Probing primordial features with the primary CMB

Probing primordial features with the primary CMB

Mario Ballardini mario.ballardini@gmail.com Department of Physics and Astronomy, University of the Western Cape, Cape Town 7535, South Africa INAF/OAS Bologna, via Gobetti 101, I-40129 Bologna, Italy
July 2, 2019
Abstract

We propose to study the imprint of features in the primordial power spectrum with the primary CMB after the subtraction of the reconstructed ISW signal from the observed CMB temperature angular power spectrum. We consider the application to features models able to fit two of the large scales anomalies observed in the CMB temperature angular power spectrum: the deficit of power at and at .

We show that if the features comes from the primordial power spectrum we should be find consistent constraints of these features model from the CMB temperature angular power spectrum removing or not the late ISW signal. Moreover, this method shows also some improvement on the constraints on the features parameters up to for models predicting a suppression of power of the quadrupole and up to for models with features at , assuming instrumental sensitivity similar to the satellite (depending on the goodness of the ISW reconstruction). Furthermore, it gives the opportunity to understand if these anomalies are attributed to early- or late-time physics.

I Introduction

Although observations show how a spatially flat CDM model with a tilted power-law spectrum of primordial density fluctuations provides a good fit to CMB temperature and polarization anisotropies Akrami:2018vks (), there are interesting hints for new physics beyond the CDM model based on slow-roll inflation in the WMAP Peiris:2003ff () and data Planck:2013jfk (); Ade:2015lrj (); Akrami:2018odb (), such as anomalies in the large angular scale pattern of CMB temperature anisotropies Peiris:2003ff (); Covi:2006ci (); Planck:2013jfk (); Benetti:2013cja (); Miranda:2013wxa (); Easther:2013kla (); Chen:2014joa (); Achucarro:2014msa (); Hazra:2014goa (); Hazra:2014jwa (); Hu:2014hra (); Ade:2015lrj (); Gruppuso:2015zia (); Gruppuso:2015xqa (); Hazra:2016fkm (); Torrado:2016sls (); Obied:2018qdr (); Akrami:2018odb ().

Anomalies in the CMB angular power spectra, as well as in the dark matter power spectrum, are predicted by several theoretically well motivated mechanisms that occur during inflation; these mechanisms support deviations from a simple power-law for the primordial power spectrum, connected with the violation of the slow-roll phase, and provide a better fit to the CMB data at .

In Fig. 1, it is plotted the comparison between the best-fit CMB temperature power spectrum for the standard CDM model and the best-fits for some features models Ade:2015lrj () which improve the fit of CMB data. Although the difference between these models, the cosmic-variance restricts our ability to discriminate between them even with a perfect measure of the CMB anisotropies.

The situation improves if well-suited data in addition to the CMB temperature anisotropies are available:

In this paper, we propose a further method to improve the current understanding of the large scales CMB anomalies based on the possibility to subtract the reconstructed integrated Sachs-Wolfe (ISW) signal from the observed CMB temperature angular power spectrum in order to constrain models with features in the primordial power spectrum with the primary CMB. After subtracting the ISW signal, possible by cross-correlating CMB maps with tracer maps of the matter density fluctuation, we have the opportunity to test the CMB angular power spectrum dominated by the SW contribution at the largest scales like the primary CMB signal generated at the last scattering surface. This technique has been already proposed and applied to real data to study the significance of anomalies at CMB maps level with and without the contamination from the late ISW signal Efstathiou:2009di (); Francis:2009pt (); Muir:2016veb (); Copi:2016hhq ().

ISW, such as CMB lensing deflection, can be considered as foreground contribution to the primary CMB signal. They can be used to further study the information content from some late-time physics (dark energy and small scales matter perturbation for instance), but they also ceil part of the primary CMB signal introducing some degeneracies between primordial paramters and other ones.

The ISW contribution to the CMB temperature fluctuations in direction is a secondary anisotropy in the CMB caused by the passage of CMB photons through evolving gravitational potential wells

(1)

where and are the gravitational potentials in the longitudinal gauge and is the visibility function. On large scales, in a dark-energy-dominated universe, CMB photons gain energy when they pass through the decaying potential wells associated with overdensities and lose energy on passing through underdensities Kofman:1985fp (). This effect mainly contributes to large angular scales and therefore at low multipoles, i.e. , since there is a little power in the potentials at late times on scales that entered the Hubble radius during radiation domination.

Figure 1: CMB temperature angular power spectrum best-fit for CDM (dashed black), cut-off (red), kink (blue), step (green) models from 2015 TT+lowP data Ade:2015lrj (). The yellow band represents the error bar from the cosmic-variance only for CDM.

Ii Primary CMB anisotropies temperature angular power spectrum

The different components that source the observed CMB temperature are assumed to have a negligible correlation with the others and drawn from a Gaussian distributions with mean zero and a known covariance matrix. Therefore, the combination will be distributed as a Gaussian with covariance matrix equals to the sum of the covariance matrices of the noise and the primordial CMB.

By knowing the ISW contribution to the CMB temperature anisotropies it is possible to reconstruct the primary CMB temperature angular power spectrum at large scales

(2)
(3)

where is the noise of the ISW angular power spectrum after the reconstruction.

ISW is generally reconstructed by cross-correlating CMB temperature angular power spectrum with LSS galaxy surveys Crittenden:1995ak () or other LSS tracers such as CMB lensing Manzotti:2014kta (), termal Sunyaev-Zeldovich Taburet:2010hb (), intensity mapping emission lines Pourtsidou:2016dzn () and clusters of galaxies Ballardini:2017xnt ().

In order to quantifity the errors from the reconstruction of the ISW signal by cross-correlating the CMB with one or more LSS tracers, we consider the standard theoretical signal-to-noise ratio (defined according to Cooray:2001ab (); Afshordi:2004kz ()) to build the noise angular power spectra for three different cases: a 3 level reconstruction of the ISW signal, compatible with the significance obtained in Ade:2015dva () by cross-correlating the temperature map with a compilation of publicly available galaxy surveys Ade:2015dva (); a 6 significance expected for next-generation of LSS galaxy surveys Raccanelli:2015lca (); Pourtsidou:2016dzn (); an ideal case with a perfect reconstruction () of the late-time ISW signal with in Eq. (3).

Iii Fisher forecast formalism

With these definitions in hand, we can proceed to perform a Fisher matrix analysis for CMB angular power spectra (temperature and -mode polarization) Knox:1995dq (); Jungman:1995bz (); Seljak:1996ti (); Zaldarriaga:1996xe (); Kamionkowski:1996ks ()

(4)

where

(5)

Here is the sum of the theoretical spectrum and the effective noise , which is given by the inverse noise weighted combination of the instrumental noise de-convolved with the beams of different frequency channels. For the temperature and polarization angular power spectra, a noise power spectrum with Gaussian beam profile Knox:1995dq () has been used

(6)

Here is the beam window function, assumed Gaussian, with ; is the full width half maximum (FWHM) of the beam in radians; and are the square of the detector noise level on a steradian patch for temperature and polarization, respectively.

Iv Models of features in the primordial power spectrum

We consider three inflation models that generate features in the primordial power spectrum (see Fig. 1): the cut-off model Contaldi:2003zv () which reproduces a suppression of power at large scales, and two models which lead to localized features in the primordial power spectrum, i.e. the kink model Starobinsky:1992ts () and the step model Dvorkin:2009ne (). Following Ballardini:2016hpi (), the fiducial spectra are centred at their best-fit parameters from 2015 TT+lowP data Ade:2015lrj () for each parameterization. The primordial power spectrum can be written as the standard power-law , modulated by the contribution dues to the violation of slow-roll

(7)
(8)

where is the amplitude of the curvature power spectrum, is the scalar spectral index and the pivot scale is fixed at Mpc.

The non-canonical contribution to for the cutoff model is given by

(9)
(10)

for the kink model by

(11)
(12)

and for the step model by

(13)
(14)

where the first- and second-order parts are

(15)
(16)

where a prime denotes , and the damping envelope is

(17)

The window functions are

(18)
(19)

See Refs. Contaldi:2003zv (); Starobinsky:1992ts (); Dvorkin:2009ne (); Ballardini:2016hpi () for a clear descripition of the models.

Figure 2: Derivatives of the CMB temperature angular power spectrum with respect to the features parameters for the cut-off (top panels), kink (central panels), step (bottom panels) models. The red dashed lines refer to the derivative of the full observed CMB temperature angular power spectrum and the blue solid lines refer to the derivative of the primary spectra.

V Results

After the ISW removal, the signal of the CMB temperature anisotropies decreases at the large angular scales. On these scales where the instrumental noise is negligible, this effect is compensated by an effective lower cosmic-variance since , i.e.

(20)

assuming a negligible . On the other hand, the primary CMB is more sensible to the variation of the cosmological parameters connected with the primordial power spectrum. It is possible to see this effect by looking at the derivatives of the CMB temperature anisotropies. In Fig. 2, the derivatives of the CMB primary anisotropies respect to the features parameters are always larger in amplitude compared to the derivatives of the observed CMB temperature anisotropies.

We consider two different configurations of CMB experiment: a reprentative of current CMB measurements by considering the 143 GHz channel full mission sensitivity and angular resolution as given in Adam:2015rua () and a CMB cosmic-variance limited experiment, both with . Results are collected in Tab. 1.

Assuming a perfect reconstruction of the ISW, we found that for the cut-off model the errors decrease by 6% on and by 16% on for an experiment with ’s sensitivity. For the kink model the errors improve by a 17% on and by 10% on . The step model is the one that benefits more from this method, the errors improve by a 27% on , by 25% on and by 19% on .

The case with the subtraction of the 3 and of the detected ISW does not lead to any improvements for both the cut-off and the kink models. Instead, for the step model there is still a reduced improvement of 5% on the amplitude and 10% on the scale parameter, even for these cases with injected noise from the ISW reconstruction.

Feature models which predict departures from the standard power-law primordial power spectrum will benefit from having better measurements of large angular scale CMB -mode polarization at the cosmic-variance level. However, the cut-off model affects the largest angular scales reproducing a suppression of power at in temperature and in the -mode polarization. For this reason, the relative improvement does not change when we consider a cosmic-variance limited CMB experiment. The instrumental noise on the -mode polarization is small even for on such scales. In this case the improvement from the subtraction of the ISW signal is very small, , even for the case of perfect ISW reconstruction for all the three considered models.

Vi Conclusion

In conclusion, this method represent a consistent way to test the origin of features in the CMB temperature angular power spectrum.

We show that the constraints on features parameters after the ISW subtraction are expected to be consistent with the ones obtained from full CMB. Moreover, this approach performs well for the step model which fits the deficit in power at , improving the contraints by on the amplitude and by on the scale parameters, even without better measurements of CMB polarization.

Finally, even if the final improvement for realistic cases of ISW subtraction could lead to small differences in terms of constraining power on the parameters of these features models, the subtraction of the ISW signal could lead to a change in the pattern of the largest scales of the CMB temperature anisotropies changing the shape of the features. For instance, if an anomaly vanishes after the subtraction of the ISW component to the CMB temperature, then a primordial explanation would be eliminated.

Model Parameters full CMB ISW ISW primary CMB
cut-off
kink
step
Table 1: 68% constraints on the features parameters for a -like CMB experiment (left) and a cosmic-variance limited one (right). Constraints are given for the standard case (full CMB) and after ISW subtraction by considering different levels of ISW detection. We report the best-fit for the features parameters from 2015 TT + lowP data Ade:2015lrj ().

Acknowledgments

We thank Xingang Chen and Roy Maartens for helpful discussions, and Fabio Finelli. This research was supported by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Technology, and was also supported by the Claude Leon Foundation. We acknowledge support by the ASI n.I/023/12/0 ”Attività relative alla fase B2/C per la missione Euclid”. We would like to thank Johns Hopkins University and Harvard University for the hospitality.

References

  • (1) Y. Akrami et al. [Planck Collaboration], arXiv:1807.06205 [astro-ph.CO].
  • (2) H. V. Peiris et al. [WMAP Collaboration], Astrophys. J. Suppl. 148 (2003) 213 doi:10.1086/377228 [astro-ph/0302225].
  • (3) P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571 (2014) A22 doi:10.1051/0004-6361/201321569 [arXiv:1303.5082 [astro-ph.CO]].
  • (4) P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A20 doi:10.1051/0004-6361/201525898 [arXiv:1502.02114 [astro-ph.CO]].
  • (5) Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].
  • (6) L. Covi, J. Hamann, A. Melchiorri, A. Slosar and I. Sorbera, Phys. Rev. D 74 (2006) 083509 doi:10.1103/PhysRevD.74.083509 [astro-ph/0606452].
  • (7) M. Benetti, Phys. Rev. D 88 (2013) 087302 doi:10.1103/PhysRevD.88.087302 [arXiv:1308.6406 [astro-ph.CO]].
  • (8) V. Miranda and W. Hu, Phys. Rev. D 89 (2014) no.8, 083529 doi:10.1103/PhysRevD.89.083529 [arXiv:1312.0946 [astro-ph.CO]].
  • (9) R. Easther and R. Flauger, JCAP 1402 (2014) 037 doi:10.1088/1475-7516/2014/02/037 [arXiv:1308.3736 [astro-ph.CO]].
  • (10) X. Chen and M. H. Namjoo, Phys. Lett. B 739 (2014) 285 doi:10.1016/j.physletb.2014.11.002 [arXiv:1404.1536 [astro-ph.CO]].
  • (11) A. Achucarro, V. Atal, B. Hu, P. Ortiz and J. Torrado, Phys. Rev. D 90 (2014) no.2, 023511 doi:10.1103/PhysRevD.90.023511 [arXiv:1404.7522 [astro-ph.CO]].
  • (12) D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, JCAP 1408 (2014) 048 doi:10.1088/1475-7516/2014/08/048 [arXiv:1405.2012 [astro-ph.CO]].
  • (13) D. K. Hazra, A. Shafieloo and T. Souradeep, JCAP 1411 (2014) no.11, 011 doi:10.1088/1475-7516/2014/11/011 [arXiv:1406.4827 [astro-ph.CO]].
  • (14) B. Hu and J. Torrado, Phys. Rev. D 91 (2015) no.6, 064039 doi:10.1103/PhysRevD.91.064039 [arXiv:1410.4804 [astro-ph.CO]].
  • (15) A. Gruppuso and A. Sagnotti, Int. J. Mod. Phys. D 24 (2015) no.12, 1544008 doi:10.1142/S0218271815440083 [arXiv:1506.08093 [astro-ph.CO]].
  • (16) A. Gruppuso, N. Kitazawa, N. Mandolesi, P. Natoli and A. Sagnotti, Phys. Dark Univ. 11 (2016) 68 doi:10.1016/j.dark.2015.12.001 [arXiv:1508.00411 [astro-ph.CO]].
  • (17) D. K. Hazra, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, JCAP 1609 (2016) no.09, 009 doi:10.1088/1475-7516/2016/09/009 [arXiv:1605.02106 [astro-ph.CO]].
  • (18) J. Torrado, B. Hu and A. Achucarro, Phys. Rev. D 96 (2017) no.8, 083515 doi:10.1103/PhysRevD.96.083515 [arXiv:1611.10350 [astro-ph.CO]].
  • (19) G. Obied, C. Dvorkin, C. Heinrich, W. Hu and V. Miranda, Phys. Rev. D 98 (2018) no.4, 043518 doi:10.1103/PhysRevD.98.043518 [arXiv:1803.01858 [astro-ph.CO]].
  • (20) M. J. Mortonson, C. Dvorkin, H. V. Peiris and W. Hu, Phys. Rev. D 79 (2009) 103519 doi:10.1103/PhysRevD.79.103519 [arXiv:0903.4920 [astro-ph.CO]].
  • (21) J. Chluba, J. Hamann and S. P. Patil, Int. J. Mod. Phys. D 24 (2015) no.10, 1530023 doi:10.1142/S0218271815300232 [arXiv:1505.01834 [astro-ph.CO]].
  • (22) F. Finelli et al. [CORE Collaboration], JCAP 1804 (2018) 016 doi:10.1088/1475-7516/2018/04/016 [arXiv:1612.08270 [astro-ph.CO]].
  • (23) D. K. Hazra, D. Paoletti, M. Ballardini, F. Finelli, A. Shafieloo, G. F. Smoot and A. A. Starobinsky, JCAP 1802 (2018) no.02, 017 doi:10.1088/1475-7516/2018/02/017 [arXiv:1710.01205 [astro-ph.CO]].
  • (24) H. Zhan, L. Knox, A. Tyson and V. Margoniner, Astrophys. J. 640 (2006) 8 doi:10.1086/500077 [astro-ph/0508119].
  • (25) Z. Huang, L. Verde and F. Vernizzi, JCAP 1204 (2012) 005 doi:10.1088/1475-7516/2012/04/005 [arXiv:1201.5955 [astro-ph.CO]].
  • (26) X. Chen, C. Dvorkin, Z. Huang, M. H. Namjoo and L. Verde, JCAP 1611 (2016) no.11, 014 doi:10.1088/1475-7516/2016/11/014 [arXiv:1605.09365 [astro-ph.CO]].
  • (27) X. Chen, P. D. Meerburg and M. Münchmeyer, JCAP 1609 (2016) no.09, 023 doi:10.1088/1475-7516/2016/09/023 [arXiv:1605.09364 [astro-ph.CO]].
  • (28) M. Ballardini, F. Finelli, C. Fedeli and L. Moscardini, JCAP 1610 (2016) 041 Erratum: [JCAP 1804 (2018) no.04, E01] doi:10.1088/1475-7516/2018/04/E01, 10.1088/1475-7516/2016/10/041 [arXiv:1606.03747 [astro-ph.CO]].
  • (29) Y. Xu, J. Hamann and X. Chen, Phys. Rev. D 94 (2016) no.12, 123518 doi:10.1103/PhysRevD.94.123518 [arXiv:1607.00817 [astro-ph.CO]].
  • (30) M. A. Fard and S. Baghram, JCAP 1801 (2018) no.01, 051 doi:10.1088/1475-7516/2018/01/051 [arXiv:1709.05323 [astro-ph.CO]].
  • (31) G. A. Palma, D. Sapone and S. Sypsas, JCAP 1806 (2018) no.06, 004 doi:10.1088/1475-7516/2018/06/004 [arXiv:1710.02570 [astro-ph.CO]].
  • (32) B. L’Huillier, A. Shafieloo, D. K. Hazra, G. F. Smoot and A. A. Starobinsky, Mon. Not. Roy. Astron. Soc. 477 (2018) no.2, 2503 doi:10.1093/mnras/sty745 [arXiv:1710.10987 [astro-ph.CO]].
  • (33) M. Ballardini, F. Finelli, R. Maartens and L. Moscardini, JCAP 1804 (2018) no.04, 044 doi:10.1088/1475-7516/2018/04/044 [arXiv:1712.07425 [astro-ph.CO]].
  • (34) J. R. Fergusson, H. F. Gruetjen, E. P. S. Shellard and M. Liguori, Phys. Rev. D 91 (2015) no.2, 023502 doi:10.1103/PhysRevD.91.023502 [arXiv:1410.5114 [astro-ph.CO]].
  • (35) J. R. Fergusson, H. F. Gruetjen, E. P. S. Shellard and B. Wallisch, Phys. Rev. D 91 (2015) no.12, 123506 doi:10.1103/PhysRevD.91.123506 [arXiv:1412.6152 [astro-ph.CO]].
  • (36) P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A17 doi:10.1051/0004-6361/201525836 [arXiv:1502.01592 [astro-ph.CO]].
  • (37) P. D. Meerburg, M. Münchmeyer and B. Wandelt, Phys. Rev. D 93 (2016) no.4, 043536 doi:10.1103/PhysRevD.93.043536 [arXiv:1510.01756 [astro-ph.CO]].
  • (38) A. Moradinezhad Dizgah, H. Lee, J. B. Muñoz and C. Dvorkin, JCAP 1805 (2018) no.05, 013 doi:10.1088/1475-7516/2018/05/013 [arXiv:1801.07265 [astro-ph.CO]].
  • (39) D. Karagiannis, A. Lazanu, M. Liguori, A. Raccanelli, N. Bartolo and L. Verde, Mon. Not. Roy. Astron. Soc. 478 (2018) no.1, 1341 doi:10.1093/mnras/sty1029 [arXiv:1801.09280 [astro-ph.CO]].
  • (40) G. Efstathiou, Y. Z. Ma and D. Hanson, Mon. Not. Roy. Astron. Soc. 407 (2010) 2530 doi:10.1111/j.1365-2966.2010.17081.x [arXiv:0911.5399 [astro-ph.CO]].
  • (41) C. L. Francis and J. A. Peacock, Mon. Not. Roy. Astron. Soc. 406 (2010) 14 doi:10.1111/j.1365-2966.2010.16866.x [arXiv:0909.2495 [astro-ph.CO]].
  • (42) J. Muir and D. Huterer, Phys. Rev. D 94 (2016) no.4, 043503 doi:10.1103/PhysRevD.94.043503 [arXiv:1603.06586 [astro-ph.CO]].
  • (43) C. J. Copi, M. O’Dwyer and G. D. Starkman, Mon. Not. Roy. Astron. Soc. 463 (2016) no.3, 3305 doi:10.1093/mnras/stw2163 [arXiv:1605.09732 [astro-ph.CO]].
  • (44) L. Kofman and A. A. Starobinsky, Sov. Astron. Lett. 11 (1985) 271 [Pisma Astron. Zh. 11 (1985) 643].
  • (45) R. G. Crittenden and N. Turok, Phys. Rev. Lett. 76 (1996) 575 doi:10.1103/PhysRevLett.76.575 [astro-ph/9510072].
  • (46) A. Manzotti and S. Dodelson, Phys. Rev. D 90 (2014) no.12, 123009 doi:10.1103/PhysRevD.90.123009 [arXiv:1407.5623 [astro-ph.CO]].
  • (47) N. Taburet, C. Hernandez-Monteagudo, N. Aghanim, M. Douspis and R. A. Sunyaev, Mon. Not. Roy. Astron. Soc. 418 (2011) 2207 doi:10.1111/j.1365-2966.2011.19474.x [arXiv:1012.5036 [astro-ph.CO]].
  • (48) A. Pourtsidou, D. Bacon and R. Crittenden, Mon. Not. Roy. Astron. Soc. 470 (2017) no.4, 4251 doi:10.1093/mnras/stx1479 [arXiv:1610.04189 [astro-ph.CO]].
  • (49) M. Ballardini, D. Paoletti, F. Finelli, L. Moscardini, B. Sartoris and L. Valenziano, arXiv:1712.02380 [astro-ph.CO].
  • (50) A. Cooray, Phys. Rev. D 65 (2002) 103510 doi:10.1103/PhysRevD.65.103510 [astro-ph/0112408].
  • (51) N. Afshordi, Phys. Rev. D 70 (2004) 083536 doi:10.1103/PhysRevD.70.083536 [astro-ph/0401166].
  • (52) P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A21 doi:10.1051/0004-6361/201525831 [arXiv:1502.01595 [astro-ph.CO]].
  • (53) A. Raccanelli, E. Kovetz, L. Dai and M. Kamionkowski, Phys. Rev. D 93 (2016) no.8, 083512 doi:10.1103/PhysRevD.93.083512 [arXiv:1502.03107 [astro-ph.CO]].
  • (54) L. Knox, Phys. Rev. D 52 (1995) 4307 doi:10.1103/PhysRevD.52.4307 [astro-ph/9504054].
  • (55) G. Jungman, M. Kamionkowski, A. Kosowsky and D. N. Spergel, Phys. Rev. D 54 (1996) 1332 doi:10.1103/PhysRevD.54.1332 [astro-ph/9512139].
  • (56) U. Seljak, Astrophys. J. 482 (1997) 6 doi:10.1086/304123 [astro-ph/9608131].
  • (57) M. Zaldarriaga and U. Seljak, Phys. Rev. D 55 (1997) 1830 doi:10.1103/PhysRevD.55.1830 [astro-ph/9609170].
  • (58) M. Kamionkowski, A. Kosowsky and A. Stebbins, Phys. Rev. D 55 (1997) 7368 doi:10.1103/PhysRevD.55.7368 [astro-ph/9611125].
  • (59) C. R. Contaldi, M. Peloso, L. Kofman and A. D. Linde, JCAP 0307 (2003) 002 doi:10.1088/1475-7516/2003/07/002 [astro-ph/0303636].
  • (60) A. A. Starobinsky, JETP Lett. 55 (1992) 489 [Pisma Zh. Eksp. Teor. Fiz. 55 (1992) 477].
  • (61) C. Dvorkin and W. Hu, Phys. Rev. D 81 (2010) 023518 doi:10.1103/PhysRevD.81.023518 [arXiv:0910.2237 [astro-ph.CO]].
  • (62) R. Adam et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A1 doi:10.1051/0004-6361/201527101 [arXiv:1502.01582 [astro-ph.CO]].
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
359369
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description