Super-Excited Initial States of Inflation Amjad Ashoorioon, INFN, Bologna Based on May 2nd, 2017 A. Ashoorioon, R. Casadio, G. Geshnizjani, H. J. Kim, arxiv:1702.06101 A. Ashoorioon, T. Koivisto, R. Casadio, JCAP 1612 (2016) no.12, 002 A. Ashoorioon, Tomi Koivisto, Phys.Rev. D94 (2016) no.4, 043009 A. Ashoorioon, K. Dimopoulos, G. Shiu, M. Sheikh-Jabbari, Phys.Lett. B737 (2014) 98-102 A. Ashoorioon, K. Dimopoulos, G. Shiu, M. Sheikh-Jabbari, JCAP 1402 (2014) 025 A. Ashoorioon, U. Danielsson, D. Chialva, JCAP 1106 (2011) 034 Origin of Mass 2017, Odense, Denmark
Introduction Much has been said about the effect of new physics, The effect is usually claimed to be subdominant and dependent on where H is the Hubble parameter during inflation, M Hand n 1 Easther, Greene, Kinney, Shiu (2001,2002) Ashoorioon, Mann (2004, 2005) M on the inflationary predictions. In some cases, it was shown the effect in the power spectrum can be quite large. Martin & Brandenberger (2003) In particular, motivated by some condensed-matter studies, they assumed! 2 = k 2 0 k 4 + 0 k 6! 2 = k 2 where 0, 0 > 0 Gluing of the solution in disjoint regions H M n Kempf, Niemeyer (2000, 2001) Kaloper, Kleban, Lawrence, et. al. (2002) Oscillatory power spectrum where the amplitude of oscillations and the second Bog. coeff. is proportional to
Introduction At the time, backreaction was assumed to constrain the excited states considerably. B 2 / 2 1 Tanaka (2001) Solving the mode equation numerically from WKB positive freq. mode z 3 0 2 0 C = P S /P B.D. 1 The correction to the power spectrum in this model could be quite large! Marozzi & Joras (2008) Later it was shown that the backreaction effect is not that constraining B. p HMP M 2 Greene, Shiu, Schalm & van der Schaar (2004) M is the scale of new physics when the modes get excited.
Outline Cosmological Perturbation Theory & Highly Excited Initial State Hemispherical Anomaly from Asymmetric Excited States Statistical Anisotropy from SO(3) non-invariant Excited States Modified Dispersion Relation from the Effective Field Theory of Inflation (EFToI) Estimation of Bogoliubov coefficients Conclusion & Plans for Future Work
Cosmological Perturbations and Excited Initial States The equation for gauge-invariant scalar perturbations u~ 00 k + k 2 z 00 u ~k =0 z a 0 z H, In a quasi-de-sitter background u = a 0 z + a 0 a( ) ' 1 H H a0 a the most generic solution to the E.O.M. in the leading order in slow-roll parameters p h i u k ( ) = k S H (1) 2 3/2 ( k )+ k S H (2) 3/2 ( k ) where the Bogoliubov coefficients satisfy the Wronskian condition S k =1 and S k =0 Bunch-Davies vacuum
Cosmological Perturbations and Excited Initial States Any excited state contains massless quanta whose positive pressure can tamper the slow-roll Inflation Derailing can be avoided if non-bd 0 p 0 non-bd H 0 The second equation, which is the stronger one, can be written as As a specific example, let us consider the crude model in which the modes get excited when k/a( ) =M S k = 0 0. p HM Pl M 2 Greene, Shiu, Schalm & van der Schaar (2004)
Cosmological Perturbations and Excited Initial States Scalar power spectrum P S = k3 2 2 u k z 2 k/h!0 Ashoorioon, Dimopoulos, Sheikh-Jabbari & Shiu (2013) P S = P BD S, Parameterization of the Parameter Space S ~ k = e i' S cosh S, S ~ k = e i' S sinh S, Let us focus on V ( )= 1 2 m2 ' 2 apple apple Using the Planck normalization for the amplitude of density perturbations: that with the help of backreation condition, yields
Cosmological Perturbations and Excited Initial States Very large values of S ( ) are phenomenologically allowed. The same could be said about tensor perturbations: v ± k ( ) = p 2 h T k H (1) 3/2 ( k )+ T k H (2) 3/2 ( k ) i
Cosmological Perturbations and Excited Initial States Violation of the consistency relation Using the same type of parameterisation as long as the bounds from backreaction are respected
Cosmological Perturbations and Excited Initial States We will have an enhancement of non-gaussianity for the local configurations f local NL =4.17 Induce scalar substantial spectral index running and blue tensor spectrum If d 2 ln S (d ln k) 2 =0 where d 2 ln S (d ln k) 2 < 0 dn S d ln k < 0 d 2 2 ln S ln S d S (d ln k) 2 =2d2 d ln k) 2 2 d ln k Ashoorioon, K. Dimopoulos, M. Sheikh-Jabbari & G. Shiu (2014) Running would be always negative d' T d T Likewise by having or one can achieve d ln k > 0 d ln k > 0 n T > 0
Hemispherical Anomaly from Asymmetric Excited States Hemispherical Asymmetry by position-dependent excitations Ashoorioon & Koivisto (2015) T (ˆx) = T iso (1 + 2A(ˆx.ˆn)) A ' 6 7% S 0 =sinh S (1 + "ˆx.ˆn)e i' S P S = P iso 1+2A(ˆx ˆn)+B(ˆx ˆn) 2 A ' " & 0.07 Quadrupolar modulation in position space proportional to B ' " 2! k 3 <<! k 1! k 2! k l=2500! k 3! k l=10! k 3 <<! k 1! k 2! k l=2500! k 3! k l=10! n! n! k 1! k 3! k 2! x 1 x! 2 x! 3 n!! x 1! x 2! x 3! n! k 2! k 1! n! k3 f max NL ' f (0) NL (1 + 2" +3"2 ) 4.81 f NL ' 1.17 f min NL ' f (0) NL (1 2" +3"2 ) 3.64
Statistical Anisotropy from SO(3) non-invariant Excited States i P S = P iso h1+m(ˆk) T (ˆk) = A, C, (odd multipoles) have to be pure imaginary numbers T iso (ˆk) h i 1+M(ˆk) Ashoorioon, Koivisto, Casadio (2016). M(ˆk) =A ˆk ˆn + B (ˆk ˆn) 2 + C (ˆk ˆn) 3 +..., dipole quadrupole Octupole Kim & Komatsu (2013), doing data analysis on the Planck 2013 data 0.03 <B<0.033 (95% C.L.) We use the following parameterization: In the where S 1 ' S ' 2 A =0 B ' 2" 2 C =0 Now from the observation constraint on B, the following constraint is obtained on " 2 2 0.015 < " 2 < 0.0165 (95% C.L.) remains indefinite in this regime from the constrains on the quadrupole moment.
3 when the preferred direction, n, makes an plane of the triangular configuration. Statistical Anisotropy from SO(3) non-invariant Excited would occur for a local configuration States /2. This variation between the miniwhen preferred direction, makes an probed by Planck and ~ corresponding to largest ~k1 the m obviously enhances for then,local concorresponding to shortest scales k3 plane of the triangular configuration. coplanar thethe preferred scale atwith which cosmicdirection variance is negligible, l ' 10. For ' 0.01 and "2 ' 0.0165 and minimum for the largest positive ~k3 he data, Eq. (32), and an inflationary n would occur for a local configuration 01 are respectively /2. This variation between the minin max um obviously enhances for the local(72) con-~k2 ~k1 fnl ' 4.3 coplanar with the preferred direction m and minimum for the largest positive the data, Eq. (32), and an inflationary min fnlrespectively ' 4.03. (73) ~k2 ~k1.01 are max when the largest wavenumould occur fnl ' 4.3 (72) (or antiparallel) to n. The minimum n the small wavenumber is parallel (or he preferred direction. The di erence~k1 min ~k3 con(73) s of fnon-gaussianity for these two NL ' 4.03. ~ L ' 0.27 which can be used to constraink2 occur when the largest wavenumeould maximum value of, one would obtain (or antiparallel) to n. The minimum fied consistency relation, the maximum nues theare small parallel (or maxwavenumber ismin fnl ' 2.96 and fnl ' 2.78. he di erence ity preferred parameter,direction. fnl takesthe intermediate min es of non-gaussianity two the conand f max dependingfor onthese the angle n ~k3 fnl ' 0.27
Statistical Anisotropy from SO(3) non-invariant Excited States Purely anisotropic Initial Condition, S 1 In this case: A =0 B = 2 cos ' S " 2 C =0 One intriguing case is when ' S = 4 B =0 no bound from quadrupole term on " 2 Still to be able to to trust the expansion in " 2 and 2 we assume 0 " 2. 1 We also set. 2 =0 bipolar bispectrum that can reach f NL ' 20 30
Dispersion Relation from the Effective Field Theory of Inflation (EFToI) In unitary gauge where the inflaton fluct. are eaten by the perturbation of the metric, the time diffeomorphism is broken. In this gauge, the most general action that respects the remaining spatial diffeomorphism is The time-diffeomorphism which is non-linearly realized can be restored using the Stueckelberg procedure t! t + 0 (x µ ) 0 (x µ )! (x µ ) then we demand that under t! t + 0 (x µ )! 0
Dispersion Relation from the Effective Field Theory of Inflation (EFToI) u = a K ij (@ i @ j + @ i g 0j ), u 00 +( 0 k 2 + 0 k 4 2 + 0 k 6 4 2 )u =0 2 In fact implementing the stueckelberg mechanism to the spatially invariant action, yields As expected terms proportional to k 6 2 appears. However terms proportional to appears too which leads to Ostrogradski ghosts. Also there will be correction of the dispersion relation from the from the presence of k 4 2 and k 2 2 1 = 2 = 4 =0 k 6 at high momenta
Dispersion Relation from the Effective Field Theory of Inflation (EFToI) Also there will be correction of the dispersion relation from the from the presence of k 4 2 and k 2 2 1 = 2 = 4 =0 k 6 at high momenta With 3 > 0, we can achieve our desired scenario where 0 > 0, 0 < 0 and 0 > 0 0 and respectively the speed of sound could be always set to one by a reparameterization d! c s d We also assume that the dispersion relation never becomes tachyonic on sub-hubble scales z 0 1 0 2 4 We also assume there is one horizon-crossing event corresponding to! 2 (k) =2H 2 For z> 1 3, there is only one turning point automatically. For 1 4 apple z apple 1 3
Estimation of Bogoliubov coefficients Corley-Jacobson Dispersion Relation In this case the exact normalized solution to the mode eq. is known Ashoorioon, Danielsson et. al. (2011) On the other hand soling the mode equation via gluing technique k apple 1 region I P S /P B.D. 1.3 1 apple k region II 1.2 1.1 P S P B.D. 1.0 0.9 γ real γ (2) real 1=O( 2 ) 0.8 0.7 (2) 1=O( ) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ϵ
Estimation of Bogoliubov coefficients Corley-Jacobson Dispersion Relation As we move towards! 0 with the one with an excited state p u X k ( ) = 2, this solution should reconcile more and more h k H (1) 3/2 ( k )+ kh (2)( k ) 3/2 i Now, if we define f( ) u X k ( ), lim!0 df ( ) d = i p 2k ( k k )= lim!0 d( u k ( )) d lim!0 d 3 f( ) d 3 = p 2 p k ( k + k )= lim!0 d 3 ( u k ( )) d 3 N (exact) k ( ) = e 4 e 2 + e 2 8 4p 1e 3/2 5 4 + i e i 4 + 4 (4 4i) p 2e 3/2 5 4 64 e 3/2 k 1/2!0 (1 + e / )sinh 2 (5 2 /8) 25 64 4 i +i 4 + 4 N (glue) k ( ) ' 2 16
Estimation of Bogoliubov coefficients Sixth Order Polynomial with an intermediate negative group velocity In terms of x k Let us estimate the number density of particles 0 x 4 1 0x 2 1 =1 2 x 2 1 In region I: In region II:
Estimation of Bogoliubov coefficients Sixth Order Polynomial with an intermediate negative group velocity For 0 =0.2
Estimation of Bogoliubov coefficients Sixth Order Polynomial with an intermediate negative group velocity Introducing the variable x k u 00 k + 0x 4 0 x 2 2 +1 x 2 u k =0. The positive frequency WKB mode in infinite past as the initial condition one can integrate the mode equation for specific values of and. 0 0 Largest enhancement in the power spectrum is obtained for P S = S P B.D. where for these values of parameters S ' 14.738 0 ' 0.2 and 0 ' 2 0 4 Now we wan to translate this to the language of excited states. Unfortunately does not saturate as! 0. Mathematica can find an implicit solution for the e.o.m. u k (x) =c 1 u (1) k (x)+c 2u (2) k (x) c 1 c 2 c 2 c 1 = i. u (i) k (1) = 1 u(i)0 k (1) = 1 and
Estimation of Bogoliubov coefficients Sixth Order Polynomial with an intermediate negative group velocity c 1 c 2 c 2 c 1 = i. c 1 = 1 p 2a and c 2 = i a p 2 where a 2 R a is determined such that the power spectrum from result. u k ( ) is the same as the numerical There will be four solutions where two of them are negative of the other ones. p We look for a solution u X h i k ( ) = k H (1) 2 3/2 ( k )+ (2)( k ) kh that produces such 3/2 value for power spectrum, it is continuous at an earlier point and it s derivative is also continuous at that point. From all 4 solutions for a, only has such a characteristic that u k ( ) and u X k ( ) reconcile after the point of integration.
Estimation of Bogoliubov coefficients Sixth Order Polynomial with an intermediate negative group velocity Re[u k ] 0.20 0.15 0.10 Re[u k excited ] Re[u k exact ] 0.05 1 2 3 4 x Im[u k ] 20 1 2 3 4 x Im[u k excited ] Im[u k exact ] -20-40 (x cross )= 1.88359 8.7681 i (x cross )=1.95519 8.80935 i N k k 2 = 80.4275
Conclusion Not always true that the effect of evolution of the modes when they have high momenta is sub-dominant. If there is modified dispersion relation with an interim phase with negative group velocity, the corrections to the power spectrum could be quite large. I also showed how one can realize these dispersion relations in the EFToI. I provided a method to Bogoliubov coefficients such that they lead to the exact estimate of the power spectrum. With the help of dispersion relation were built.! 2 = k 2 0 k 4 + 0 k 6 excited states with k & 9 If perturbations start from such super-excited states, M. few 10H
Conclusion Generally, we will have enhancement of the non-gaussianity in the local shape for such super-excited states. Position-dependent modulation of these states can be used to explain the observed hemispherical anomaly. Rotational Symmetry Breaking Excited Initial States can create power spectrum with statistical anisotropy and anisotropic bispectrum. Bispectrum in such modified dispersion relations is what we are investigating. Effect of (rk) 2 on tensor perturbations EOM is under investigation too.
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