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Vektör ve skalar alanlı bir şişme modeli

An Inlationary model with vector and scalar fields

  1. Tez No: 46394
  2. Yazar: TEVFİK AÇIKTEPE
  3. Danışmanlar: PROF.DR. JAN KALAYCI
  4. Tez Türü: Doktora
  5. Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1995
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 45

Özet

ÖZET Bu doktora çalışmasında çok alanlı şişme modellerine ilişkin olarak geliştirilmiş analitik bir metod tanıtılmış ve hybrid şişme potansiyeli ile etkileşen, biri skalar diğeri vektör parçacık olan iki alanlı bir modele uygulanmıştır. Bu potansiyelde vektör parçacık minimumlardan birine hızla inerken (“waterfall”) skalar parçacığın global minimumuna yavaşça yuvarlanması ile evrenin şişme dönemine girebileceği gösterilmiştir. Bir vektör parçacığın varlığı, uzay zamanda bir anizotropluğa neden olduğundan uzay zaman metriği FRW yerine Bianchi-I seçilmiş ve karşılık gelen hareket denklemleri, ener j i -momentum tensörü verilmiştir. Hacimsel genleşme parametresi, (H) ve yeni bir alan olarak ortaya çıkan shear (a) kullanılarak bu denklemlerin hidrodinamik karşılıkları da verilmiştir. Denklemlerin belli bir rejimde şişmeli bir çözümü bulunabilmiştir. Hareket denklemlerinin \ = 0 şartını veren klasik bir çözümü de sunulmuştur. Bianchi-I uzay zamanlarında normal şişme {H = 0) elde etmek mümkün değildir. Bu tezde de, sunulan çözüm H * 0 'dır ve süper şişmeye karşılık gelmekte dir. iv

Özet (Çeviri)

AN INFLATIONARY MODEL WITH VECTOR AND SCALAR FIELDS SUMMARY Inflation gives a mechanism which surmounts standart cosmology's problems such as horizon, homogeniety, flatness, monopoles etc. These problems can be solved if the universe is assumed to have begun with natural initial conditions and shortly afterwords to have undergone a de Sitter phase. During the de Sitter phase the scale factor,R, expands exponantially and reduces the space curvature and increases the particle horizon. This enermous expansion dilutes the density of relic particles also. This phase is named as inflation and pointed out that it could solve all the basic, problems of the standard scenario. The basic idea behind the inflationary scenario is that the early stage of the universe (at time 10“A5-10-35 s) there was an epoch during which the energy density of the universe was dominated by an almost constant potential term of massive scalar field, V(). The evolving universe remains in the supercooled metastable state =0 for a long time. Its temperature falls off, the energy- momentum tensor gradually becomes equal to T^n = g^ V(0), and the universe expands exponantially. The scalar field gradually changes, the potential decreases. The energy-momentum tensor of a massive scalar field has the form and takes the form of a perfect fluid with an energy density and pressure given by P = Too = \¥ + \R~Ht) (V4»2 + v()2 - V(*) 2 o If the potential term is significally larger than the kinetic and gradent terms, the equation of state of matter fields is p * -p « V((|>) For matter dominated universe equation of state is p=constant, this indicates V(c|>) «constant and hence scarcely varies during this period. From the solution of Friedman equation R « exp ^8%p/3Mp t)It follows that the universe expands exponantially that is it enters a de Sitter phase. From the conservation of energy momentum Tfvv = 0, the equation of motion is $ + 3tf(j> + y'(4>) = 0 Here H, plays the role of a friction term and is given by H* [ (8it/3Mp) p] 1/2. This equation of motion is the same as that of a ball rolling down a hill with a friction. This equation has two important different regimes, each of has a simple analytical solution 1) The slow rolling regime where the friction term dominates with conditions, 3# * -V1, (j)2 > V (assuming £2 = 1) and 2 + V(«|») 3Mp 2 Differentiating the first with respect to time and using the second gives 2H = --§* 4> Ml This allows us to use 4> as a time variable. Therefore it is possible to eliminate the time dependence in the Friedman equation. Obtaining {Hf)2_12]iH2= 32% v{^ Ml Ml M2 ^ 4rc We can now define the slow roll parameters,1 MZ «?/... m; h”This parameters measure how accurate the slow-roll approximation would be at a given value of cj>. The condition for inflation, â > 0, is precisely equivalent to e() < 1. The number of e-foldings N between scalar field values and end is given by, tf(4>/4>soJs Ln *(4>son) son' _ 4ic M2 X JetAtS rw \ m2p { jmr and the Hubble parameter is ff(4>) =Hsoa exp {-J rson 47Ie^ d») VI iThe construction of an analytical formalism, that deals with anisotropic inflation in the Bianchi-I type spacetime is achieved by observing that Bianchi-I inflation closely related to double inflation in the flat FRW model. The Einstein field equations for a set of N homogeneous scalar fields 4>j(t) interacting through a potential V(4>j) can be written as (l)2= 4(l % ti+ri*j)) r2 N The time dependence of H arises only through its dependence on j, such that H(t) = H(4>j (t)). Therefore the above system may be expressed as a set of first order nonlinear partial differential equations 3k2H2-2 f) (^.)=k4^(4>J ,“ d InR(t) H~dtThe first equation has a similar structure to the Hamilton- Jacobi equation and can be analytically solved when there exist seperable solutions of the form H(j) = nNm”i ^((j)^). The metric for the Bianchi-I model may be written as ds2=-dt2+a2 ( t) (dx2+dy2) + b2 ( t) dz2 The expansion of this anisotropic model when it contains a perfect fluid source is governed by the equations Vlll3H+3H2+2a2 +-K2 (p+3p) =0 2 - v»2rı -ht2 3Hd=K^p+a ö+3Ho=0 p+3H(p+p) =0 where 3H=(4+- ) ve 03-^.(4--) b a yj Jb a We will now consider a multiple inflation model where one of the fields is a vector field, while the other is a scalar field. The Lagrangian of the model is We may assume Bianchi-I metric for our purposes. Einstein field equations are 4+(- -| ) 4 + 2 4^=0 z a b z d% z a i? d ? o a a b M2 2 iP- 2 Â+Â+ÂÂ= -UL (AÜ + AÂa-ır) ajbaij m22 i?2 2 ıxa a jvfj 2 £2 2 dç The equations of motion can be written in hydrodynamical form as 3H2=K2Mp+02 3H + 3H2 + 2o2 + AK2#2(p + 3p) = 0 b + 3Ha=- ^K2M|(p-pz) 2 p + 3#(p+p) + ^/3a(pz-p)=0 Our purpose is to find an inflationary solution to above equations with the potential y)= ±X(¥-M2)2 + Ajny + -I^VÇ2 The parameters are chosen as i > 0 and A'< 1, this states that, inflation may end by a rapid rolling of the field £ to the local minimum at £ = 0 triggered by the slow rolling of the field (j>. Since^/ = ° tne equations of motion becomes $ + 3H$ + m2$ = 0 ijr + 3#ijf = 0 3H2 = k2(- 2 + - ıjr2 + - XMA + - m2$2 ) 2 2 T 4 2 where ilr is defined as ijr = -£- a. We can use the slow rolling K approximation ?-£? ) = A(^)B(4>) giving 3A') we get V = *i + - \ -f [In (P + n/P2 + 1) - yj and as a result of this the expansion is described as H = -±-{1 + 4> s/3 2m2 4 2 Y Finally, since a = N -| Ki|r we get XI2>(4>) = İ3İ(2P + 2y/P2 + 1 + A) 1/3 and a() = a, [ P(P + v^2 + D2 ]"1/3 where at and bj/s are constants coming from the initial conditions. As aresult the time dependance of (J> will be given in an integral form J OK t->2tj-2. ~2Jv2/-i, r%2\ with V = -XMA + - iZ222 4 2 Xll

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