Red, Straight, no bends: primordial power spectrum reconstruction from CMB and large-scale structure
PPrepared for submission to JCAP
Red, Straight, no bends: primordialpower spectrum reconstructionfrom CMB and large-scale structure
Andrea Ravenni a,b,c
Licia Verde c,d,e,f and Antonio J. Cuesta c a Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova, viaMarzolo 8, I-35131, Padova, Italy b INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy c Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí iFranquès 1, E08028 Barcelona, Spain d ICREA (Institució catalana de recerca i estudis avançats) e Radcliffe Institute for Advanced Study, Harvard University, MA 02138, USA f Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, NorwayE-mail: [email protected], [email protected], [email protected]
Abstract.
We present a minimally parametric, model independent reconstruction of theshape of the primordial power spectrum. Our smoothing spline technique is well-suited tosearch for smooth features such as deviations from scale invariance, and deviations from apower law such as running of the spectral index or small-scale power suppression. We usea comprehensive set of the state-of the art cosmological data:
Planck observations of thetemperature and polarisation anisotropies of the cosmic microwave background, WiggleZ andSloan Digital Sky Survey Data Release 7 galaxy power spectra and the Canada-France-HawaiiLensing Survey correlation function. This reconstruction strongly supports the evidence fora power law primordial power spectrum with a red tilt and disfavours deviations from apower law power spectrum including small-scale power suppression such as that inducedby significantly massive neutrinos. This offers a powerful confirmation of the inflationaryparadigm, justifying the adoption of the inflationary prior in cosmological analyses.
Keywords: cosmology: cosmic microwave background, Large-scale structure – cosmology:Large-scale structure – cosmology: power spectrum a r X i v : . [ a s t r o - ph . C O ] M a y ontents All recent cosmological observations are in excellent agreement with the standard Λ CDMmodel: a spatially flat cosmological model, with matter-energy density dominated by a cos-mological constant and cold dark matter, where cosmological neutrinos are effectively masslessand where the primordial power spectrum of adiabatic perturbations is a (almost scale in-variant) power law. State-of-the art cosmological observations such as those of the
Planck satellite [1], measuring cosmic microwave background (CMB) anisotropies, provided us withvery precise measurements of the parameters of the standard cosmological model [2].Most cosmological analyses assume a power-law primordial power spectrum with a fixedspectral index, and deviations from this assumption are often in the form of a “running” ofthe spectral index.A nearly scale invariant power spectrum is a generic prediction of the simplest modelsof inflation, but there are models with (small) deviations from this prediction (e.g., [3–6]).Small deviations from scale invariance constitute a critical and generic prediction of inflation.For this reason a model-independent reconstruction of the primordial power spectrum (PPS)shape can be a powerful test of inflationary models.Here we perform a minimally parametric reconstruction of the PPS using smoothingspline interpolation in combination with cross validation. This approach follows [7–9].The idea is simple: we choose a functional form that allows a great deal of freedom inthe shape of the deviations from a power-law. Because most models predict the PPS to besmooth, among the possible choices we use a smoothing spline. The ensuing challenge is toavoid over-fitting the data; a complex function that fits the data set extremely well is of nointerest if we are simply fitting statistical noise. To prevent over-fitting we use cross-validationand a roughness penalty. The roughness penalty is an additional parameter that penalises ahigh degree of structure in the functional form. By performing cross-validation as a functionof this penalty, we can judge the amount of freedom in the smoothing spline that the datarequire, without fitting the noise.The
Planck collaboration has performed an analysis with the same goals in mind, butwith different methods [10]. They carried out both a parametric search for deviations from a– 1 –ower law, using a set of theoretically motivated shapes for the PPS, and a minimally para-metric analysis to reconstruct the PPS. In all cases there is no strong evidence for deviationsfrom a power law.Our analysis differs from that of the
Planck collaboration and from others existing in theliterature as we analyse jointly a comprehensive set of state-of-the-art experiments probingthe matter power spectrum and the latest
Planck measurements.Because we assume standard late-time evolution of density perturbations and considerboth early-time observables (CMB) and late-time ones (i.e., large-scale structure), our re-construction is also sensitive to late-time effects on structure formation. In particular anon-negligible neutrino mass would suppress the growth of structures below the neutrinofree-streaming scale, inducing an “effective” loss of small scale power in our reconstructedPPS. Reconstructing in a model-independent way a possible neutrino signature on the shapeof the matter power spectrum is of particular importance as [11–16] claims that relatively largeneutrino masses ( Σ ν (cid:38) . eV) could solve the tension between CMB and local measurements,whilst other studies [17–24] rule out this possibility.The rest of the paper is organised as follows: in section 2 we briefly summarise themethodology adopted, the data chosen and how they are analysed. In section 3 we presentour findings; we discuss and present the conclusions in section 4. We perform a minimally-parametric reconstruction of the primordial power spectrum basedon the method presented in [7] and further refined in [8, 9]. Here we only briefly summarisethe approach; it is based on the cubic smoothing spline technique (for details see [25]). Inthis approach to recover a smooth function f ( x ) , given its value f i only on a set of n points x i , hereafter knots , one fits the pairs ( x i , f i ) with a cubic spline s ( x ) . The spline, its first,and second derivatives are continuous on the knots by definition. The full function is thenuniquely defined by the values at the knots and two boundary conditions. We choose torequire that the jump in the third derivative across the first and last knots is forced to zero.In our application the resulting spline function is the reconstructed primordial powerspectrum. The f i are free parameters we wish to determine and we place the knots equallyspaced in log k as it is the most conservative choice to recover deviations from a power law.The whole s ( k ) is used as the PPS to compute the observables and evaluate the likelihood ofthe parameters f i . Including the roughness penalty, the effective likelihood becomes − log( L ) = − log( L exp ) + α p (cid:90) ln k f ln k i (cid:0) s (cid:48)(cid:48) (ln k ) (cid:1) d ln k (2.1)where s (cid:48)(cid:48) denotes the second derivative of s with respect to ln k , k i and k f are respectivelythe position of the first and of the last knots, α p is a weight that controls the penalty, and L exp is the likelihood given by the experiments.The roughness penalty effectively reduces the degrees of freedom, disfavouring jaggedfunctions that “fit the noise”. As α p goes to infinity, one effectively implements linear re-gression; as α p goes to zero one is interpolating. The use of cubic spline — instead of otherpossible interpolating functions — is motivated by the fact that such a function minimisesthe roughness penalty for a given set of knots ( f i , x i ) .– 2 –n generic applications of smoothing splines, cross-validation is a rigorous statisticaltechnique for choosing the optimal roughness penalty [25]. Cross-validation (CV) quantifiesthe notion that if the PPS has been correctly recovered, we should be able to accuratelypredict new independent data. To make the problem computationally manageable, we adoptthe following. We split the data set in two halves A and B . A Markov chain Monte Carlo(MCMC) parameter estimation analysis (for a given roughness penalty) is carried out on A ,finding the best fit model. Then the − log likelihood of B given the best fit model for A , CV AB , is computed and stored. This is repeated by switching the roles of the two halves,obtaining CV BA . The sum CV AB + CV BA , gives the “CV score” for that penalty weight. Withthis construction, the smoothing parameter that best describes the entire data set is the onethat minimises the CV score. The cross validation data sets are described below (see table1). We choose to use 5 knots equally spaced in log k between k = 10 − Mpc − and k =1 Mpc − , i.e., k = 10 − Mpc − , k = 1 . × − Mpc − , k = 3 . × − Mpc − , k = 5 . × − Mpc − , k = 1 Mpc − (see figure 1 bottom panel for knots placementvisualisation). The number and position of the knots is held fixed throughout the analysis.As discussed in reference [8], beyond a minimum number of knots, there is a trade-off betweenthe number of knots and the penalty, and the form of the reconstructed function does notdepend significantly on the number of knots beyond this minimum number. As the main goalof this work is to explore, in a minimally parametric way, smooth deviations from a powerlaw, a few ( > ) knots are sufficient.The basic cosmological parameters, ω b = Ω b h , ω c = Ω c h , h , and τ reio — physical bary-onic matter density parameter, physical cold dark matter parameter, dimensionless Hubbleparameter and optical depth to last scattering surface — are varied in the MCMC along-side the values f i of the reconstruction at the knots. A flat geometry is assumed so that Ω m + Ω Λ = 1 .The prediction for cosmological observables, the calculation of the likelihood and theMCMC parameter inference are implemented using the standard Boltzmann code CLASS[26] and its Monte Carlo code, Monte Python (MP) [27], suitably modified. Even though we reconstruct the primordial power spectrum, we are sensitive to late-timecosmological effects. Our main focus is on massive neutrinos: the presence of non-negligiblymassive neutrinos would distort our reconstruction in a way that is predictable due to thelinearity of the growth functions [28] (see Appendix). Thus in the analysis we will assumemassless neutrinos.Of course neutrino masses do not actually affect the physical PPS. But assuming stan-dard gravity, standard growth of structure, and massless neutrinos in the analysis, wouldyield a reconstructed PPS with an artificial distortion, if neutrino masses were not negligible.In fact a detectable signature of massive neutrinos in the real data would appear as a powersuppression feature in the reconstructed PPS. Of course a detection of power suppressioncannot be univocally interpreted as signature of neutrino masses; other particles beyond thestandard model could easily share the same properties of neutrinos when it comes to dampingperturbations or it could be a real feature in the PPS.– 3 – -10 -10 -10 -10
1 10 100 1000 l ( l + ) C l lPlanck low lPlanck high lSDSS DR7WiggleZPlanck lensingCFHTLenS fi ducial LCDM 100 1000 1000010 -4 -3 -2 -1 m P ( k ) k [h/Mpc] Figure 1 . Comoving scales covered by the experiments used in our analysis. The vertical dashed lineshow the limit of the linear scales. The triangles show the position of the knots. The leftmost one isnot visible in the plot.
We use a comprehensive set of power spectra obtained from observations of CMB and of largescale structure (including both weak gravitational lensing and galaxies redshift surveys) asfollows: • Planck power spectra of temperature and polarisation of the CMB. The
Planck collab-oration released in 2013 the temperature data from the first half of the mission [29].We complement the
Planck low (cid:96) from (cid:96) = 2 to (cid:96) = 49 , and the high (cid:96) angular powerspectrum. We use the temperature and polarisation data up to (cid:96) = 2500 and we referto this as PlanckCMB15. • Beside the CMB power spectrum,
Planck reconstructed the CMB lensing potential [31],which contains information on the amplitude of large scale structure integrated fromrecombination to present time. We will refer to it as PlanckLens. • The Canada-France-Hawaii Lensing Survey (CFHTLenS) [32] provides the two pointcorrelation function of the tomographic weak lensing signal. http://class-code.net http://baudren.github.io/montepython.html – 4 –un A B
Table 1 . Cross-validation datasets A and B for the various runs. The reconstruction (Rec.) involveall the experiments together. • The WiggleZ Dark Energy Survey (WiggleZ), through the measurement of position andredshift of 238,000 galaxies, mapped a volume of one cubic gigaparsec over seven regionsof the sky up to a redshift z (cid:46) . The corresponding galaxy power spectrum is presentedin [33]. • The Sloan Digital Sky Survey collaboration, in Data release 7 (SDSS DR7), used asample of luminous red galaxies to reconstruct the halo density field and its powerspectrum roughly between k = 0 . h /Mpc and k = 0 . h /Mpc [34].In figure 1 we show the scales probed by each experiment along with the location of theknots. We now describe the cross validation set up. In order to constrain both the shape of the PPSand the cosmological parameters, we have to consider CMB primary data in all CV runs. Be-cause of time constraints PlanckCMB2013 is used in the set up CV runs but PlanckCMB2015is used in the final run. This choice is conservative, favouring slightly more freedom (lowerpenalty) to the reconstructed PPS. Besides these, we have 4 other experiments: 2 measuringweak lensing and 2 using galaxy catalogues. We perform 3 CV runs in a pyramidal scheme assummarised in table 1. We start performing in parallel two different cross-validation analysison two pairs of experiments where each pair is formed by a weak lensing experiment and bya galaxy catalogue. The dependence of the CV score on α p was mapped by sampling several α p values. The results of these preliminary runs show no unexpected behaviour or tension,i.e., the reconstructed PPS shows no significant deviation from a power-law, and the shapeof the CV score is the same for both run 1.1 and run 1.2. Knowing this, we then combinethe large scale structure data to have one weak lensing and one galaxy survey in each CVset. The best roughness penalty found from this CV is used in the final run which includesall experiments (this is called “Rec.” run in the table). The penalty parameter value to usein the reconstruction is determined by the CV score of run 2 alone: its dependence on α p isillustrated in figure 2. The fact that the shape of the three CV scores — from run 1.1, 1.2,and 2 — shown in figure 2 is very similar, indicate robustness and that there are no significanttensions between the datasets.The CV score has a fairly well defined “wall” for high penalties , but is quite constantunder a certain threshold at α p ∼ . For high α p the penalty starts being the dominantcontribution to the likelihood, so the behaviour in the limit of high α p is expected. On theother hand, if small values of the penalty were to lead to overfitting, the CV score shouldincrease as α p decreases. This is not what we see and can be understood as follows. CMBangular power spectra are always included in the analysis and in this limit, it is the statisticalpower of these data (not the penalty) that drives the smoothness of the reconstruction and– 5 – -6 -4 -2 C V s c o r e α p CV run 1.1CV run 1.2CV run 2
Figure 2 . CV score as a function of α p for the cross-validation run 2. A different arbitrary offset hasbeen subtracted from each CV score. therefore the CV score. In other words, for low values of the penalty below α p ∼ , alldatasets are well consistent with the Planck-inferred PPS reconstruction: the CMB dataalone disfavour unnecessarily wiggly shapes, even when there is a low penalty.Since there is not a well defined minimum for the CV score, we opt for presenting twodifferent cases. One is more conservative, in the sense that it has a stronger penalty thatallows only small deviations from the concordance power-law model. For this one we choose α p = 1 .The other leaves more freedom to the data, as we choose a more relaxed penalty α p =0 . . A reconstruction with α p (cid:28) . is pretty much uninformative. In fact recall that thefree parameters in our MCMC runs are the physical baryon density ω b , the physical cold darkmatter density ω cdm , the rescaled Hubble parameter h , the optical depth to reionization τ reio ,and the value of the five knots of the spline that we used to parametrize the shape of the PPS.At such low penalty values the reconstruction transfers in part the features of the radiationtransfer function and the effect of the optical depth to reionization into the PPS opening updegeneracies in parameter space. Here we present the results with the latest
Planck likelihood (2015 release) and all the largescale structure power spectrum data (
Planck
Lensing 2015, WiggleZ, CFHTLenS, and SDSSDR7), with the two different roughness penalties ( α p = 1 and α p = 0 . ) justified above.As discussed in refs. [17, 35–38] there is a tension between the inferred matter powerspectrum amplitude from CMB and from CFHTLenS, which may arise from possible system-atic errors in the photometric redshifts of CFHTLens. For this reason we present results firstwithout and then with CFHTLens. In figure 3a and 3b we show the reconstructed PPS for α p = 1 and α p = 0 . respectively.The colour-bars on the upper side show the scales probed by each experiment as in figure 1,– 6 – .82.02.22.42.62.83.0 0.001 0.01 0.1 P ( k ) x k 1/Mpc (a) α p = 1 . P ( k ) x k 1/Mpc (b) α p = 0 . . Figure 3 . Reconstructed PPS. The best fit reconstruction is shown in white. Errors are shown byplotting in dark blue (light blue) 400 spline picked at random among the 68.27% most likely points(points in the range 68.27% - 95.45%) in the MCMC. The red (pale red) region shows the 68% (95%)confidence intervals for
Planck green for PlanckLens, red for WiggleZ, gold for SDSS DR7. PlanckCMB15 covers the wholeplot. The best fit reconstruction is shown in yellow and errors are shown by plotting in darkblue (light blue) a random sample of 400 reconstructions chosen among the 68.27% most likelypoints (points in the range 68.27% - 95.45%) in the MCMC. The 95.5% confidence regionsappear to coincide with the 68.3%: this is because the reconstructed spectra are simply morewiggly and are not allowed to deviate more, and consistently across scales, from the best fit.In the figure the red and pale red regions show the 68 and 95% confidence intervals forthe standard power law Λ CDM
Planck
Planck parametric fit at all scales. For the less con-servative penalty this is also true on scales corresponding to (cid:96) > . This did not happenwith the previous generation of cosmological data (see [9]) where the reconstructed PPS wassignificantly less constrained than with a power law fit.The additional freedom in the PPS allowed by the lower penalty α p = 0 . is used onscales corresponding to low CMB multipoles (cid:96) < . These scales are dominated by cosmicvariance and are known to be lower than the standard Λ CDM prediction e.g., [29, 39–41] andrefs therein.In figure 4a and 4b we also show the reconstructed n ( k ) ≡ d ln P ( k ) /d ln k ( α p = 1 and α p = 0 . ) for ease of comparison with the standard power law results. We find noevidence that any scale dependence of the power spectrum spectral slope is necessary, whichis in agreement with previous analyses. However with this new data set we find that n = 1 ishighly disfavoured by the data, in particular for α p = 1 the significance of the departure fromscale invariance is comparable with that obtained when adopting the “inflation–motivated”power-law prior. Even for the more flexible reconstruction, not even one point of the morethan × MCMC points falls near scale invariance.The results shown in Figs. 3 and 4 offer a powerful confirmation of the inflationary Recall that the quantity that was actually reconstructed using cross-validation to find the optimal penaltyis in reality the power spectrum. – 7 – n ( k ) k 1/Mpc (a) α p = 1 . n ( k ) k 1/Mpc (b) α p = 0 . . Figure 4 . Power spectrum spectral index of the reconstructed PPSs. The white line corresponds tothe best fit reconstruction. Errors are shown by plotting in dark blue (light blue) 400 reconstructionsrandomly selected from the 68.27% most likely points (points in the range 68.27% - 95.45%) in theMCMC. The red (pale red) region shows 68% (95%) confidence intervals for the power law
Planck n ( k ) ≡ , i.e., scale invariance. P ( k ) / P ( k ) Λ C D M k 1/Mpc (a) α p = 1 . P ( k ) / P ( k ) Λ C D M k 1/Mpc (b) α p = 0 . . Figure 5 . Reconstructed PPS divided by the
Planck P ( k ) . The redlines show the small-scales power suppression effect due to massive neutrinos. The upper line is the Σ m ν = 0 eV theoretical prediction based on the conditional best fit to Planck H data, the lower line is the same with Σ m ν = 0 . eV. paradigm, justify the adoption of the inflationary prior in cosmological analyses.Finally in figs. 5a and 5b we show the ratio of the reconstructed PPS to the best fit Planck Σ m ν = 0 eV and . eV. The two models are theconditional (i.e., keeping Σ m ν fixed at the required value) best fit to the data ( Planck H data). Clearly models with Σ m ν > . eV are highly disfavoured by the data even with this minimally parametric reconstruction:not a single step of a × size MCMC goes near the Σ m ν = 0 . eV line. This of coursedoes not exclude the — admittedly contrived — case with a arbitrarily large neutrino mass– 8 – .82.02.22.42.62.83.0 0.001 0.01 0.1 P ( k ) x k 1/Mpc (a) α p = 1 . P ( k ) x k 1/Mpc (b) α p = 0 . . Figure 6 . Reconstructed PPS. Refer to figure 3 for explanation and colour code. In addition, thepurple line shows the scales covered by CFHTLenS. n ( k ) k 1/Mpc (a) α p = 1 . n ( k ) k 1/Mpc (b) α p = 0 . . Figure 7 . Power spectrum spectral index of the reconstructed PPSs. Refer to figure 4 for explanationand colour code. In addition, the purple line shows the scales covered by CFHTLenS. inducing a small scale power suppression which is cancelled by a compensating boost of thePPS on the same scales. Occam’s razor disfavours this scenario.
The reconstructed P ( k ) , n ( k ) and P ( k ) relative to the power law best fit are shown in figures 6,7, and 8 using the same conventions as in figures 3, 4, and 5.Comparison with the results of section 3.1 (in figures 3, 4, 5) shows that qualitativelythe reconstructions are very similar, there is no strong evidence for deviations from the powerlaw behaviour and scale invariance is still excluded. However quantitatively some differencesmay be appreciated. Adding the CFHTLenS datasets has the effect of lowering the overallPPS normalisation (clearly visible by comparison with figure 3).The ratio with Planck power law best-fit in figure 8 highlights how, independently fromour choice of datasets, high neutrino masses are disfavoured. Quantitatively the Σ m ν > . eV bound is excluded at more than 95% confidence if we assume a power law PPS, as discussedin section 3.1.For completeness we also report the recovered values and errors for all the model pa-rameters in table 2 and table 3 for the two penalties α p = 1 and α p = 0 . respectively.– 9 – .08.09.01.01.11.2 0.001 0.01 0.1 P ( k ) / P ( k ) Λ C D M k 1/Mpc (a) α p = 1 . P ( k ) / P ( k ) Λ C D M k 1/Mpc (b) α p = 0 . . Figure 8 . Reconstructed PPS relative to
Planck
Without CFHTLenS − ln L min = 6727 . Param best-fit mean ± σ
95% lower 95% upper ω b .
226 2 . +0 . − . .
199 2 . ω cdm . . +0 . − . . . h . . +0 . − . . . τ . . +0 . − . . . +9 K .
788 2 . +0 . − . .
507 2 . +9 K .
546 2 . +0 . − . .
395 2 . +9 K .
307 2 . +0 . − . .
231 2 . +9 K .
072 2 . +0 . − . .
028 2 . +9 K .
872 1 . +0 . − . .
802 2 . With CFHTLenS − ln L min = 6777 . Param best-fit mean ± σ
95% lower 95% upper ω b .
244 2 . +0 . − . .
207 2 . ω cdm . . +0 . − . . . h . . +0 . − . . . τ . . +0 . − . . . +9 K .
721 2 . +0 . − . .
463 2 . +9 K .
512 2 . +0 . − . .
363 2 . +9 K .
281 2 . +0 . − . .
213 2 . +9 K .
065 2 . +0 . − . .
021 2 . +9 K .
869 1 . +0 . − . .
802 2 . Table 2 . Best fit, mean and confidence intervals for the MCMC parameters in the reconstructionwith α p = 1 The degeneracies among the parameters for the PPS value at the knots can be appre-ciated in the triangle plots of figure 9a for α p = 1 and figure 9b for α p = 0 . . Correlationswith and among the cosmological parameters not shown are negligible. As expected, higherpenalty induce correlations among the knots which are stronger between neighbouring ones.Interestingly the only cosmological parameter that correlates with the knots is τ reio ,– 10 –ithout CFHTLenS − ln L min = 6726 . Param best-fit mean ± σ
95% lower 95% upper ω b .
239 2 . +0 . − . .
194 2 . ω cdm . . +0 . − . . . h . . +0 . − . . . τ . . +0 . − . .
04 0 . +9 K .
968 2 . +0 . − . .
014 3 . +9 K .
159 2 . +0 . − . .
769 2 . +9 K .
272 2 . +0 . − . .
191 2 . +9 K .
148 2 . +0 . − . .
024 2 . +9 K .
933 1 . +0 . − . .
773 2 . With CFHTLenS − ln L min = 6776 . Param best-fit mean ± σ
95% lower 95% upper ω b .
235 2 . +0 . − . .
203 2 . ω cdm . . +0 . − . .
116 0 . h .
686 0 . +0 . − . . . τ . . +0 . − . .
04 0 . +9 K .
241 2 . +0 . − . .
09 3 . +9 K .
294 2 . +0 . − . .
787 2 . +9 K .
28 2 . +0 . − . .
176 2 . +9 K .
106 2 . +0 . − . .
019 2 . +9 K .
899 1 . +0 . − . .
772 2 . Table 3 . Best fit, mean and confidence intervals for the MCMC parameters in the reconstructionwith α p = 0 . which show degeneracy with the knots at higher k (figure 9c and 9d). This behaviour ishowever not unexpected. The τ reio parameter only affects the CMB and in particular itsmain effect is to suppress the temperature power spectrum at multipoles (cid:96) (cid:38) . Withour choice for the location of the knots, the most affected knots are therefore K and K .Improved polarisation data at low (cid:96) should reduce this degeneracy. The figure excluding theCFHTLenS dataset is qualitatively very similar and thus is not shown here. The analysis of the latest cosmological data [2] indicates a highly significant deviation fromscale invariance of the primordial power spectrum (PPS) when parameterized by a powerlaw or by a spectral index and a “running”. This offers a powerful tool to discriminateamong theories for the origin of perturbations and among inflationary models. In fact, thedeviation from scale invariance of the PPS is a critical prediction of inflation and is the onlysignature that is generic to all inflationary models. It is therefore a vital test of the inflationaryparadigm.One may wonder if a strong theory prior on the form of the power spectrum, such as thepower law prescription, can lead to artificially tight constraints or even a spurious detectionof a deviation from scale invariance, if the adopted model were not a good fit to the data.Here we have built on the work of [7, 8] to reconstruct the PPS with a minimallyparametric approach, using the cross-validation technique as the smoothness criterion. We– 11 – .32.52.8 + K + K + K +9 K +9 K + K +9 K +9 K +9 K (a) α p = 1 . + K + K + K +9 K +9 K + K +9 K +9 K +9 K (b) α p = 0 . . + K +9 K τ + K +9 K (c) α p = 1 . + K +9 K τ + K +9 K (d) α p = 0 . . Figure 9 . Triangular plots for the run with all the datasets combined. We refer to the value of thespline function evaluated at the i -th knot as K i . consider a comprehensive set of state-of-the art cosmological data including probes of theCosmic Microwave Background, and of large scale structure via gravitational lensing andgalaxy redshift surveys. While the spline reconstruction used here is best suited for smoothfeatures in the PPS, it is also sensitive to sharp features if they have high enough signal-to-noise.We find that there is no evidence for deviations from a power law PPS, and that errorsof the reconstructed PPS are comparable with errors obtained with a power law fit. Theseresults should be compared with those presented in [9], to appreciate the increase in statisti-cal power brought about by the latest generation of experiments. In fact with current dataa scale-invariant power spectrum is highly disfavoured even with this minimally parametricreconstruction. In particular for our conservative choice of smoothness penalty parametervalues the significance of the departure from scale invariance is comparable with that ob-tained when adopting the “inflation–motivated” power-law prior. Constraints no longer relax– 12 –ignificantly when generic forms of the PPS are allowed.Because of its flexibility, our reconstruction would be able to detect the tell-tale signatureof small scale power suppression induced by free streaming of neutrino if they are sufficientlymassive. Of course in reality the suppression happens in the late-time power spectrum, notin the primordial one. But as we do not include the effect of neutrino masses in the mattertransfer function, the reconstruction would recover an “effective” small scale damping. Ourreconstruction detects no such signature, ruling out a model with a power law PPS and sumof neutrino masses of . eV or larger.Our results, which recover in a model independent way a power law power spectrumwith a small but highly significant red tilt, offer a powerful confirmation of the inflationaryparadigm, justifying adoption of the inflationary prior in cosmological analyses. Acknowledgments
Planck .72.12.5 10 -3 -2 -1 P h y s i c a l P ( k ) x k [1/Mpc] Λ CDM (A.1) In the first case the initial condition isa power-law PPS. -3 -2 -1 P ( k ) / P ( k ) Λ C D M k 1/MpcM ν =0.06eVM ν =0.2eV M ν =0.4eV (B.1) In the second case the neutrino dampingis considered in the initial conditions. -3 -2 -1 P ( k ) / P ( k ) Λ C D M k 1/MpcM ν =0.06eVM ν =0.2eV M ν =0.4eV (A.2) Then the evolution takes into accountthe presence of massive neutrino. -3 -2 -1 M o c k P ( k ) x k [1/Mpc]M ν =0.4eV (B.2) The initial condition is a power-law mul-tiplied by the neutrino power suppression.Then the evolution equation with masslessneutrino is used. -2 -1 m P ( k ) k h/MpcInitial ConditionsStandard (R.1) The two methods generate the same ob-servables in the scales of interest. Here weshow the non-linear matter power spectrum. -2 -1 r e l a t i v e e rr o r k h/Mpc (R.2) The error do not exceed 1% at scaleslarger than k = 0 . h/ Mpc and is due to non-linearities.
Figure 10 . How a non-zero neutrino mass would induce a power suppression in the reconstructedpower spectrum. Both method A.1–A.2 and B.1–B.2 give the same result R.1–R.2 at linear level.
Appendix: Reconstruction sensitivity to non-primordial effects
Figure 10A.1–10R.2 visualises the concept — exploited here — that in the reconstructed powerspectrum the effect of a non-zero neutrino mass is degenerate with a power suppression. Thisis a good approximation especially on scales where the evolution is linear or mildly non-linear, i.e., k < . h /Mpc. Consider a Λ CDM universe with massive neutrinos, whereall the cosmological parameters are known and with a power law PPS at the end of inflation– 14 –figure 10A.1). From these initial conditions we evolve the perturbations assuming massiveneutrino (different values for the total mass are shown). On small physical scales neutrinofree streaming [42] suppresses power 10A.2) yielding a resulting power spectrum shown infigure 10R.1. Now we can think of an alternative method: implement the neutrino powersuppression (figure 10B.1) directly on the initial PPS as a deviation from a power law as shownin figure 10B.2. This initial power spectrum is then evolved assuming massless neutrinos. Thelinearity of the perturbation evolution equations guarantees that the generated matter andCMB power spectra would be the same as in the first case (figure 10R.1). In figure 10R.2 wecan appreciate the fact that discrepancies in the prediction made in the two cases come fromnon-linearities. For example, when considering neutrino with Σ m ν = 0 . eV we expect a smallscale suppression in the linear power spectrum of 15% (figure 10A.2). The differences dueto non linearities exceed 1% only above k = 0 . h/ Mpc, and are never more than 4% at thescales of interest. This means that in principle this approach we should be able to distinguishthe effect for Σ m ν ≥ . eV, even though it would prevent us to obtain an unbiased measurein case of detection.Another source of error that might contribute is given by the use of spline with a limitednumber of knots. If the number of knots, or their position, is not suitably chosen, one couldbe unable to reconstruct a given signal. It is not our case, with our choice of knots we haveverified that we can reconstruct any neutrino power suppression with a − accuracy. References [1] Planck Collaboration, P. A. R. Ade, N. Aghanim, et al.,
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