WMAP 3-year constraints on the inflationary expansion Fabio Finelli INAF/IASF-BO – Istituto di Astrofisica Spaziale e Fisica Cosmica di Bologna INAF/OAB – Osservatorio Astronomico di Bologna also supported by INFN Bologna Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

MOTIVATIONS Inflation is the dominant paradigm. Inflationary predictions survived many observations: nearly flat spatial sections, spectrum of gaussian nearly scale invariant curvature fluctuations are now firm points of our cosmological model. WMAP1 has been able to first detect large scale curvature perturbations (TE) and to constrain inflationary models: excityng times are coming with future CMB and LSS data. Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

OUTLINE 1. Constraints on inflation at first order in HFF Analyze WMAP3 data (plus CMBsmall and LSS data) with different methods of comparing inflationary predictions with observations wtr to WMAP team: 1. Constraints on inflation at first order in HFF 2. Second order in HFF (i.e. including running) 3. Relaxing theoretical priors a. Tensor-to-scalar ratio as a free parameter b. Blue spectrum for GW Based on work in collaboration with Maria Rianna and Reno Mandolesi, astro-ph/0608277 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Curvature perturbation, gravitational waves 1st Hubble crossing 2nd Standard Inflation Curvature perturbation, gravitational waves Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Inflation driven by scalar field Einstein-Klein-Gordon action: S = R d 4 x p ¡ g 1 6 ¼ G + £ 2 ¹ º @ Á V ( ) ¤ Á ( t ) = + ± x ; Homogeneous EOM V ; Á ( ) + Ä 3 H _ = T = ½ Á _ 2 + V ( ) Pressure and energy densities T i = p Á _ 2 ¡ V ( ) Kinematic configuration: slow-roll V ( Á ) > _ 2 V ( Á ) À _ 2 3 H _ Á + V ; ( ) ' Ä a > H 2 ' 8 ¼ G 3 V ( Á ) Accelerated expansion Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Standard Inflationary Predictions Gravitational Waves Scalar Perturbations Q (Q is the g.i fluctuation associated to the Klein-Gordon inflaton) Both where Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Courtesy by W. Kinney Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

² = ´ = V M V M 2 Á p l Á 2 p l Courtesy by W. Kinney Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

² ' ² ' 4 ¡ ´ 1 2 Horizon Flow Functions (Schwarz et al, 2001) ² = = ( d l n H ¡ N _ 2 Ä a = ( 1 ¡ ² ) H 2 ² = H ( N i ) ² 2 ´ 1 H d t = ¡ Ä 3 ² n + 1 = d l j N acceleration ² 1 < Single-field models: Implicit relation can be inverted assuming first two HFF small V = 3 H 2 8 ¼ G ¡ 1 ² ¢ ² 1 ' 6 ¼ G ³ V ; Á ´ 2 V ; Á = ¡ 3 H 2 ( 4 ¼ G ) 1 ² + 6 ¢ ² 2 ' 1 4 ¼ G · ³ V ; Á ´ ¡ ¸ V ; Á 3 H 2 = ² 1 ¡ + 5 6 ² 1 ' ² 2 ' 4 ¡ ´ Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

The zoo plot in (2,1) plane Courtesy by Schwarz et al., JCAP 0408003 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

CosmoMC CAMB Getdist Markov Chain Monte Carlo A.Lewis & S.Bridle CosmoMC Phys. Rev. D 66 (2002) http://cosmologist.info/cosmomc Markov Chain Monte Carlo CAMB Getdist Parallelized Einstein-Boltzmann code Statistical analysis Data CMB LSS: 2dF (SDSS) Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

b = ¡ 2 ( C + 1 ) ² r = 1 6 ² [ + C ] = e x p h b + l n ³ ´ : i P ( k First order HFF Expansion of the ln of the power spectra in terms of ln(k/k*): when you sample HFF there is no danger that the PS go negative (even more important at second order in HFF). P ( k ) ¤ = e x p h b + 1 l n ³ ´ : i Leach, Liddle, Martin, Schwartz (2002) P T ( k ¤ ) = 1 6 H 2 ¼ m p l P S ( k ¤ ) = H 2 ¼ ² 1 m p l b T = ¡ 2 ( C + 1 ) ² b S = ¡ 2 ( C + 1 ) ² b T 1 = n ¡ 2 ² b S 1 = n ¡ 2 ² r = 1 6 ² [ + C 2 ] C ´ ° E + l n 2 ¡ ¼ : 7 9 6 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Effect of gravity waves on CMB anisotropies in temperature and polarization = ¡ 8 n T ( S 1 ) : 3 2 l ( + 1 ) C = 2 ¼ £ ¹ K ¤ Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

¤ f ! ; H z 1 A ² g k = : 1 M p c R ´ = ¡ f ( h ; ­ : ) n ' 5 ² Flat CDM ¤ Data used CMB: ACBAR, VSA, CBI, BOOMERANG03 (B03), WMAP1 or WMAP3 LSS: 2dF e SDSS (used separately) 7-dim Parameter space f ! b ; c H z r e 1 A s ² 2 g k ¤ = : 1 M p c ¡ Although this pivot scale does not coincide with the ones used by WMAP (0.002 and 0.05), it is placed close to the range of k probed by the whole dataset. f ! b ; c H 1 A S ( k ¤ ) z r e ² 2 g Derived parameters R 1 ´ C T S = ¡ f ( h ; ­ ¤ : ) n ' 5 ² Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

r = 1 6 ² ² > : 5 < ­ h 1 ; 9 4 < H 1 ; z 3 1 < ² ; : 5 KG inflation r = 1 6 ² ² 1 > WMAP1 WMAP3 ACBAR+CBI+VSA+B03+2dF + WMAP1/WMAP3 Priors: : 5 < ­ b h 2 1 ; c 9 4 < H 1 ; z r e 3 1 ¡ 4 < ² 2 ; : 5 Errors reduced with respect to the first year, tau and ns significantly decreased (see Spergel et al. 06) Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

R < : 3 ² < : 7 R < : 7 ² < : 3 < 1 : 4 £ < 1 : £ WMAP1 WMAP3 R 1 < : 3 ² 1 < : 7 R 1 < : 7 ² 1 < : 3 H i n f l m P < 1 : 4 £ ¡ 5 H i n f l m P < 1 : £ ¡ 5 V m 4 P l < : 2 5 £ 1 ¡ V m 4 P l < : 1 £ ¡ m 2 P l 1 6 ¼ ³ V ; Á ´ < : 3 m 2 P l 1 6 ¼ ³ V ; Á ´ < : 7 V 1 4 i n f l < 2 : 7 £ 6 G e V 1 4 i n f l < 2 : 3 £ 6 G e WMAP1 WMAP3 WMAP1 WMAP3 n S < 1 2 ¾ Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Comparison with WMAP team results Tensor-to-scalar ratio (without marginalizing on n) WMAPII+CMBsmall+2dF WMAPII+2dF ² 1 < : 7 r : 2 < 3 Spergel et al. (2006) r : 1 < 2 7 Scalar spectral index (marginalized over all other 6 params) WMAPII only n S = : 9 8 7 + 1 ¡ 3 n S = : 9 6 § 1 7 Spergel et al. (2006) PS: do not quote the number in the abstract of the WMAPII paper since there are no tensor Spergel et al. (2006) n S = : 9 5 1 + ¡ Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

5 9 < p : 3 1 7 < ¸ 9 2 m Á = ¸ V / Á ² ' V ( Á ) = ¤ e x p ³ ´ 4 Large field chaotic inflation WMAP1 WMAP3 V / Á ° ² 1 ' ° 4 ¢ N 2 With WMAP3 models with an exit from inflation (2>0) are definitively preferred! Power-law inflation WMAP1 WMAP3 V ( Á ) = ¤ 4 e x p ³ ¸ m P l ´ ¸ = q 1 6 ¼ p a ( t ) / p ² 1 = p ¡ ² i = ¸ 2 5 9 < p : 3 1 7 < ¸ 9 2 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

2dF02 vs SDSS WMAP3+CMBsmall+2dF02 WMAP3+CMBsmall+SDSS 2dF allows more gravitational waves than SDSS (this applies also to subsequent releases -2dF05- and to the case with running) Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

The effect of CMBsmall Is Harrison-Zeldovich (nS = 1,r = 0) inside or not 2? WMAP3+SDSS WMAP3+CMBsmall+SDSS The answer depends on systematics and foregrounds, but also on the inclusion of CMBsmall (ACBAR, VSA, CBI, B03). CMBsmall prefers less power on small scales: not only HZ is disfavoured at 2, but also nS = 1 line. NB: Our results for WMAP3+SDSS fully agree with Kinney et al. 2006, Easther & Peiris 2006. Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Running of the spectral index <->Second order in HFF In inflationary context both running and tensor need to be included. Running is included both in the scalar and in the tensor spectral index (second order in SR parameters as predicted by Kosowsky & Turner (1995)). By using HFFs it is just needed an extra parameter: ² 3 A self-consistent expression at second order in HFFs is needed also for amplitudes and spectral indices, not just runnings. d ² 2 N ´ 3 ' » V + 6 1 ¡ 8 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Green`s function method (GFM) Second order in HFF I Green`s function method (GFM) Stewart & Gong (2001), Leach, Liddle, Martin, Schwartz (2002) P ( k ) ¤ = e x p h b + 1 l n ³ ´ 2 : i b S 1 = n ¡ b S 2 = ® b T 1 = n b T 2 = ® b S 1 = ¡ 2 ² ( C + 3 ) b S 2 = ¡ ² 1 3 b T 1 = ¡ 2 ² ( C + ) b T 2 = ¡ ² 1 R = 1 6 ² h + C 2 ³ ¡ ¼ 5 ´ 8 4 3 i C ´ ° E + l n 2 ¡ ¼ : 7 9 6 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Method of comparison equations (MCE) Second order in HFF II Method of comparison equations (MCE) Casadio,FF,Kamenschik,Luzzi,Venturi (2006) P ( k ) ¤ = e x p h b + 1 l n ³ ´ 2 : i b S 1 = n ¡ b S 2 = ® b T 1 = n b T 2 = ® b S 1 = ¡ 2 ² ( D + 3 ) b S 2 = ¡ ² 1 3 b T 2 = ¡ ² 1 b T 1 = ¡ 2 ² ( D + ) R = 1 6 ² h + D 2 ³ 9 ) ¡ ¢ ¼ 4 8 ´ 3 i D ´ 1 3 ¡ l n ¼ : 7 2 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

V ( Á ) = m V ( Á ) = h 1 ¡ a r c t n 5 i Á = ¡ : 3 M ² ² ² 2 V ( Á ) = h 1 ¡ 2 ¼ a r c t n 5 M p l i Wang, Mukhanov & Steinhardt (1997) Á ¤ = ¡ : 3 M p l ² 2 ² 1 ² 2 3 Solid: numerical Dashed: 1st HFF approximation Dotted: 2nd HFF approximation Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

f ! ; H z 1 A ² g WMAP1 WMAP3 f ! b ; c H 1 A S ( k ¤ ) z r e ² 2 3 g z r e 1 A s ² 2 3 g WMAP1 WMAP3 + CMBsmall + SDSS f ! b ; c H 1 A S ( k ¤ ) z r e ² 2 3 g GFM formula This result agrees with Leach and Liddle (2003) for WMAPI and with Martin and Ringeval (2006) for WMAPII. Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

f ! ; H z 1 A ² g ² ; < ² < 1 WMAP3+CMBsmall+SDSS MCE GFM z r e 1 A s ² 2 3 g MCE GFM PS: the analytic PS requires ² 1 2 ; 3 < and not ² 3 < 1 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

WMAP team Usefulness of our k_*! ® = ¡ : 6 § 2 8 ® = ¡ : 7 § 2 6 ® = ¡ WMAP1/WMAP3+ACBAR+CBI+VSA+B03+SDSS WMAP1/WMAP3+ACBAR+CBI+VSA+B03+SDSS Scalar running (marginalized over all other 7 params) WMAP1 WMAP3 WMAPII+CMBsmall+2dF ® S = ¡ : 6 § 2 8 ® S = ¡ : 7 § 2 6 WMAPII+CMBsmall+SDSS WMAP team ® S = ¡ : 8 7 + 4 1 3 WMAPII+B03+ACBAR Usefulness of our k_*! Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

f ! b ; c H z r e 1 A s ² 2 d N g Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

¢ Â = l n L ¡ £ 5 6 7 1 + 3 ' 4 for 2df02 Second order HFF Second order HFF parametrization has one more parameter --> constraints on cosmological parameters present in 1st order parametrization degrade Higher r, more blue ns are allowed, HZ is within 2, inflationary models with ns~1 are allowed at 1. for 2df02 ¢ Â 2 = l n L r u ¡ o £ 5 6 7 1 + 3 ' 4 Second order HFF First order HFF WMAP3+CMBsmall+2dF Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Tensor-to-scalar ratio as a free parameter Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

NEC (null energy condition) violating Inflation Hubble parameter growing in time _ H = ¡ 8 ¼ G 3 ( ½ + p ) > NEC_viol ² 1 < n T = ¡ 2 ² 1 > K-G prior: - 1 ¡ 2 < ² 4 ACBAR+CBI+VSA+ B03+ 2dF + WMAP1/WMAP3 p = 1 2 g ¹ º @ Á ¡ V ( ) Â ² 1 > ¡ : 3 2 WMAP1 ² 1 = ¡ p ² i = ¸ 2 WMAP3 beyond 2 ¾ with WMAP3 Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Conclusions:I First order in HFF Models with an exit from inflation (2 >0) preferred by WMAP3 WMAP3 shows how quartic model and HZ are disfavoured wrt the massive model. With our combination of data also inflationary models with n 1 are around the 2 contours (due to CMBsmall) Still early to decide if more or less tensors wrt to the consistency relation are preferred by data (less tensors are also predicted by non-minimal coupling of the inflaton): very important to predict the tensor-to-scalar ratio, otherwise difficult to constrain the potential. KG is well inside the present constraints. Models predicting a blue spectrum for tensors are equally good fit to the current data as standard inflationary models. Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

Conclusions:II Runnings of the spectral indices (second order in HFFs) Do not cut out running evidence with priors when using HFFs! Statistical evidence for running increased from WMAPI to WMAPII: it is zero at 2 or beyond. If is not primordial and is due to systematics then these have not been understood yet Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006