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Title: Planck intermediate results: LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters

Abstract

The six parameters of the standard ΛCDM model have best-fit values derived from the Planck temperature power spectrum that are shifted somewhat from the best-fit values derived from WMAP data. These shifts are driven by features in Planck the Planck temperature power spectrum at angular scales that had never before been measured to cosmic-variance level precision. We have investigated these shifts to determine whether they are within the range of expectation and to understand their origin in the data. Taking our parameter set to be the optical depth of the reionized intergalactic medium Τ, the baryon density ω b, the matter density ω m, the angular size of the sound horizon θ∗ the spectral index of the primordial power spectrum, n s, and A se -2τ (where A s is the amplitude of the primordial power spectrum), we have examined the change in best-fit values between a WMAP-like large angular-scale data set (with multipole moment < 800 in the Planck temperature power spectrum) and an all angular-scale data set (< 2500 Planck temperature power spectrum), each with a prior on τ of 0.07 ± 0.02. We find that the shifts, in units of the 1σ expected dispersion for each parameter,more » are { Δτ,ΔA se - ,Δn s,Δω m,Δω b,Δθ∗ } = { -1.7,-2.2,1.2,-2.0,1.1,0.9 }, with a χ2 value of 8.0. We find that this χ2 value is exceeded in 15% of our simulated data sets, and that a parameter deviates by more than 2.2σ in 9% of simulated data sets, meaning that the shifts are not unusually large. Comparing ℓ < 800 instead to ℓ > 800, or splitting at a different multipole, yields similar results. We examined the ℓ < 800 model residuals in the ℓ > 800 power spectrum data and find that the features there that drive these shifts are a set of oscillations across a broad range of angular scales. Although they partly appear similar to the effects of enhanced gravitational lensing, the shifts in ΛCDM parameters that arise in response to these features correspond to model spectrum changes that are predominantly due to non-lensing effects; the only exception is Τ, which, at fixed A se -2τ, affects the ℓ > 800 temperature power spectrum solely through the associated change in A s and the impact of that on the lensing potential power spectrum. We also ask, "what is it about the power spectrum at ℓ < 800 that leads to somewhat different best-fit parameters than come from the full range?" We find that if we discard the data at ℓ < 30, where there is a roughly 2σ downward fluctuation in power relative to the model that best fits the full range, the ℓ < 800 best-fit parameters shift significantly towards the ℓ < 2500 best-fit parameters. In contrast, including ℓ < 30, this previously noted "low-deficit" drives n s up and impacts parameters correlated with n s, such as ω m and H 0. As expected, the ℓ < 30 data have a much greater impact on the ℓ < 800 best fit than on the ℓ < 2500 best fit. So although the shifts are not very significant, we find that they can be understood through the combined effects of an oscillatory-like set of high-residuals and the deficit in low-power, excursions consistent with sample variance that happen to map onto changes in cosmological parameters. Finally, we examine agreement between PlanckTT data and two other CMB data sets, namely the Planck lensing reconstruction and the TT power spectrum measured by the South Pole Telescope, again finding a lack of convincing evidence of any significant deviations in parameters, suggesting that current CMB data sets give an internally consistent picture of the ΛCDM model.« less

Authors:
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Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
Contributing Org.:
Planck Collaboration
OSTI Identifier:
1525255
Grant/Contract Number:  
AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
Astronomy and Astrophysics
Additional Journal Information:
Journal Volume: 607; Journal ID: ISSN 0004-6361
Publisher:
EDP Sciences
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS

Citation Formats

Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., Benabed, K., Bersanelli, M., Bielewicz, P., Bonaldi, A., Bonavera, L., Bond, J. R., Borrill, J., Bouchet, F. R., Burigana, C., Calabrese, E., Cardoso, J. -F., Challinor, A., Chiang, H. C., Colombo, L. P. L., Combet, C., Crill, B. P., Curto, A., Cuttaia, F., de Bernardis, P., de Rosa, A., de Zotti, G., Delabrouille, J., Di Valentino, E., Dickinson, C., Diego, J. M., Doré, O., Ducout, A., Dupac, X., Dusini, S., Efstathiou, G., Elsner, F., Enßlin, T. A., Eriksen, H. K., Fantaye, Y., Finelli, F., Forastieri, F., Frailis, M., Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K., Génova-Santos, R. T., Gerbino, M., González-Nuevo, J., Górski, K. M., Gratton, S., Gruppuso, A., Gudmundsson, J. E., Herranz, D., Hivon, E., Huang, Z., Jaffe, A. H., Jones, W. C., Keihänen, E., Keskitalo, R., Kiiveri, K., Kim, J., Kisner, T. S., Knox, L., Krachmalnicoff, N., Kunz, M., Kurki-Suonio, H., Lagache, G., Lamarre, J. -M., Lasenby, A., Lattanzi, M., Lawrence, C. R., Le Jeune, M., Levrier, F., Lewis, A., Liguori, M., Lilje, P. B., Lilley, M., Lindholm, V., López-Caniego, M., Lubin, P. M., Ma, Y. -Z., Macías-Pérez, J. F., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Maris, M., Martin, P. G., Martínez-González, E., Matarrese, S., Mauri, N., McEwen, J. D., Meinhold, P. R., Mennella, A., Migliaccio, M., Millea, M., Miville-Deschênes, M. -A., Molinari, D., Moneti, A., Montier, L., Morgante, G., Moss, A., Narimani, A., Natoli, P., Oxborrow, C. A., Pagano, L., Paoletti, D., Partridge, B., Patanchon, G., Patrizii, L., Pettorino, V., Piacentini, F., Polastri, L., Polenta, G., Puget, J. -L., Rachen, J. P., Racine, B., Reinecke, M., Remazeilles, M., Renzi, A., Rocha, G., Rossetti, M., Roudier, G., Rubiño-Martín, J. A., Ruiz-Granados, B., Salvati, L., Sandri, M., Savelainen, M., Scott, D., Sirignano, C., Sirri, G., Stanco, L., Suur-Uski, A. -S., Tauber, J. A., Tavagnacco, D., Tenti, M., Toffolatti, L., Tomasi, M., Tristram, M., Trombetti, T., Valiviita, J., Van Tent, F., Vielva, P., Villa, F., Vittorio, N., Wandelt, B. D., Wehus, I. K., White, M., Zacchei, A., and Zonca, A. Planck intermediate results: LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters. United States: N. p., 2017. Web. doi:10.1051/0004-6361/201629504.
Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., Benabed, K., Bersanelli, M., Bielewicz, P., Bonaldi, A., Bonavera, L., Bond, J. R., Borrill, J., Bouchet, F. R., Burigana, C., Calabrese, E., Cardoso, J. -F., Challinor, A., Chiang, H. C., Colombo, L. P. L., Combet, C., Crill, B. P., Curto, A., Cuttaia, F., de Bernardis, P., de Rosa, A., de Zotti, G., Delabrouille, J., Di Valentino, E., Dickinson, C., Diego, J. M., Doré, O., Ducout, A., Dupac, X., Dusini, S., Efstathiou, G., Elsner, F., Enßlin, T. A., Eriksen, H. K., Fantaye, Y., Finelli, F., Forastieri, F., Frailis, M., Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K., Génova-Santos, R. T., Gerbino, M., González-Nuevo, J., Górski, K. M., Gratton, S., Gruppuso, A., Gudmundsson, J. E., Herranz, D., Hivon, E., Huang, Z., Jaffe, A. H., Jones, W. C., Keihänen, E., Keskitalo, R., Kiiveri, K., Kim, J., Kisner, T. S., Knox, L., Krachmalnicoff, N., Kunz, M., Kurki-Suonio, H., Lagache, G., Lamarre, J. -M., Lasenby, A., Lattanzi, M., Lawrence, C. R., Le Jeune, M., Levrier, F., Lewis, A., Liguori, M., Lilje, P. B., Lilley, M., Lindholm, V., López-Caniego, M., Lubin, P. M., Ma, Y. -Z., Macías-Pérez, J. F., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Maris, M., Martin, P. G., Martínez-González, E., Matarrese, S., Mauri, N., McEwen, J. D., Meinhold, P. R., Mennella, A., Migliaccio, M., Millea, M., Miville-Deschênes, M. -A., Molinari, D., Moneti, A., Montier, L., Morgante, G., Moss, A., Narimani, A., Natoli, P., Oxborrow, C. A., Pagano, L., Paoletti, D., Partridge, B., Patanchon, G., Patrizii, L., Pettorino, V., Piacentini, F., Polastri, L., Polenta, G., Puget, J. -L., Rachen, J. P., Racine, B., Reinecke, M., Remazeilles, M., Renzi, A., Rocha, G., Rossetti, M., Roudier, G., Rubiño-Martín, J. A., Ruiz-Granados, B., Salvati, L., Sandri, M., Savelainen, M., Scott, D., Sirignano, C., Sirri, G., Stanco, L., Suur-Uski, A. -S., Tauber, J. A., Tavagnacco, D., Tenti, M., Toffolatti, L., Tomasi, M., Tristram, M., Trombetti, T., Valiviita, J., Van Tent, F., Vielva, P., Villa, F., Vittorio, N., Wandelt, B. D., Wehus, I. K., White, M., Zacchei, A., & Zonca, A. Planck intermediate results: LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters. United States. doi:10.1051/0004-6361/201629504.
Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., Benabed, K., Bersanelli, M., Bielewicz, P., Bonaldi, A., Bonavera, L., Bond, J. R., Borrill, J., Bouchet, F. R., Burigana, C., Calabrese, E., Cardoso, J. -F., Challinor, A., Chiang, H. C., Colombo, L. P. L., Combet, C., Crill, B. P., Curto, A., Cuttaia, F., de Bernardis, P., de Rosa, A., de Zotti, G., Delabrouille, J., Di Valentino, E., Dickinson, C., Diego, J. M., Doré, O., Ducout, A., Dupac, X., Dusini, S., Efstathiou, G., Elsner, F., Enßlin, T. A., Eriksen, H. K., Fantaye, Y., Finelli, F., Forastieri, F., Frailis, M., Franceschi, E., Frolov, A., Galeotta, S., Galli, S., Ganga, K., Génova-Santos, R. T., Gerbino, M., González-Nuevo, J., Górski, K. M., Gratton, S., Gruppuso, A., Gudmundsson, J. E., Herranz, D., Hivon, E., Huang, Z., Jaffe, A. H., Jones, W. C., Keihänen, E., Keskitalo, R., Kiiveri, K., Kim, J., Kisner, T. S., Knox, L., Krachmalnicoff, N., Kunz, M., Kurki-Suonio, H., Lagache, G., Lamarre, J. -M., Lasenby, A., Lattanzi, M., Lawrence, C. R., Le Jeune, M., Levrier, F., Lewis, A., Liguori, M., Lilje, P. B., Lilley, M., Lindholm, V., López-Caniego, M., Lubin, P. M., Ma, Y. -Z., Macías-Pérez, J. F., Maggio, G., Maino, D., Mandolesi, N., Mangilli, A., Maris, M., Martin, P. G., Martínez-González, E., Matarrese, S., Mauri, N., McEwen, J. D., Meinhold, P. R., Mennella, A., Migliaccio, M., Millea, M., Miville-Deschênes, M. -A., Molinari, D., Moneti, A., Montier, L., Morgante, G., Moss, A., Narimani, A., Natoli, P., Oxborrow, C. A., Pagano, L., Paoletti, D., Partridge, B., Patanchon, G., Patrizii, L., Pettorino, V., Piacentini, F., Polastri, L., Polenta, G., Puget, J. -L., Rachen, J. P., Racine, B., Reinecke, M., Remazeilles, M., Renzi, A., Rocha, G., Rossetti, M., Roudier, G., Rubiño-Martín, J. A., Ruiz-Granados, B., Salvati, L., Sandri, M., Savelainen, M., Scott, D., Sirignano, C., Sirri, G., Stanco, L., Suur-Uski, A. -S., Tauber, J. A., Tavagnacco, D., Tenti, M., Toffolatti, L., Tomasi, M., Tristram, M., Trombetti, T., Valiviita, J., Van Tent, F., Vielva, P., Villa, F., Vittorio, N., Wandelt, B. D., Wehus, I. K., White, M., Zacchei, A., and Zonca, A. Mon . "Planck intermediate results: LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters". United States. doi:10.1051/0004-6361/201629504. https://www.osti.gov/servlets/purl/1525255.
@article{osti_1525255,
title = {Planck intermediate results: LI. Features in the cosmic microwave background temperature power spectrum and shifts in cosmological parameters},
author = {Aghanim, N. and Akrami, Y. and Ashdown, M. and Aumont, J. and Baccigalupi, C. and Ballardini, M. and Banday, A. J. and Barreiro, R. B. and Bartolo, N. and Basak, S. and Benabed, K. and Bersanelli, M. and Bielewicz, P. and Bonaldi, A. and Bonavera, L. and Bond, J. R. and Borrill, J. and Bouchet, F. R. and Burigana, C. and Calabrese, E. and Cardoso, J. -F. and Challinor, A. and Chiang, H. C. and Colombo, L. P. L. and Combet, C. and Crill, B. P. and Curto, A. and Cuttaia, F. and de Bernardis, P. and de Rosa, A. and de Zotti, G. and Delabrouille, J. and Di Valentino, E. and Dickinson, C. and Diego, J. M. and Doré, O. and Ducout, A. and Dupac, X. and Dusini, S. and Efstathiou, G. and Elsner, F. and Enßlin, T. A. and Eriksen, H. K. and Fantaye, Y. and Finelli, F. and Forastieri, F. and Frailis, M. and Franceschi, E. and Frolov, A. and Galeotta, S. and Galli, S. and Ganga, K. and Génova-Santos, R. T. and Gerbino, M. and González-Nuevo, J. and Górski, K. M. and Gratton, S. and Gruppuso, A. and Gudmundsson, J. E. and Herranz, D. and Hivon, E. and Huang, Z. and Jaffe, A. H. and Jones, W. C. and Keihänen, E. and Keskitalo, R. and Kiiveri, K. and Kim, J. and Kisner, T. S. and Knox, L. and Krachmalnicoff, N. and Kunz, M. and Kurki-Suonio, H. and Lagache, G. and Lamarre, J. -M. and Lasenby, A. and Lattanzi, M. and Lawrence, C. R. and Le Jeune, M. and Levrier, F. and Lewis, A. and Liguori, M. and Lilje, P. B. and Lilley, M. and Lindholm, V. and López-Caniego, M. and Lubin, P. M. and Ma, Y. -Z. and Macías-Pérez, J. F. and Maggio, G. and Maino, D. and Mandolesi, N. and Mangilli, A. and Maris, M. and Martin, P. G. and Martínez-González, E. and Matarrese, S. and Mauri, N. and McEwen, J. D. and Meinhold, P. R. and Mennella, A. and Migliaccio, M. and Millea, M. and Miville-Deschênes, M. -A. and Molinari, D. and Moneti, A. and Montier, L. and Morgante, G. and Moss, A. and Narimani, A. and Natoli, P. and Oxborrow, C. A. and Pagano, L. and Paoletti, D. and Partridge, B. and Patanchon, G. and Patrizii, L. and Pettorino, V. and Piacentini, F. and Polastri, L. and Polenta, G. and Puget, J. -L. and Rachen, J. P. and Racine, B. and Reinecke, M. and Remazeilles, M. and Renzi, A. and Rocha, G. and Rossetti, M. and Roudier, G. and Rubiño-Martín, J. A. and Ruiz-Granados, B. and Salvati, L. and Sandri, M. and Savelainen, M. and Scott, D. and Sirignano, C. and Sirri, G. and Stanco, L. and Suur-Uski, A. -S. and Tauber, J. A. and Tavagnacco, D. and Tenti, M. and Toffolatti, L. and Tomasi, M. and Tristram, M. and Trombetti, T. and Valiviita, J. and Van Tent, F. and Vielva, P. and Villa, F. and Vittorio, N. and Wandelt, B. D. and Wehus, I. K. and White, M. and Zacchei, A. and Zonca, A.},
abstractNote = {The six parameters of the standard ΛCDM model have best-fit values derived from the Planck temperature power spectrum that are shifted somewhat from the best-fit values derived from WMAP data. These shifts are driven by features in Planck the Planck temperature power spectrum at angular scales that had never before been measured to cosmic-variance level precision. We have investigated these shifts to determine whether they are within the range of expectation and to understand their origin in the data. Taking our parameter set to be the optical depth of the reionized intergalactic medium Τ, the baryon density ωb, the matter density ωm, the angular size of the sound horizon θ∗ the spectral index of the primordial power spectrum, ns, and Ase-2τ (where As is the amplitude of the primordial power spectrum), we have examined the change in best-fit values between a WMAP-like large angular-scale data set (with multipole moment < 800 in the Planck temperature power spectrum) and an all angular-scale data set (< 2500 Planck temperature power spectrum), each with a prior on τ of 0.07 ± 0.02. We find that the shifts, in units of the 1σ expected dispersion for each parameter, are { Δτ,ΔAse- 2τ,Δns,Δωm,Δωb,Δθ∗ } = { -1.7,-2.2,1.2,-2.0,1.1,0.9 }, with a χ2 value of 8.0. We find that this χ2 value is exceeded in 15% of our simulated data sets, and that a parameter deviates by more than 2.2σ in 9% of simulated data sets, meaning that the shifts are not unusually large. Comparing ℓ < 800 instead to ℓ > 800, or splitting at a different multipole, yields similar results. We examined the ℓ < 800 model residuals in the ℓ > 800 power spectrum data and find that the features there that drive these shifts are a set of oscillations across a broad range of angular scales. Although they partly appear similar to the effects of enhanced gravitational lensing, the shifts in ΛCDM parameters that arise in response to these features correspond to model spectrum changes that are predominantly due to non-lensing effects; the only exception is Τ, which, at fixed Ase-2τ, affects the ℓ > 800 temperature power spectrum solely through the associated change in As and the impact of that on the lensing potential power spectrum. We also ask, "what is it about the power spectrum at ℓ < 800 that leads to somewhat different best-fit parameters than come from the full range?" We find that if we discard the data at ℓ < 30, where there is a roughly 2σ downward fluctuation in power relative to the model that best fits the full range, the ℓ < 800 best-fit parameters shift significantly towards the ℓ < 2500 best-fit parameters. In contrast, including ℓ < 30, this previously noted "low-deficit" drives ns up and impacts parameters correlated with ns, such as ωm and H0. As expected, the ℓ < 30 data have a much greater impact on the ℓ < 800 best fit than on the ℓ < 2500 best fit. So although the shifts are not very significant, we find that they can be understood through the combined effects of an oscillatory-like set of high-residuals and the deficit in low-power, excursions consistent with sample variance that happen to map onto changes in cosmological parameters. Finally, we examine agreement between PlanckTT data and two other CMB data sets, namely the Planck lensing reconstruction and the TT power spectrum measured by the South Pole Telescope, again finding a lack of convincing evidence of any significant deviations in parameters, suggesting that current CMB data sets give an internally consistent picture of the ΛCDM model.},
doi = {10.1051/0004-6361/201629504},
journal = {Astronomy and Astrophysics},
number = ,
volume = 607,
place = {United States},
year = {2017},
month = {11}
}

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Minimal anisotropy of the microwave background radiation in the gravitational instability picture
journal, July 1984


Planck 2013 results. XV. CMB power spectra and likelihood
journal, October 2014


Planck 2015 results : VIII. High Frequency Instrument data processing: Calibration and maps
journal, September 2016


Acoustic Signatures in the Cosmic Microwave Background
journal, November 1996

  • Hu, Wayne; White, Martin
  • The Astrophysical Journal, Vol. 471, Issue 1
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Cosmological parameters from CMB and other data: A Monte Carlo approach
journal, November 2002


A Measurement of the Cosmic Microwave Background Damping tail from the 2500-Square-Degree Spt-Sz Survey
journal, November 2013


Planck 2015 results : XV. Gravitational lensing
journal, September 2016


Likelihood analysis of CMB temperature and polarization power spectra
journal, May 2008


Suppressing the CMB quadrupole with a bounce from the contracting phase to inflation
journal, May 2004


Cosmic Black-Body Radiation and Galaxy Formation
journal, January 1968

  • Silk, Joseph
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Four-Year [ITAL]COBE[/ITAL] DMR Cosmic Microwave Background Observations: Maps and Basic Results
journal, June 1996

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  • The Astrophysical Journal, Vol. 464, Issue 1
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Reionization and Cosmic Microwave Anisotropies
journal, December 1993

  • Sugiyama, Naoshi; Silk, Joseph; Vittorio, Nicola
  • The Astrophysical Journal, Vol. 419
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COMPARING PLANCK AND WMAP : MAPS, SPECTRA, AND PARAMETERS
journal, February 2015


NINE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE ( WMAP ) OBSERVATIONS: FINAL MAPS AND RESULTS
journal, September 2013

  • Bennett, C. L.; Larson, D.; Weiland, J. L.
  • The Astrophysical Journal Supplement Series, Vol. 208, Issue 2
  • DOI: 10.1088/0067-0049/208/2/20

A Measurement of Secondary Cosmic Microwave Background Anisotropies from the 2500 Square-Degree Spt-Sz Survey
journal, January 2015


A two-fluid approximation for calculating the cosmic microwave background anisotropies
journal, November 1994

  • Seljak, Uros
  • The Astrophysical Journal, Vol. 435
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Planck and the local Universe: Quantifying the tension
journal, September 2013


TEASING: a fast and accurate approximation for the low multipole likelihood of the cosmic microwave background temperature
journal, November 2009


Quantifying concordance in cosmology
journal, May 2016


H0 revisited
journal, March 2014

  • Efstathiou, G.
  • Monthly Notices of the Royal Astronomical Society, Vol. 440, Issue 2
  • DOI: 10.1093/mnras/stu278

Planck 2013 results. XVII. Gravitational lensing by large-scale structure
journal, October 2014


Planck 2015 results : II. Low Frequency Instrument data processings
journal, September 2016


Planck 2015 results : IX. Diffuse component separation: CMB maps
journal, September 2016


KiDS-450: cosmological parameter constraints from tomographic weak gravitational lensing
journal, November 2016

  • Hildebrandt, H.; Viola, M.; Heymans, C.
  • Monthly Notices of the Royal Astronomical Society, Vol. 465, Issue 2
  • DOI: 10.1093/mnras/stw2805

Anisotropies in the cosmic microwave background: an analytic approach
journal, May 1995

  • Hu, Wayne; Sugiyama, Naoshi
  • The Astrophysical Journal, Vol. 444
  • DOI: 10.1086/175624

Planck 2015 results : XX. Constraints on inflation
journal, September 2016


Dependence of the cosmic microwave background lensing power spectrum on the matter density
journal, October 2014

  • Pan, Z.; Knox, L.; White, M.
  • Monthly Notices of the Royal Astronomical Society, Vol. 445, Issue 3
  • DOI: 10.1093/mnras/stu1971

Planck intermediate results : XLVII.
journal, December 2016


The Atacama Cosmology Telescope: CMB polarization at 200 < ℓ < 9000
journal, October 2014

  • Naess, Sigurd; Hasselfield, Matthew; McMahon, Jeff
  • Journal of Cosmology and Astroparticle Physics, Vol. 2014, Issue 10
  • DOI: 10.1088/1475-7516/2014/10/007

Neutrinos Help Reconcile Planck Measurements with the Local Universe
journal, February 2014


Perturbations of a Cosmological Model and Angular Variations of the Microwave Background
journal, January 1967

  • Sachs, R. K.; Wolfe, A. M.
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