Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/russian/psas/ffk2013/tfiles/Gorbunov_FFK-2013.pdf Äàòà èçìåíåíèÿ: Mon Nov 25 12:35:12 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 03:18:50 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: êàëëèñòî

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Imprints of the Early Universe dynamics in gravity wave spectrum
Dmitry Gorbunov
Institute for Nuclear Research of RAS, Moscow

Workshop on Precision Physics and Fundamental Physical Constants CAO RAS, Pulkovo, Russia

Dmitry Gorbunov (INR)

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Outline

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Outline

1

Problems of the Big Bang Theory

2

Inflationar y stage and reheating

3

Sensitive to reheating observables in GW

Dmitry Gorbunov (INR)

Imprints of early time dynamics in GW

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Problems of the Big Bang Theory

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Outline

1

Problems of the Big Bang Theory

2

Inflationar y stage and reheating

3

Sensitive to reheating observables in GW

Dmitry Gorbunov (INR)

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Problems of the Big Bang Theory

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Initial singularity problem
a a
2

H2 =

8 G , 3

p = w , w > -

1 3

dust:
=

p=0
const , a(t ) = const · (t - ts ) a3
2/3

singular at t = ts
, (t ) = const (t - ts )
2

t s = 0 , H (t ) =

2 3 11 a (t ) = , = H2 = a 3t 8 G 6 G t 2

radiation:
=

p = 1 3
const , a(t ) = const · (t - ts ) a4
1/2

singular at t = ts
, (t ) = const (t - ts )
2

ts = 0 , H (t ) =
Dmitry Gorbunov (INR)

a 1 3 31 (t ) = , = H2 = a 2t 8 G 32 G t 2
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Imprints of early time dynamics in GW


Problems of the Big Bang Theory

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Entropy problem
µ T for equation of state p = p( ) of the primordial plasma we obtain -3d (ln a) = d = d (ln s) p+
µ0

a = 0 - + 3 ( + p) = 0 a

entropy is conser ved in a comoving volume sa3 = const For the visible par t of the Universe: At the "Bang" for the Planck-size volume:
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW

3 SBB s ,0 · lPl 100
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Ss

,0

3 · lH 10

88


Problems of the Big Bang Theory

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Horizon problem lH (t )
a distance covered by photon emitted at t = 0 size of the causally connected par t, that is the visible par t of the Universe ("inside horison") ds2 = dt 2 - a2 (t ) dx 2 = a2 ( ) d 2 - dx lH (t ) =a(t ) dx =a(t )
2 t

ds2 = 0 c dt t 1/H (t ) a(t )

d =a(t )
0

a (t ) t , 0 < < 1 , Lphys a

lH0 /lH,r (t0 ) l
Dmitry Gorbunov (INR)

H0

/l

H,r

(tr ) a(tr )/a0 Hr /H0 a(tr )/a0

1 + zr

30
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Problems of the Big Bang Theory

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Flatness problem
Take non-flat 3-dim manifold (general case) Curvature contribution to the total energy density behaves as curv (t ) 1/a2 (t ) Then at present: 0.01 >
cur v

=



cur v

(t0 )



c

10-4 â 10-4 â

curv (t0 ) = 10 rad (t0 )
2 T curv (T ) 2 T0 tot (T )

-4

â

a2 (t0 ) curv (t ) a2 (t ) rad (t )

For hypothetical Planck epoch T MPl 1019 GeV one gets 0.01 >
cur v

10

60

â

curv (MPl ) tot (MPl )

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Problems of the Big Bang Theory

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Heavy relics problem (monopole problem)
Let's introduce new stable par ticle X of mass MX Imagine: at moment tX they appear in the early Universe with small velocities (e.g. nonrelativistic) and small density nX (tX ) nrad (tX ) Since nX a-3 nrad then nX (t )/nrad (t ) const

M n (t ) X (t ) X · X X a(t ) rad (t ) T (t ) nrad (tX )
Radiation dominates at least while 1 eV T 3 MeV Therefore even for MX = 10 TeV we must require nX (tX )/nrad (tX ) 10-12 !!! In some SM extenstions it is difficult to avoid heavy relics production: gravitational production, MX H , phase transitions. . .
3 Example: monopoles, produced "one per horizon volume", nX (tX ) = 1/lH (tX ) = H 3 (tX ); Then for its present contribution:

X =
Dmitry Gorbunov (INR)

X 10 c

17

â

MX 10 GeV
16

TX 10 GeV
16

3

g 100
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Imprints of early time dynamics in GW


Problems of the Big Bang Theory

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All the Hot Big Bang puzzles above are problems of the initial state of our Universe

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Problems of the Big Bang Theory

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2.7 K
4.1 K

TODAY
accelerated expansion matter domination

14 by
7.1 by

0.26 eV 0.76 eV

recombination matter domination radiation domination

370 ty 57 ty

e+p H +

80 keV primordial nucleosynthesis 1 MeV 2.5 MeV 200 MeV neutrino decoupling QCD phase transition

3 min 1s 0.1 s 10 µ s

3H 2H

p + p 2H +

+ 4He 7Li + + 2H n + 3He

confinementfree quarks

100 GeV bar yogenesis

Electroweak phase transition

0.1 ns

hot Universe dark matter production reheating

inflation Dmitry Gorbunov (INR) Imprints of early time dynamics in GW 10.10.2013, FFK-13 10 / 32


Inflationary stage and reheating

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Outline

1

Problems of the Big Bang Theory

2

Inflationar y stage and reheating

3

Sensitive to reheating observables in GW

Dmitry Gorbunov (INR)

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Inflationary stage and reheating

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Inflationary solution of Hot Big Bang problems
conformal time
no initial singularity in dS space all scales grow exponentially, including the radius of the 3-sphere the Universe becomes exponentially flat any two par ticles are at exponentially large distances no heavy relics no traces of previous epochs! no par ticles in post-inflationar y Universe to solve entropy problem we need post-inflationar y reheating
Dmitry Gorbunov (INR)

par ticle horizon

conformal time

space coordinate

casually connected regions

inflationar y expansion space coordinate

Imprints of early time dynamics in GW

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Inflationary stage and reheating

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Unexpected bonus: generation of per turbations
Quantum fluctuations of wavelength of a free massless field have an amplitude of 1/ In the expanding Universe: a inflation: lH 1/H = const, so modes "exit horizon" Ordinar y stage: lH 1/H t , lH / , modes "enter horizon" Evolution at inflation inside horizon: < lH a 1/ 1/a

outside horison: > lH a = const = Hinfl !!!

got "classical" fluctuations: =

quantum

â eN

e

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Inflationary stage and reheating

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Key observable: matter (and tensor) per turbations
CMB is isotropic, but "up to corrections, of course..." 1 Ear th movement with respect to CMB T dipole 10-3 T More complex anisotropy:
T T
10
-17

10

Wavelength (cm) 1.0

0.1

10 Hz-1) sr
-1

-18

2

I ( W m

10-4

10

-19

2.726 K blackbody 10
-20

-2

There were matter inhomogenities / T /T at the stage of recombination (e + p + H ) =

10

-21

FIRAS DMR UBC LBL-Italy Princeton Cyanogen

COBE satellite COBE satellite sounding rocket White Mt. & South Pole ground & balloon optical

2

h

1

h

0

h

10

-22

23

h

RA 22
h

1

Jeans instability in the system of gravitating par ticles at rest = / galaxies (CDM halos)

10 100 Frequency (GHz)

1000
4
o h

3

h

21 -39 o -42 o -45 Dec 50 40 30 20 10

h

60

There are neither sources no mechanisms to produce the initial inhomogeneities, if we the Universe is described by GR and SM we must modify the theory!

South 12434 galaxies

cz (1000 km/s)

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Inflationary stage and reheating

I N R

2.7 K
4.1 K

TODAY
accelerated expansion matter domination

14 by
7.1 by

0.26 eV 0.76 eV

recombination matter domination radiation domination

370 ty 57 ty

e+p H +

80 keV primordial nucleosynthesis 1 MeV 2.5 MeV 200 MeV neutrino decoupling QCD phase transition

3 min 1s 0.1 s 10 µ s

3H 2H

p + p 2H +

+ 4He 7Li + + 2H n + 3He

confinementfree quarks

100 GeV bar yogenesis

Electroweak phase transition

0.1 ns

hot Universe dark matter production reheating

inflation Dmitry Gorbunov (INR) Imprints of early time dynamics in GW 10.10.2013, FFK-13 15 / 32


Inflationary stage and reheating

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Role of reheating:

Opens Hot Big Bang stage Helps to solve entropy problem

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Inflationary stage and reheating

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Reheating exploits interactions between inflaton and SM

Either already existing gravity in R 2 -model SM-interaction in the Higgs-inflation Or some new specially designed for this purpose Higgs-por tal for any scalar inflaton: H H 2

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Inflationary stage and reheating

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Inflation & Reheating: simple realization with Higgs
¨ X +3H X + V (X ) = 0 Xe > M
Pl

generation of scale-invariant scalar (and tensor) per turbations from exponentially stretched quantum fluctuations of X

/ 10-5 requires e.g. for V = X 4 : 10-13 reheating ? renormalizable? the only choice: H HX 2 "Higgs por tal" larger

X
Chaotic inflation, A.Linde (1983)

larger Treh
2

quantum corrections



No scale, no problem
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW 10.10.2013, FFK-13 18 / 32


Inflationary stage and reheating

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Studying the reheating stage can help to

Explore the mechanism of SM par ticles production operating in the early Universe Distinguish between otherwise similar inflationary models e.g.: Higgs-inflation vs. R 2 -inflation Understand late-times cosmology e.g.: Dark matter production at reheating Bar yogenesis via Affleck­Dine mechanism Probe other new physics e.g.: in R 2 -inflation Treh counts the number of scalars lighter than 1013 GeV

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Inflationary stage and reheating

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What do we know about reheating?
From obser vation (BBN) we know that Most probably, (BAU) In a par ticular inflationary model 2 2 2 g T 4 < Vinf = 3Hinf MP 30 Upper limits on Hinf from searches for GW: contribution to expansion rate at BBN rad = N 0.5 GW (BBN ) < total (BBN ) r < 0.15 Hence one obtains Treh
Dmitry Gorbunov (INR)

T

Treh > 1 MeV reh > 100 GeV

CMB limits on tensor modes Hinf < 4 â 10 MP
-4

4 â 1016 GeV
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Imprints of early time dynamics in GW


Inflationary stage and reheating

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Details of reheating (GW can reflect!)
Instant or Continious?
rad + inf : change the Universe expansion rate often the effect can be absorbed by a shift in Treh

Homogeneous or not (e.g. structured)
spatial size inhomogeneities in matter of present size l0 lH · are unobser vable. . . 102 GeV a0 0.01 pc â areh Treh l lH MP /T
2 reh

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Inflationary stage and reheating

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GW can help

Gravity waves freely propagate Instant or Continious? Homogeneous or not? Its spectrum saves all information about their production and later Universe expansion

However, in practice it works only for specific classes of models

Dmitry Gorbunov (INR)

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Sensitive to reheating observables in GW

I N R

Outline

1

Problems of the Big Bang Theory

2

Inflationar y stage and reheating

3

Sensitive to reheating observables in GW

Dmitry Gorbunov (INR)

Imprints of early time dynamics in GW

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Sensitive to reheating observables in GW

I N R

General test of reheating
Accurate fit to the power spectrum of scalar (also tensor) per turbations
In fact, spectra are a bit tilted, as H PR (k ) = AR k k
infl

slightly evolves , PT (k ) = A
R T

ns -1

k k

n

T

.

CMB anisotropy measurement determines A q = k /a0 0.002/Mpc, which fixes the number of e-foldings left Ne For tensor per turbations one introduces r

at present scales

AT AR

Works for any inflationary model ! However, the sensitivity to T
reh

is logarithmic only !

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Sensitive to reheating observables in GW

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The power spectra of primordial per turbations
The same potential, the same at the end of inflation
e.g. F.Bezrukov, D.G., M.Shaposhnikov (2008)
V 2 M 0.20
2 P

n

s

1-

8(4N + 9) , (4N + 3)2

r

192 (4N + 3)2

0.15

But WMAP observes different N in the two models: k /a0 = 0.002/Mpc exits horizon at different moments 1 log 3 2 30 27
1/4

0.10

N=

- log

(k / a 0 ) V + log 1/3 1/4 T0 g0 Ve MP

1/2

0.05

2

4

6

8

10

12

MP

1 1 Ve 1013 GeV - log 13 - log 3 Treh 10 GeV 3 The difference is N 57 - 1 log 3 1013 F.Bezrukov, D.G. (2011) GeV , NR 2 = 54.37, NH = 57.66 Treh

R 2 -inflation: ns = 0.964, Higgs-inflation: ns = 0.966, Planck(?), CMBPol(1-2 )

r = 0.0036, r = 0.0032. NLO is needed
D.G., A.Tokareva (2013)

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Sensitive to reheating observables in GW

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Recent analysis (Planck) of cosmlogical data
0.25 Planck +WP Planck +WP+highL Planck +WP+BAO Natural Inflation Power law inflation SB SUSY R
2 2 2/3

Tensor-to-Scalar Ratio (r ) 0.05 0.10 0.15 0.20

Convex Concave

V V V V 0.936 0.944 0.952 0.960 0.968 0.976 0.984 0.992 1.000 Primordial Tilt (ns )

3

0.00

1303.5062

Ne = 50 - 60
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Dmitry Gorbunov (INR)


Sensitive to reheating observables in GW

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Break in spectrum of primordial GW

Notice that for postinflationary stage with p = w at w < 1/3 : GW /U 1/a , at RD : GW /U const
reh

One expects a break ("knee") in inflationary GW spectrum at (T

)

Likewise one expects grows of per turbations! which may enter nonlinear regime and star ts to form halos made of inflaton

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Sensitive to reheating observables in GW

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Gravity waves from inflation and inflaton clumps
Notice that at MD : GW /U 1/a , at RD : GW /U const
reh

One expects a break ("knee") in inflationary GW spectrum at (T at MD : / a e.g.,R -inflation :
2
F.Bezrukov, D.G. (2011)

)

areh 107 ainf

scalar per turbations enter nonlinear regime GW from: collapses at formation of clumps merging of clumps evaporation of clumps (inflaton decays) Since GW /U 1/a, the strongest signal in present GW spectrum is expected at (Treh )
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW

K.Jedamzik, M.Lemoine, J.Mar tin (2010)

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Sensitive to reheating observables in GW

I N R

Help in distinguishing the models: R 2 with H HR

D.G., A.Tokareva (2013)

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Sensitive to reheating observables in GW

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Actually we observe rather narrow range

Obser vable range: kmax 105 kmin N
e

10

We can't describe small scales: for a long time they are in nonlinear regime With GW we can probe per turbations at other scales! (impor tant for exotic models of inflation)
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW 10.10.2013, FFK-13 30 / 32


Sensitive to reheating observables in GW

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Exotic models: production at preheating

Preheating: ver y effective production of ultrarelativistic, firestly noninteracting par ticles (bose enhancement, coherence, etc)

Ultrarelativistic noninteracting par ticles sources GW
J.-F.Dufaux, A.Bergman, G.Felder, L.Kofman and J.-Ph.Uzan (2007)

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Sensitive to reheating observables in GW

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Conclusion
Study of GW signal allows to probe primordial spectrum at small scales (for scalar modes it is obscured by structure formation) test the postinflating physics, including the reheating mechanism (distinguish between quite similar inflationary models, e.g. R 2 and Higgs-inflation) We badly need new experiments (space missions) to detect GW !!!
Presently achieved sensitivities in cosmic photons ( , X -rays, radio wave bands), cosmic neutrinos (e.g. ICECUBE), cosmic rays (e.g. CASCADE GRANDE, AUGER) to the flux from logarithmic scale range are at similar level of erg/cm2 /s, while for GW the sensitivities achieved (e.g. LIGO, VIRGO) are about TEN orders worse

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Backup slides

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Mode evolution
Amplitude remains constant, while superhorizon, e.g. k /a < H Subhorizon Inhomogeneities of DM star t to grow at MD-stage, CDM / T 0.8 eV Smaller objects (first stars, dwarf galaxies) are first to form
CDM

a from

Subhorizon Inhomogeneities of bar yons join those of DM only after recombination, CDM /CDM a from Trec 0.25 eV

at recombination B /B T /T 10-4 and would grow only by a factor Trec /T0 10 without DM Subhorizon Inhomogeneities of photons / oscillate with constant amplitude at RD and with decreasing amplitude at MD, thus we can measure TRD /MD /Trec Phase of oscillations decoupled after recombination depends on the wave-length, recombination time and sound speed / cos k
tr 0

3

vs dt a(t )

= cos(kl

sound

)

T ( , ) =



alm Ylm ( , ) ,

alm a

lm

= Cl 2 Dl /(l (l + 1))
10.10.2013, FFK-13 35 / 32

Dmitry Gorbunov (INR)

Imprints of early time dynamics in GW


I N R

CMB measurements (Planck) H0 ,
Angular scale
90 6000 5000


DM

, B , , R , ns
0.1


18



1



0.2



0.07



D [µK2]

4000 3000 2000 1000 0 2 10 50 500 1000 1500 2000 2500

Multipole moment,
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW 10.10.2013, FFK-13 36 / 32


I N R

Power spectrum of per turbations
In the Minkowski space-time: fluctuations of a free quantum field are gaussian
0

its power spectrum is defined as
0

dq P (q ) 2 (x ) = q P = q /(2 )

dq q 2 q (2 )2

We define amplitude as (q ) Cast the solution in terms solves the equation

In the expanding Universe momenta q = k /a gets redshifted (x, t ) = c (t ) + (x, t ) , (x, t ) e±i ¨ + 3H + q = k /a q = k /a k2 + V (c ) = 0 a2
kx

(k, t )

H as in Minkowski space-time H for inflaton = const

Matching at tk : q (tk ) = k /a(tk ) = H (tk ) Hk gives (q ) =
2 Hk Hk P (q ) = 2 (2 )2

amplification Hk /q = e

Ne (k )

!!!

Hk const = Hinfl hence (almost) flat spectrum
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I N R

Transfer to matter per turbations: simple models
Illustration: Local delay(advance) t in evolution due to impact of of all modes with > H : = c t , t

at the end of inflation -H , then H c Hence, / is also gaussian. Power spectrum of scalar per turbations PR (k ) = H 2 c
2 2

,

Analogously for the tensor per turbations: each of the two polarizations of the gravity waves solves the free scalar field equation! PT (k ) =
2 16 Hk 2 MPl

calculated at t = tk : H = k /a H

k

To the leading order no k -dependence: both spectra are "flat"
Dmitry Gorbunov (INR) Imprints of early time dynamics in GW

(scale-invariant)!
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I N R

Light inflaton nonminimally coupled to gravity

SX

SM

=

-g d 4 x LSM + LXH + Lext + Lgrav , 1224 m X - X - HH - X 2X 4 R,
2 2

LXH = Lgrav

1 µ X µ X + 2 M2 + X 2 =- P 2

,

g

µ

~ g

µ

= 2 g

µ

,

2 2 = 1 + X 2 /MP ,

m = mh 2 = Outcome: =


= 2

. 2

U (X ) =

X4 4 const = 2 MP 44 dX = dX

at X .

2 v 2 2 . = 2 m easier to test! , Br(B )

X X :

2 2 + 6 2 X 2 /MP 4

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0.3 0.25 0.2 =0.001 0.15 0.1 0.05 0 0.93 =0.01 =0.1 0.94 0.95 0.96 ns 100 10-1
-2

0.3 WMAP9, 95% 68% =0 WMAP9+BAO, 95% 68% 0.25 0.2 0.15 0.1 0.05 0 10-5 r

r

0.97

0.98

0.99

1

10-4

10-3

10-2

10-1

100

10-9

10-10

10

10-11

10-3 10-4 10-5 0.93 10-12



0.94

0.95

0.96 ns

0.97

0.98

0.99

1

10-13 10-5

10-4

10-3

10-2

10-1

100

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I N R

Higgs-inflation
F.Bezrukov, M.Shaposhnikov (2007)

M2 d 4 x -g - P R - H HR + LSM 2 In a unitar y gauge H T = 0 , (h + v )/ 2 (and neglecting v = 246 GeV) S= M 2 + h2 (µ h)2 h R+ - d 4 x -g - P 2 2 4
4

S=

slow roll behavior due to modified kinetic term even for 1 Go to the Einstein frame: g with canonically normalized : MP d = dh
2 MP + (6 + 1) h 2 MP 2

2 2~ (MP + h2 ) R MP R

µ

=

-2

~ g

µ

,

2 = 1 +

h2 2 MP

+ h

2

, U ( ) =

4 MP h4 ( ) . 2 + h2 ( ))2 4(MP

we have a flat potential at large fields:
Dmitry Gorbunov (INR)

U ( ) const

@

h

MP /


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I N R

U()
M
P

M / /4

42

log() MP /

MP /

exact h 3/2 h2 /MP 6 log( h/MP ) MP / log(h)

M / /16

42

v /4 0 0 v WMAP

4

Reheating by Higgs field
after inflation: MP / < h < MP /

0 0

end

exponentially flat potential! @
4 MP 4 2

h

MP / :
2

effective dynamics : L=

h
2

2

U ( ) =

2 1 - exp - 3MP

2 1 MP µ µ - 2 2 6

Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields!
Dmitry Gorbunov (INR)

NO NEW d.o.f. Different reheating temperature. . .

0812.3622, 1111.4397

from WMAP-normalization: 47000 â
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Imprints of early time dynamics in GW


I N R

U()
M
P

M / /4

42

log() MP /

MP /

exact h 3/2 h2 /MP 6 log( h/MP ) MP / log(h)

M / /16

42

v /4 0 0 v WMAP

4

Reheating by Higgs field
after inflation: MP / < h < MP /

0 0

end

exponentially flat potential! @
4 MP 4 2

h

MP / :
2

effective dynamics : L=

h
2

2

U ( ) =

2 1 - exp - 3MP

2 1 MP µ µ - 2 2 6

Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields!
Dmitry Gorbunov (INR)

NO NEW d.o.f. Different reheating temperature. . .

0812.3622, 1111.4397

from WMAP-normalization: 47000 â
10.10.2013, FFK-13 42 / 32

Imprints of early time dynamics in GW


F.Bezrukov, D.G., M.Shaposhnikov, 0812.3622

I N R

M

P

g 2 MP | (t )| 2 mW ( ) = 26 mt ( ) = yt MP | (t )| sign (t ) 6

log() MP /

MP /

exact h 3/2 h2 /MP 6 log( h/MP ) MP / log(h)

reheating via W + W - , ZZ production at zero crossings then nonrelativistic gauge bosons scatter to light fermions W +W
-

Reheating by Higgs field
after inflation: MP / < h < MP /

f¯ f

Hot stage star ts almost from T = MP / 1014 GeV: effective dynamics : L= 1 µ µ - 2 6 h2
2 MP 2 1 /4



2

3.4 â 10

13

GeV < Tr < 9.2 â 10

13

0.125

GeV

Advantage: NO NEW interactions to reheat the Universe inflaton couples to all SM fields! Dmitry Gorbunov (INR) Impr

ns = 0.967 , r = 0.0032F.Bezr
ints of early time dynamics in GW

ukov, D.G., 43 / 32

10.10.2013, FFK-13