid:20)(cid:90) dτ d x (cid:104)R k R k R k R k H I, (cid:105) (cid:21) . (C10)However, a quick inspection of the scalar exchange contribution tells us that the contribution of the scalarexchange is proportional to f NL ≈ B β s since, at most, there are two cubic vertex proportional to Bβ s . Aswe will now see, this contribution is suppressed by a factor B with respect to the leading contribution of thecontact interaction which is proportional to Bβ s – e.g. look at the term ˙ s in S which will bring twice β s down.Now, to simplify the computation of the interaction Hamiltonian we will assume that only the terms whichare proportional to (1 − c s ) contribute to the third order Lagrangian and, thus, the third order Lagrangian6is only proportional to B . This means that the terms in the fourth order interaction Hamiltonian that comefrom L are always squared and so proportional to B . In this way, we can neglect the terms coming from L and L and the interaction Hamiltonian is, for our purposes, given by H I, = −L + O ( B ) . (C11)We again select the highest contribution in terms of β s and evaluate the integral near the sharp feature andin the regular tetrahedron configuration ( k = k = k = k = k and | k + k | = | k + k | = k ). Then wefind (cid:104)R k R k R k R k (cid:105) eq ≈ − Bβ s Im (cid:34) (cid:90) dx e iKc s τ (cid:16) (cid:18) kτβ s (cid:19) F + 2 i (cid:18) kτβ s (cid:19) dFdx − (cid:18) kτβ s (cid:19) d Fdx (cid:17)(cid:35) × P R , M k (2 π ) δ ( k + k + k + k ) (C12) ≈ − √ πBβ s (cid:18) Kτ f β s (cid:19) e − K τ f β s sin ( Kτ f ) × P R , M k (2 π ) δ ( k + k + k + k )(C13)where now K = k + k + k + k and P R , = H π (cid:15)c s M . So we have g eq NL = − √ πBβ s (cid:18) Kτ f β s (cid:19) e − K τ f β s sin ( Kτ f ) (C14)where we used the normalization of [79] in order to compare with [60], that is (cid:104)R k R k R k R k (cid:105) = 216100 g NL ( k , k , k ) k + k + k + k k k k k × P R , M (2 π ) δ ( k + k + k + k ) . (C15) [1] Uros Seljak, “Gravitational lensing effect on cosmic microwave background anisotropies: A Power spectrumapproach,” Astrophys. J. , 1 (1996), arXiv:astro-ph/9505109 [astro-ph].[2] Antony Lewis and Anthony Challinor, “Weak gravitational lensing of the CMB,” Phys. Rept. , 1–65 (2006),arXiv:astro-ph/0601594 [astro-ph].[3] Duncan Hanson, Anthony Challinor, and Antony Lewis, “Weak lensing of the CMB,” Gen. Rel. Grav. ,2197–2218 (2010), arXiv:0911.0612 [astro-ph.CO].[4] N. Aghanim et al. (Planck), “Planck 2018 results. VI. Cosmological parameters,” (2018), arXiv:1807.06209[astro-ph.CO].[5] Erminia Calabrese, Anze Slosar, Alessandro Melchiorri, George F. Smoot, and Oliver Zahn, “Cosmic MicrowaveWeak lensing data as a test for the dark universe,” Phys. Rev. D77 , 123531 (2008), arXiv:0803.2309 [astro-ph].[6] Julian B. Mu˜noz, Daniel Grin, Liang Dai, Marc Kamionkowski, and Ely D. Kovetz, “Search for CompensatedIsocurvature Perturbations with Planck Power Spectra,” Phys. Rev.
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