1. 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

2. 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

3. 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/
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

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

5. Inflation driven by scalar field
Einstein-Klein-Gordon action: S = R d 4 x p g 1 6 G + 2 @ Á V ( ) Á ( t ) = + x ;
Homogeneous EOM V ; Á ( ) + Ä 3 H _ = T = 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 G 3 V ( Á )
Accelerated expansion
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

6. 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

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

8. 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

9. ² ' ² ' 4 ¡ ´ 1 2 Horizon Flow Functions (Schwarz et al, 2001) ² = = (d l n H N _ 2 Ä a = ( 1 ) H 2 = H ( N i ) 2 1 H d t = Ä 3 n + 1 = d l j N
acceleration 1
<
Single-field models:
Implicit relation can be inverted assuming first two HFF small V = 3 H 2 8 G 1 1 ' 6 G V ; Á 2 V ; Á = 3 H 2 ( 4 G ) 1 + 6 2 ' 1 4 G V ; Á V ; Á 3 H 2 = 1 + 5 6 1 ' 2 ' 4
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

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

11. CosmoMC CAMB Getdist Markov Chain Monte Carlo
A.Lewis & S.Bridle
CosmoMC
Phys. Rev. D 66 (2002)
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

12. 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 : i
Leach, Liddle, Martin, Schwartz (2002) P T ( k ) = 1 6 H 2 m p l P S ( k ) = H 2 1 m p l b T = 2 ( C + 1 ) b S = 2 ( C + 1 ) b T 1 = n 2 b S 1 = n 2 r = 1 6 [ + C 2 ] C E + l n 2 : 7 9 6
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

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

14. ¤ 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 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 g
Derived parameters R 1 C T S = f ( h ; : ) n ' 5
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

15. r = 1 6 ² ² > : 5 < ­ h 1 ; 9 4 < H 1 ; z 3 1 < ² ; : 5
KG inflation r = 1 6 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 ; : 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

16. R < : 3 ² < : 7 R < : 7 ² < : 3 < 1 : 4 £ < 1 : £
WMAP1
WMAP3 R 1
< : 3 1
< : 7 R 1
< : 7 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 V ; Á
< : 3 m 2 P l 1 6 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
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

17. Comparison with WMAP team results
Tensor-to-scalar ratio (without marginalizing on n)
WMAPII+CMBsmall+2dF
WMAPII+2dF 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

18. 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 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 m P l = q 1 6 p a ( t ) / p 1 = p i = 2 5 9
< p : 3 1 7
< 9 2
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

19. 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

20. 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

21. 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: 3
A self-consistent expression at second order in HFFs is needed also for amplitudes and spectral indices, not just runnings. d 2 N 3 ' V + 6 1 8
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

22. 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 2 : i b S 1 = n b S 2 = b T 1 = n b T 2 = b S 1 = 2 ( C + 3 ) b S 2 = 1 3 b T 1 = 2 ( C + ) b T 2 = 1 R = 1 6 h + C 2 5 8 4 3 i C E + l n 2 : 7 9 6
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

23. 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 2 : i b S 1 = n b S 2 = b T 1 = n b T 2 = b S 1 = 2 ( D + 3 ) b S 2 = 1 3 b T 2 = 1 b T 1 = 2 ( D + ) R = 1 6 h + D 2 9 ) 4 8 3 i D 1 3 l n : 7 2
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

24. V ( Á ) = m V ( Á ) = h 1 ¡ a r c t n 5 i Á = ¡ : 3 M ² ² ²2 V ( Á ) = h 1 2 a r c t n 5 M p l i
Wang, Mukhanov & Steinhardt (1997) Á = : 3 M p l 2 1 2 3
Solid: numerical
Dashed: 1st HFF approximation
Dotted: 2nd HFF approximation
Galileo Galilei Institute for Theoretical Physics, Oct. 16, 2006

25. f ! ; H z 1 A ² g WMAP1 WMAP3 f ! b ; c H 1 A S ( k ¤ ) z r e ² 2 3 gz r e 1 A s 2 3 g
WMAP1
WMAP3
+ CMBsmall + SDSS f ! b ; c H 1 A S ( k ) z r e 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

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

27. 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

28. f ! b ; c H z r e 1 A s 2 d N g
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29. ¢ 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

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

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

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

33. 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

34. 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