Vektör ve skalar alanlı bir şişme modeli
An Inlationary model with vector and scalar fields
- Tez No: 46394
- Danışmanlar: PROF.DR. JAN KALAYCI
- Tez Türü: Doktora
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Uzay, Vektörler, Şişme özellikleri, Space, Vectors, Swelling properties
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
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(<» p = R~Ht)Tu = -i<j)2 - -ii?”2(t) (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 <J> 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#<j> * -V1, (j)2 > V (assuming £2 = 1) and <£ < V1, corresponds to inflation. 2) The rapid oscillation regime where the friction term is almost negligible, corresponds to reheating period that takes place at the end of the inflation. The slow- rolling phase must last long enough to obtain sufficient inflation. (N k 70) yields Na f H dt = f -?<*fr = K2 f -ZÜL cftj) * yo İ 1 9 I V'(<b) *o According to the“Old”scenario inflation occurs before phase transition from“false”vacuum state. The tunneling from the de Sitter inflating phase to the“true”vacuum occurs through a percolation of bubbles. Since the nucleation rate of the true vacuum bubbles could not complete with the constant expanssion rate of the universe there was no way to exit from inflation while maintaining the homogeneity. In“New”inflation scenario, inflation occurs during the symmetry-breaking phase transition so that the whole observed universe contained in a single bubble. In this model the exit from the false vacuum region is governed by the classical equation of motion so model is free from the“graceful-exit”problem.“Chaotic”inflation models are not based on a phase transition. They only require a classical scalar field with a power-law potential. There are too many models out of the classification as“old”,“new”, or“chaotic”. Some of them are power-law inflation, R2 inflation, Brans-Dicke inflation“Hybrid”inflation etc. Equations of motion of a single field inflationary model are as usual VIH2 = -^r(4<İ>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(<J>) < 1. The number of e-foldings N between scalar field values <J> and <t>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>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 <ftj _ 2 dH dt ~k2 34>, ”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>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<Ç,4>)= ±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(- <j>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 ?-£? <. H and the Hamil ton- Jacobi equation takes the formK2 3\(r 4 2 This is indeed a separable equation as#(i|r,4>) = A(^)B(4>) giving 3A'<*) = ±{tem.)* + l 6\Jr the solution is A(i|r) = -|r cosh[ y.+.A Kdjr-^) ] 73 On the other hand integrating the equations cftj) 1^?20d İni? ^?XM4 + -l^A* and dfr __2 d InJ? k tanh [ y± + \~2 4 Kdlr-ılr*)] we get sinh [ yx + \ -f Kdlr-ilrj)] Ş^A-Af4 3K2»2 4m* If we call the right hand side as P(4>) 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(<f>) = 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 + - iZ22<J>2 4 2 Xll
Benzer Tezler
- A programmable CMOS analog vector quantizer chin
Programlanabilir CMOS analog vektör kuantalayıcı tümdevresi
DEVRİM YILMAZ AKSIN
Yüksek Lisans
İngilizce
1999
Elektrik ve Elektronik Mühendisliğiİstanbul Teknik ÜniversitesiPROF. DR. UĞUR ÇİLİNGİROĞLU
- Tek esnekli ferroelektrik kristalde sınır tabakası denklemlerinin çözümleri
The Solution of boundry layer Equations in the uniaxial crystal
SALİM CEYHAN
- Genelleştirilmiş kovaryant türevin weyl geometrisinde bazı uygulamaları
Some application of prolonged covariant derivative in weyl geometry
ELİF CANFES ÖZKARA
- Conformal mapping of nets in n-dimensional weyl hypersurfaces
n-boyutlu weyl hiperyüzeyleri üzerindeki şebekelerin konform tasviri
Z.İNANÇ AKDENİZCİ
