Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://nuclphys.sinp.msu.ru/conf/epp10/Galtsov.pdf Äàòà èçìåíåíèÿ: Sat Sep 7 17:14:20 2013 Äàòà èíäåêñèðîâàíèÿ: Fri Feb 28 02:29:12 2014 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: universe

Inflation without scalar fields
16 Lomonosov Conference on elementary particle physics
August 22-28 June 2013

D.V. Gal'tsov
Department of Theoretical Physics Mos8cow State University Russia

Inflation without scalar fields ­ p. 1/2


Scalar inflaton
Two mechanisms of inflation widely accepted: Effective cosmological constant via scalar field potential (Guth, Linde,...). Problem: identification of inflaton with known physical fields Modified Einstein equation (Starobinski,...). Problem: Relation to fundamental principles Question: Is it possible to find inflation/dark energy mechanism using known physical fields and Einstein gravity?

Inflation without scalar fields ­ p. 2/2


SU(2) homogeneous and isotropic field
A very natural vector triplet is provided by the SU(2) Yang-Mills. Unlike the U(1) case, classical SU(2) vector field configuration exists which is compatible with space homogeneity and isotropy of the FRW metric ds2 = a2 ( ) d 2 - dl
2 k

(where k = -1, 0, 1 stand for open, flat and closed spatial geometry) without averaging. This was shown for k=1,0 by Cervero and Jacobs 78', Henneaux 82', Hosotani 84'; and for all k by Gal'tsov and Volkov 91' (some later work on the EYM FRW cosmology: Ber tolami, Mourao, Moniz, Cavaglia, de Alfaro, Filippov, Kapetanakis, Koutsoumbas, Lukas, Mayr, Ding). The configuration is parametrized by a single scalar function h(t) and contains both color electric and magnetic components
a a Ei = h i , a a Bi = (k - h2 ) i .

The reduced YM lagrangian in the conformal frame reads L= 3 8 a h2 - (k - h2 )
2

4

so the electric par t corresponds to the kinetic term in the reduced action, while the magnetic
n t on s h u a ar fi par t -- to the potential term. It is wor th noting that this configuration is unique uIpflatio witcoatlsicnlg. el

ds ­ p. 3/2


Cold matter for hot Universe (
P = /3 w i th = 3 8 a
4

Gal'tsov and Volkov 91'

)

The standard YM action is conformally invariant, so the above configuration in the context of FRW cosmology gives rise to the EOS of the photon gas h2 + (k - h2 )
2

Thus one obtains the hot Universe driven by cold matter. The YM equations reduce to EOM of a par ticle in the potential well V = (k - h2 )
2

which in the closed case is a double well potential. Two minima correspond to neighboring topologically different vacua. Depending on the "par ticle" total energy, it (classically) oscillates either around a single vacuum, or around both of them. In the flat and open cases the potential has one minimum at h = 0 with V = 0 for flat and V = 1 for open. The Chern-Simons 3-form e2 2ie 3 = Tr A dA - AA A ,, 8 2 3 satisfying d3 = S U (2) S 3 ):
e2 8 2

TrF F , is non-trivial in the case k = 1 (winding number of the map NC
S

=
S3

3 =
S

1 (h + 1)2 (2 - h) 4
S

Vacuum h = -1 is topologically trivial: NC

= 0, vacuum h = 1 is non-trivial: NC

=1

Inflation without scalar fields ­ p. 4/2


Closed FRW Universe V ( )

WH -1 INSTANTON 1

C
w

Inflation without scalar fields ­ p. 5/2


EYM sphalerons
Self-gravitating YM field gives rise to par ticle-like solutions analogous to sphalerons in the Weinberg-Salam theory (Gal'tsov and Volkov 91'). The role of Higgs is played by gravity. Creation and decay of these sphalerons generates a transition of the YM field from one vacuum to another and is accompanied by the fermion number non-conservation In the EYM cosmology an unstable state w = 0 (w-par ticle sitting at the top of the barrier between two vacua) is cosmological sphaleron (Gibbons and Steif 93'). It has topological charge NC S = 1/2 like sphalerons in the Weinberg-Salam theory. The EOM ¨ h = 2h(1 - h2 ) is solved by h = 0, this correspond to the h-par ticle total energy h2 + (1 - h2 )2 = 1 The YM field in this case is purely magnetic. A more general solution with the same energy describes rolling down of w-par ticle (sphaleron decay): h = 2/ cosh( 2 ), where is conformal time. Rolling down to vacuum h = 0 takes an infinite time, while the corresponding full cosmological evolution a =
4 g g2

sin takes finite time. Thus

cosmological sphaleron is quasi stable. This conclusion is not modified if a positive cosmological constant is added (Ding 94')

Inflation without scalar fields ­ p. 6/2


Instanton and wormholes

Donets and Gal'tsov 92'

Instanton corresponds to tunneling between the neighboring vacua h = ±1. It is a self-dual Euclidean YM configuration for which the stress tensor is zero, therefore conformally flat gravitational field can be added just as a background. Tunneling solutions at higher excitation levels 0 < C 1 are not self-dual. In flat space-time they are known as meron (C = 1, NC S = 1/2), which is the Euclidean counterpar t of the cosmological sphaleron, and the so-called nested merons 0 < C < 1, 1/2 < NC S < 1. The energy-momentum tensor of the meron is non-zero, and in flat space this solution is singular. When gravity is added, the singularity at the location of a meron expands to a wormhole throat, and consequently, the Euclidean topology of the space-time transforms to that of a wormhole. Topological charge of the meron wormholes is zero, the charge of the meron being swallowed by the wormhole (Hosoya and Ogura 89', Verbin and Davidson 90', Gupta et al 90', Rey 90'). The total action for these wormholes is divergent because of slow fall-off of the meron field at infinity, so the amplitude of creation of a baby universe associated with the Euclidean wormholes is zero. However, when a positive cosmological constant is added (inflation) the action becomes finite due to compactness of the space. Such solutions can be interpreted as describing tunneling between the de Sitter space and the hot FRW universe.

Inflation without scalar fields ­ p. 7/2


The wormhole action
In the Euclidean regime the EOM-s for EYM theory are separable and admit the integrals (in conformal time) h2 - (h2 - 1)2 = -C, a2 + (a4 /3 - a2 ) = -C /(e2 m2 l ) P

Solutions describe independent tunneling for the YM variable h, and the cosmological radius a with different periods Th , Ta depending on the excitation level C . To be wormholes, they must obey a quantization condition nh Th = na Ta with two integers (Verbin and Davidson 89') For a specific value of the cosmological constant = 32 me 4 Pl
2

it was found that Ta = Th for all C [0, 1] (Donets, Gal'tsov 92'). In par ticular, for C = 1 (meron limit) the radius a becomes constant (Euclidean static Einstein Universe). For C = 1, 0 the solutions describe creation of baby universes. Remarkably, under above conditions, the total action (gravitational plus YM) is precisely zero SY M + Sg r = 0 Thus, the pinching off of baby universes occurs with probability one.

Inflation without scalar fields ­ p. 8/2


Non-Abelian Born-Infeld
Open sing theory suggests the replacement of the Maxwell Lagrangian density tr L=-
-g Fµ F µ 16

by the Born-Infeld one 2 L= - det(g 4

µ

+F

µ

/ ) -



-g ,

being the critical BI field strength ( = 1/2 in string theory). In four dimensions this is equivalent to ~ 2 Fµ F µ (Fµ F µ )2 (R - 1), R = 1 + L= - 4 2 16 4 In the non-Abelian case F µ is matrix valued, and the prescription is more complicated (symmetrized trace) 2 STr LS T r = - det(gµ + Fµ / ) - -g 4 But qualitatively similar results are obtained with an ordinary trace prescription equivalent to a ~a summation over color indices in the field invariants Fµ F aµ , Fµ F aµ . In FRW 2 cosmology ds2 = N 2 dt2 - a2 dl3 one finds (Gal'tsov, Dyadichev 03') L Here
S tr

1 - 2K 2 + 2V 2 - 3V 2 K 2 = -N a , 2 + V 2 - K2V 2 1-K
3

L = -N a V
2

3

1 - 3K 2 + 3V 2 - 9K 2 V
2

2

h2 K = 2 2 2, aN
2

(h2 - k) = 2 a4

Inflation without scalar fields ­ p. 9/2


NBI cosmology
Homogeneous and isotropic NBI cosmology with an ordinary trace lagrangian turns out to be completely solvable by separation of variables (Dyadichev, Gal'tsov, Zorin, Zotov,02'). It lieads to an interesting EOS: c - , P= 3 c + where c = /4 is the critical energy density, corresponding to vanishing pressure. For larger energies the pressure becomes negative, its limiting value being P = -/3 (EOS of the ensemble of non-interacting isotropically distributed straight Nambu-Goto strings (indicating on the stringy origin of the NBI lagrangian). In the low-energy limit the YM EOS: P = /3 is recovered. Thus, the NBI FRW cosmology smoothly interpolates between the string gas cosmology and the hot Universe. The energy density is = c a4 + 3(h2 - k)2 -1 a4 - 3h4 ,

Inflation without scalar fields ­ p. 10/2


From the YM (NBI) equation one obtains the following evolution equation for the energy density: a ( + 2c ) = -2 , a + c which can be integrated to give a4 ( + 2c ) = const. From this relation one can see that the behavior of patterns: 1) for large energy densities ( c ) the small densities c one has a radiation law Remarkably, the equation for the scale factor a can

( 1)

( 2)

the NBI field interpolates between two energy density scales as a-2 ; 2) for a-4 . be decoupled (g = G):

2g a(a2 + k) a=- ¨ 2g a2 + 3(a2 + k) and admits the first integral 3 a2 + k
2

+ 4g a

2

a2 + k = C ,

which allows to draw phase por traits for different k

Inflation without scalar fields ­ p. 11/2


Dynamics of YM field
The NBI equation for h can be integrated once, giving (in conformal time) h2 a4 =4 a4 - (k - h2 )2 a + 3a 0 which is then solved in terms of Jacobi elliptic functions:
12 10 8 6 4 2 0 ­2 ­4 0 5 10 15 20 25

4 0

The main difference with the ordinary EYM cosmology relates to small values of a. One can see that near the singularity (a 0) the YM oscillations in the NBI case slow down, while in the ordinary YM cosmology the frequency remains constant in the conformal gauge. Near the singularity h = h0 +
b0 t2 6(k+b2 ) 0

+ +O(t4 ) where = ±

3(k + b2 )2 - 4g 2 (k - h2 )2 , h0 0 0

is a free parameter, b0 is a metric expansion parameter.

Inflation without scalar fields ­ p. 12/2


DE as conformal symmetry breaking
Conformal symmetry breaking in NBI theory demonstrates the occurrence of negative pressure, but its value P = -/3 is still insufficient to serve as DE. Meanwhile, a stronger violation of conformal symmetry may provide EOS with -1. Such violation can be of different nature Quantum corrections Non-minimal coupling to gravity Dilaton and other coupled scalar fields, including Higgs String theory corrections Assuming the Lagrangian to be an arbitrary function L(F , G ) of invariants F = -F
a µ

F

aµ

~a /2 and G = -Fµ F

aµ

/4,

one finds the pressure and the energy density (conformal time) P = = L+ L L [2(k - h2 )2 ) - h2 ] - 3 w(k - h2 ) a F G L 2 L -L + 6 h +3 h(k - h2 ) a-4 F F 2
-4

Inflation without scalar fields ­ p. 13/2


For a simple estimate consider power-low dependence L F (F /µ2 )
-1

where µ has dimension of mass. Then in the electric (kinetic) dominance (E) regime one obtains P 3 - 2 w= = 3(2 - 1) For cer tain thi possible. In the n o t b e re a ch e d , Another form of

s quantity may be arbitrarily close to -1 or even less. A E-phantom regime is magnetic (potential) dominance (B) regime w = 4 /3 - 1 the value -1 can but an admissible DE regime is also possible. the lagrangian ( suggested by quantum corrections) is L F ln(F /µ2 )

Then the energy density is = 3 T [ln(F /µ2 ) + 2] + V ln(F /µ2 ) a and
-4

P (T + V ) ln(F /µ2 ) + 2(2V - T ) w= = 3(T + V ) ln(F /µ2 ) + 6T

where T = h2 , V = (k - h2 )2 . It is easy to see that W -1 for ln(F /µ2 ) -1. In this case DE regime is transient. The phantom regime is also possible. Thus, DE conditions for the YM homogeneous and isotropic field can arise indeed as a result of breaking of the conformal symmetry.

Inflation without scalar fields ­ p. 14/2


Weinberg-Salam cosmology
This model may be qualified as "realistic": it describes dynamics of the phase transition with both YM and Higgs homogeneous condensates. The lagrangian includes interacting the SU(2) YM and the complex doublet Higgs: L
m

=

1 Tr(F 2e2

µ

F

µ

)+

1 (Dµ ) Dµ - ( - v 2 )2 . 2 4

Complex doublet Higgs (unlike the real triplet) admits homogenous and isotropic configurations, and only in the spatially closed universe, in which case A=e 1 - h(t) U dU 2
-1

,

= f (t)e

i(t)

U 0 ,

where U = exp(irn),

with three real functions h(t), f (t), (t). The residual symmetry of this ansatz is O(1) (reflection h to -h.) The phase rotation of Higgs (t) can be integrated out (it gives no potential) using the EOM d f 2 a3 2j N =0= 2 3 dt N fa with constant internal momentum number j . The corresponding stress tensor then acquires an additional isospin centrifugal potential term Vj = j f
2

2 a6

.

Inflation without scalar fields ­ p. 15/2


One-dimensional reduced action
The resulting one-dimensional lagrangian contains Scalar kinetic and potential terms f2 , T= 2N 2 V = (f 2 - 1)2 , 4 Vj = j
2 6

f 2a

;

Gauge field kinetic and potential terms 3h2 K= , 2 a2 2N Interaction potential Vin
t

3(h2 - 1)2 W= ; 4 2a

3f 2 (h + 1)2 = . 4a2

The interaction term breaks the residual O(1) invariance: the YM h = ±1 minima are no more equivalent dynamically. The corresponding energy density and pressure read : = T + V + Vj + K + W + Vin p = T - V + Vj +
t

K W Vin + - 3 3 3

t

Inflation without scalar fields ­ p. 16/2


The Friedman equation for the scale factor reads a ¨ = - (2T + 2Vj + K - V + W ), a 3 so cosmic acceleration can be generated only by the Higgs potential term allows the accelerated expansion. Dominance of different potentials gives the following EOS: V Vj W Vin
t

w = -1 w=1 w = 1/3 w = -1/3

Cosmological constant Stiff matter Radiation String gas

An analytic solution exist when the scalar field is frozen in the local maximum of the Higgs potential and the phase is not rotating : f = 0, j = 0, e- t 2 (4 - 2 2 ) + 2 + C2 e t . a= 4 4 C2
2

with 2 = /3, 2 = 4C1 /3, and C1 , C2 being the integration constants. Depending on parameters, it may describe both an exponential expansion, or contractions to the singularity. If = 2 one has the static solution with a0 = 2/ .

Inflation without scalar fields ­ p. 17/2


Dynamical system for j = 0
The evolution is described by the six-order dynamical system with stationary points satisfying the equations 3 f (h + 1)2 + (f 2 - 1)f = 0, 2a2 32 6 f (h + 1) + 2 (h2 - 1)h = 0, 2 a 3 a(f 2 - 1)2 - 3 (h2 - 1)2 = 0 4 2a with the constraint a2 = 3. The above static solution corresponds to the point f = h = 0, a4 = 6. It described the false vacuum Higgs, the interaction with YM being switched off. Correspondingly, the critical points of YM are the minima h = ±1 and the maximum h = 0 from which only the the last one satisfy the constraint. For l = 8/3 > 1, there is another critical point l-1 f2 = h = , a=2 1-w l+1 This is neither false, nor true Higgs vacuum: interaction with YM shifts the critical point to some intermediate value 0 < f0 < 1. Similarly, the critical points of YM are shifted to h± = (1 ± minimum.
2 1 - a2 f0 )/2. From the constraint equation it follows that for l < 3 one has a

local maximum, for l = 3 maximum and minimum coincide, for l > 3 there is a local

Inflation without scalar fields ­ p. 18/2


Eingenvalues
Linearisation around the above two critical points give the stability matrices
3l(l-1) 8

0
3l 2

0

M1 =

0 0

0

0 , l , 2



6l l2 -1 M2 = - (l+1)2 - l-1 2



-

3l2 (l-1) 4(l+1)

3l l-1 - 4 l+1 3l(3-l) 2(l+1) 2(l+1)(l-1) 4l

3l2 l-1 2 2(l+1) 3 2l(l-1) - 2(l+1)3/2 1 2



.

The eigenvalue of det(M1 - µ2 I ) = 0 for Higgs is µ2 = 3l(l - 1)/8, for l > 1 one has a f node, for l < 1 a focus. This means that Vint supersedes V , so the effective potential has minimum for f = 0 instead of maximum. For YM we obtain µ2 = 3l/2 > 0, hence a node at f h = 0. For the scale factor µ2 = l/2 > 0, also a node describing exponential expansion a and/or contraction. Critical value a0 corresponds to compensation of inflationary Higgs potential V and deflationary YM potential W , so if star t with a > a0 , a new accelerated expansion period begins, if a < a0 it will collapse. For M2 one has a cubic equation for µ2 . One root is always real and negative (focus). Two others are complex on the interval l [1.8, 7.5], with equal real par ts and imaginary differing by sign, i.e. four spiral orbits from one focus to another differing by directions and orientation (left/right winding). For 1 < l < 1.8 roots are real and positive (nodes), while for l > 7.5 negative (elliptic orbits). Thus for l > 7.5 one has foci in the full six-dimensional phase space, hence the system is stable.

Inflation without scalar fields ­ p. 19/2


Metastable w = -1/3 state
Physically, the motion near the stable critical points corresponds to the EOS w = -1/3 (string gas). The system can spend rather long time in such a state until deviations from linearity become significant. Then it exhibits transition either to inflation with w -1 or to a dust-like state w 0. The value of the Hubble parameter of inflation depends on the value of Higgs in the metastable phase and it is not defined by the full height of the Higgs potential like in the usual inflaton picture. In par ticular, on the lower bound l = 3 for existence of the min = 1/ 2, metastable string phase (YM field near minimum of the potential) we get f0 hence a weaker inflation. For zero isospin angular momentum j the value of the resulting Hubble constant of inflation decreases with increasing duration of the metastable phase. The scale factor during the string gas era exhibits small oscillations around the value corresponding to the static Einstein universe. The relative value of the isospin centrifugal potential is weighted by the isospin momentum j . With growing j duration of the metastable phase increases, and the Hubble parameter of inflation grows up, since the system approach the authentic false vacuum. For large enough j the Hubble constant of inflation practically does not depend on the duration of quasistatic universe.

Inflation without scalar fields ­ p. 20/2


Numerical experiments
Numerical experiments confirm expectations that the system exhibits non-thermal phase transition from the state of usual matter w > 0 to inflation w -1 through the metastable stage with the EOS of string gas w -1/3. This plot shows evolution of YM-Higgs-gravity coupled system star ting from some state w > 0 with l=0.3 (lambda=0.033), j=0.037. This is followed by long stage of quasistable oscillations of w around the value w -1/3 after which the system star ts to inflate. The slope of the logarithm of the scale factor is an effective Hubble constant, its value depends on duration of the metastable phase: typically the longer is metastable phase, the smaller is Hubble parameter of inflation. After a finite time of inflation the system returns to w > 0 state (not shown).
ln a/a(0)

h f w t

Inflation without scalar fields ­ p. 21/2


Evolution of Higgs
During the metastable w = -1/3 stage Higgs oscillates with small amplitude till transition to inflationary phase when it is stabilized on some non-zero value. The accelerated expansion at the this stage is due to interaction between the gauge and scalar fields which dynamically 'raises up' the Higgs field minima. Inflation end when Higgs reaches the true vacuum f = 1 after some oscillations.

y

x

During the inflationary stage the interaction term is suppressed like a neglected.

-2

a n d ca n b e

Inflation without scalar fields ­ p. 22/2


Time-developed Multiverse
Inflation period stops when Higgs reaches the true vacuum, as usual. Fur ther evolution corresponds to EOS with 0 < w < 1 and leads to contraction either to singularity, or to bounce, after which the new cycle of expansion begins

a

f

h

t

New initial conditions depend on the state of YM components, which oscillate rather independently. As a result, each new cycle star ts with different (chaotically distributed) initial conditions. The overal picture can be viewed as Multiverse developed in time.

Inflation without scalar fields ­ p. 23/2


Concluding remarks
SU(2) YM triplet perfectly fits to homogeneous and isotropic cosmological metrics. The same for complex doublet Higgs (but not for real triplet) Including YM condensates into cosmological scenarios leads to novel features of cosmological phase transitions, not fully explored yet: YM oscillations introduce the second periodicity, classical oscillations can mimic (par t) of CMB, YM dynamics may be responsible for a star t up and an end of periods of acceleration. Both string (Born-Infeld) and standard model (Weinberg-Salam) motivated scenarios demonstrate special role of the EOS w = -1/3 corresponding to zero acceleration. In par ticular, WS model predicts possibility of long metastable phase of almost zero acceleration before inflation

Global evolution exhibits cyclic structure with non-identical cycles whose parameters are chaotically distributed.

Inflation without scalar fields ­ p. 24/2