A n' s dn s /d ln k. Harrison-Zel' dovich

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1 4 3 4 Vol 43 No Journal of Shanghai Normal University Natural Sciences Astrophysics Aug DOI /J 1SSN O A B - BICEP2 1 B - B - BICEP r = % CL 7 0σ r = 0 BICEP2 Planck 2 Planck + WP + highl n s = ± r < % CL n' s dn s /d ln k n s = ± r < 0 26 n' s = % CL n' s B - V chaotic inflation V ~ n6 natural inflation V ~ V 0 cos /f Harrison-Zel' dovich n s - 1 ~ CG ZR shnu edu cn shnu edu cn

2 O 10-2 n' s ~ O e-folding N N 20 ~ M - 2 pl = 8πG = V V = λ a n 2n λ = M2 pl 2 V' ( V ) 1 2 V η = M 2 pl V ξ = V'V ' M4 pl 2 V 2 n s n' s n s - 1 2η n s ' 16 η ξ 4 ξ η ξ = α + β - α η α β 1 5 n 2 a n a n = 1 Σn-1 4n 2n - 1 n - 1 k = 1 a n-k 2β - 3α k 2 + αn + α - β k a k - 4 k + 1 n - k 2n - 2k - 1 n - k - 1 a k a 1 λ a 1 = 1 β = 0 a n n 2 = 0 V ~ 2 V ~ e V ~ Σ n /n Taylor Taylor α β λ SUGRA K hler V = e K K i j- D i W D W W - 3 W 2 D i W = i W + i K W 5 7 K = g 1 φ + φ φ + φ 2 + X 2 - c X X shift φ 槡 2Imφ X

3 4 393 W = XΣ h n n + W α α β 1 V V ~ = b n 2n η = -2 c n 2n ξ = -2 d n 2n b 0 = 4 c 0 = 2 d 0 = 2βn 1 b n c n d n Σ n k = 1 b n = 4 n + 1 a n+1 + Σ n k = 1 [ ] 10 4 k + 1 n - k + 1 a k +1 a n-k +1 - Σ k +1 a m a k +2 -m b 11 n-k c n = 2 n + 1 2n + 1 a n+1 - Σ n k = 1 d n = 8 n + 2 2n + 3 n + 1 a n+2 + m = 1 a k +1 c n-k [ ] 12 8 k + 1 n - k + 2 2n - 2k + 3 n - k + 1 a k +1 a n-k +2 - Σ k +1 a m a k +2 -m d 13 n-k 3 e-folding f 0 = 1 t end end V N = Hdt = - t V' f n = a n+1 - Σ n k = 1 d = 2 4 Σ k = 0 f n n + 1 2n m = 1 k + 1 a k +1 f n-k n 1-2n end end b n 2n end end end ~ M pl

4 ( ) k P s = A s k * n s -1 +n s 'ln k /k * /2 17 A s n s 3 n s ' = dn s /d ln k n s ' = 16 η α - 2 β - α η 18 r r α 2 n' s ~ O ~ O 10-2 r r n s n' s n s r = Planck WMAP Planck + WP highl Planck BAO ± Planck + WP ± Planck + WP + lensing ± Planck + WP + highl ± Planck + WP + BAO n' s = ± Planck + WP n' s = ± Planck + WP + lensing n' s = Planck + WP + highl n' s = Planck + WP + BAO 19 Background Imaging of Cosmic Extragalactic Polarization B2 Planck WP r = Supernova Legacy Survey SNLS r = Planck Union BAO BICEP2 r = BAO r = B α β 3 /M pl O 10 Planck A s k = Mpc - 1 = λ n ( ) λ 3π2 2 a n 2 α β 2-1 r A s

5 α β 1σ 2σ α β r 0 2 n' s O 10-2 n s a n b n c n d n f n a n 6 V 5 26 a k x k b k x k k = 0 k = 0 = k = 0 c k x k c n = Σ n k = 0 a k b n-k 6 11 ~ 13 a n b n c n d n 2n 2na n 2n = 2n 2n - 1 a n 2n = 2n 4n n + 1 2n + 1 a n+1 2n = c n 2n b n 2n 21 α 2 α - β α b n - 2 α - β c n - 2 d n 2n = d n 2n 2n 2α n + 2 α - β a n - 8n n + 1 2n + 1 a n+1 2n - 8 α - β n 2 a n 2n a n αb n - 2 α - β c n - 2 d ~ a 1 λ a 1 = 1 5

6 a 1 = a 2 a 3 = β = β 5β - 3α 2880 a 4 = β ( 10α2 2-23αβ + 15β ) a 5 = β ( - 210α α 2 β - 330αβ β ) b 0 = 4 31 b 1 = β 3 32 b 2 = β 5β - 4α 240 b 3 = β ( 18α αβ + 13β ) b 4 = - β ( 20α3-36α 2 β + 5αβ β ) b 5 = - β ( - 42α4 + 43α 3 β + 120α 2 β 2-175αβ β ) c 0 = 2 37 c 1 c 2 = 5β = β 15β - 14α 480 c 3 = β ( 162α αβ + 47β ) c 4 = β ( - 110α3 + 81α 2 β + 110αβ β ) c 5 = β ( 1092α α 3 β α 2 β αβ β ) d 0 = 2β 43 d 1 = β 5β - 3α 12 44

7 4 397 d 2 = β ( 10α2 2-24αβ + 15β ) 480 d 3 = β ( - 126α α 2 β - 234αβ β ) d 4 = - β ( - 90α α 3 β + 34α 2 β 2-180αβ β ) d 5 = - β ( 924α5 + 74α 4 β α 3 β α 2 β αβ β ) f 0 = 1 49 f 1 = - β f 2 = β f 3 = β ( - 30α2 2-29αβ + 25β ) f 4 = β ( 42α α 2 β - 111αβ 2 3-4β ) f 5 = - β ( 630α α 3 β α 2 β αβ β ) ADE P A R AIKIN R W BARKATS D et al BICEP2 Collaboration BICEP2 I Detection of B-mode polarization at degree angular scales J /OL arxiv astro - ph CO 2ADE P A R AGHANIM N ARMITAGE-CAPLAN C et al Planck Collaboration Planck 2013 results XXII Constraints on inflation J /OL arxiv astro - ph CO 3GUTH A H The inflationary universe A possible solution to the horizon and flatness problems J Phys Rev D LINDE A D A new inflationary universe scenario A possible solution of the horizon flatness homogeneity isotropy and primordial monopole problems J Phys Lett B ALBRECHT A STEINHARDT P J Cosmology for grand unified theories with radiatively induced symmetry breaking J Phys Rev Lett LINDE A D Chaotic inflation J Phys Lett B FRESSE K FRIEMAN J A OLINTO A V Natural inflation with pseudo-nambu-goldstone bosonsj Phys Rev Lett GONG Y The challenge for single field inflation with BICEP2 result J /OL arxiv gr - qc 9CHUNG D J H SHIU G TRODDEN M Running of the scalar spectral index from inflationary models J Phys Rev D EASTHER R PEIRIS H Implications of a running spectral index for slow roll inflation J JCAP ~ 8

8 astro - ph / FENG B LI M Z ZHANG R J et al An inflation model with large variations in spectral index Phys Rev D astro - ph / KOBAYASHI T TAKAHASHI F Running spectral index from inflation with modulations J JCAP ~ J J MA Y Z WANG Y Reconstructing the local potential of inflation with BICEP2 data J /OL arxiv astro - ph CO 16CHOUDHURY S MAZUMDAR A Reconstructing inflationary potential from BICEP2 and running of tensor modesj / OL arxiv hep - th 17NAKAYAMA K TAKAHASHI F YANAGIDA T T Polynomial chaotic inflation in supergravity J Journal of Cosmology and Astroparticle Physics ~ 18 18KOBAYASHI T SETO O Polynomial inflation models after BICEP2 J /OL arxiv astro - ph CO 19CHENG C HUANG Q G Constraints on the cosmological parameters from BICEP2 Planck and WMAP J /OL arxiv astro - ph CO 20CHENG C HUANG Q G Constraint on inflation model from BICEP2 and WMAP 9-year data J /OL arxiv astro - ph CO 21LI H XIA J Q ZHANG X Global fitting analysis on cosmological models after BICEP2 J /OL arxiv astro - ph CO 22WU F LI Y LU Y et al Cosmological parameter fittings with the BICEP2 data J /OL arxiv astro - ph CO 23LIZARRAGA J URRESTILLA J DAVERIO D et al Can topological defects mimic the BICEP2 B-mode signal J /OL arxiv astro - ph CO 24HO C M HSU S D Does the BICEP2 observation of cosmological tensor modes imply an era of nearly planckian energy densities J /OL arxiv hep - ph 25ADE P A R AGHANIM N ARMITAGE-CAPLAN C et alplanck Collaboration Planck 2013 results XVI Cosmological parameters J /OL arxiv astro - ph CO 26GRADSHTEYN I S RYZHIK I M JEFFREY A et al Table of integrals series and productsm 7th edition USA Elsevier Academic Press 2007 Power series inflation and primordial gravitational waves FENG ChaojunLI Xinzhou Shanghai United Center for Astrophysics SUCA Shanghai Normal UniversityShanghai China Abstract The tensor-to-scalar ratio can be measured by observing the polarization of photos in the cosmic microwave background It shows very strong power on constraining inflation models However for a single field inflation model it can not predict both a large tensor-to-scalar ratio and a large running of the scalar power spectral index To solve this problem the authors propose an inflation model with power series potential It is found that this model is consistent with latest observations Key words tensor-to-scalar ratio spectral index running of spectral index power series inflation

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