Astro 4PT Lecture Notes Set 1. Wayne Hu
|
|
- Christiana Hart
- 5 years ago
- Views:
Transcription
1 Astro 4PT Lecture Notes Set 1 Wyne Hu
2 References Reltivistic Cosmologicl Perturbtion Theory Infltion Drk Energy Modified Grvity Cosmic Microwve Bckground Lrge Scle Structure Brdeen (1980), PRD Kodm & Sski (1984), Prog. Th. Phys. Supp., 78, 1 Mukhnov, Feldmn, Brndenberger (1992), Phys. Reports, 215, 203 Mlik & Wnds (2009), Phys. Reports, 475, 1
3 Covrint Perturbtion Theory Covrint = tkes sme form in ll coordinte systems Invrint = tkes the sme vlue in ll coordinte systems Fundmentl equtions: Einstein equtions, covrint conservtion of stress-energy tensor: G µν = 8πGT µν µ T µν = 0 Preserve generl covrince by keeping ll degrees of freedom: 10 for ech symmetric 4 4 tensor
4 Metric Tensor Useful to think in lnguge since there re preferred sptil surfces where the stress tensor is nerly homogeneous In generl this is n Arnowitt-Deser-Misner (ADM) split Specilize to the cse of nerly FRW metric g 00 = 2, g ij = 2 γ ij. where the 0 component is conforml time η = dt/ nd γ ij is sptil metric of constnt curvture K = H 2 0(Ω tot 1). (3) R = 6K 2
5 Metric Tensor First define the slicing (lpse function A, shift function B i ) g 00 = 2 (1 2A), g 0i = 2 B i, A defines the lpse of proper time between 3-surfces wheres B i defines the threding or reltionship between the 3-coordintes of the surfces This bsorbs 1+3=4 degrees of freedom in the metric, remining 6 is in the sptil surfces which we prmeterize s g ij = 2 (γ ij 2H L γ ij 2H ij T ). here (1) H L perturbtion to the sptil curvture; (5) H ij T trce-free distortion to sptil metric (which lso cn perturb the curvture)
6 Curvture Perturbtion Curvture perturbtion on the 3D slice δ[ (3) R] = 4 2 ( 2 + 3K ) H L i j H ij T Note tht we will often loosely refer to H L s the curvture perturbtion We will see tht mny representtions hve H T = 0 It is esier to work with dimensionless quntity First exmple of 3-sclr - SVT decomposition
7 Mtter Tensor Likewise expnd the mtter stress energy tensor round homogeneous density ρ nd pressure p: T 0 0 = ρ δρ, T 0 i = (ρ + p)(v i B i ), T i 0 = (ρ + p)v i, T i j = (p + δp)δ i j + pπ i j, (1) δρ density perturbtion; (3) v i vector velocity, (1) δp pressure perturbtion; (5) Π ij n nisotropic stress perturbtion So fr this is fully generl nd pplies to ny type of mtter or coordinte choice including non-linerities in the mtter, e.g. sclr fields, cosmologicl defects, exotic drk energy.
8 Counting DOF s 20 Vribles (10 metric; 10 mtter) 10 4 Einstein equtions Conservtion equtions +4 Binchi identities 4 Guge (coordinte choice 1 time, 3 spce) 6 Degrees of freedom Without loss of generlity these cn be tken to be the 6 components of the mtter stress tensor For the bckground, specify p() or equivlently w() p()/ρ() the eqution of stte prmeter.
9 Homogeneous Einstein Equtions Einstein (Friedmnn) equtions: ( 1 ) 2 d = K dt + 8πG 2 3 ρ [= 1 ( 1 ) 2 ȧ ] d 2 dt 2 = 4πG 3 (ρ + 3p) [= 1 2 d dη so tht w p/ρ < 1/3 for ccelertion Conservtion eqution µ T µν = 0 implies ρ ρ = 3(1 + w)ȧ overdots re conforml time but eqully true with coordinte time ȧ ]
10 Homogeneous Einstein Equtions Counting exercise: 20 Vribles (10 metric; 10 mtter) Homogeneity nd Isotropy Einstein equtions Conservtion equtions +1 Binchi identities 1 Degree of freedom without loss of generlity choose rtio of homogeneous & isotropic component of the stress tensor to the density w() = p()/ρ().
11 Accelertion Implies Negtive Pressure Role of stresses in the bckground cosmology Homogeneous Einstein equtions G µν = 8πGT µν imply the two Friedmnn equtions (flt universe, or ssociting curvture ρ K = 3K/8πG 2 ) ( 1 d dt ) 2 = 8πG 3 ρ 1 d 2 = 4πG (ρ + 3p) dt 2 3 so tht the totl eqution of stte w p/ρ < 1/3 for ccelertion Conservtion eqution µ T µν = 0 implies ρ ρ = 3(1 + w)ȧ so tht ρ must scle more slowly thn 2
12 Questions regrding Drk Energy Coincidence: given the very different sclings of mtter nd drk energy with, why re they comprble now? Stbility: why doesn t negtive pressure imply ccelerted collpse? or why doesn t the vcuum suck? Answer: stbility is ssocited with stress (pressure) grdients not stress (pressure) itself. Exmple: the cosmologicl constnt w Λ = 1, constnt in time nd spce no grdients. Exmple: sclr field where w = p/ρ δp/δρ = sound speed non-dibtic stress To quntify the perturbtion response let s exmine the norml mode decomposition
13 Sclr, Vector, Tensor In liner perturbtion theory, perturbtions my be seprted by their trnsformtion properties under 3D rottion nd trnsltion. The eigenfunctions of the Lplcin opertor form complete set 2 Q (0) = k 2 Q (0) S, 2 Q (±1) i = k 2 Q (±1) i V, 2 Q (±2) ij = k 2 Q (±2) ij T, Vector nd tensor modes stisfy divergence-free nd trnsverse-trceless conditions i Q (±1) i = 0 i Q (±2) ij = 0 γ ij Q (±2) ij = 0
14 Vector nd Tensor Quntities A sclr mode crries with it ssocited vector (curl-free) nd tensor (longitudinl) quntities A vector mode crries nd ssocited tensor (trce nd divergence free) quntities A tensor mode hs only tensor (trce nd divergence free) These re built from the mode bsis out of covrint derivtives nd the metric Q (0) i = k 1 i Q (0), Q (0) ij = (k 2 i j γ ij)q (0), Q (±1) ij = 1 2k [ iq (±1) j + j Q (±1) i ],
15 Sptilly Flt Cse For sptilly flt bckground metric, hrmonics re relted to plne wves: Q (0) = exp(ik x) Q (±1) i = i (ê 1 ± iê 2 ) i exp(ik x) 2 Q (±2) 3 ij = 8 (ê 1 ± iê 2 ) i (ê 1 ± iê 2 ) j exp(ik x) where ê 3 k. Chosen s spin sttes, c.f. polriztion. For vectors, the hrmonic points in direction orthogonl to k suitble for the vorticl component of vector
16 Sptilly Flt Cse Tensor hrmonics re the trnsverse trceless guge representtion Tensor mplitude relted to the more trditionl h + [(e 1 ) i (e 1 ) j (e 2 ) i (e 2 ) j ], h [(e 1 ) i (e 2 ) j + (e 2 ) i (e 1 ) j ] s h + ± ih = 6H ( 2) T H (±2) T proportionl to the right nd left circulrly polrized mplitudes of grvittionl wves with normliztion tht is convenient to mtch the sclr nd vector definitions
17 Perturbtion k-modes For the kth eigenmode, the sclr components become A(x) = A(k) Q (0), H L (x) = H L (k) Q (0), δρ(x) = δρ(k) Q (0), δp(x) = δp(k) Q (0), the vectors components become 1 B i (x) = B (m) (k) Q (m) i, v i (x) = m= 1 nd the tensors components H T ij (x) = 2 m= 2 H (m) T (k) Q (m) ij, Π ij (x) = 2 m= 2 1 m= 1 v (m) (k) Q (m) Π (m) (k) Q (m) ij, Note tht the curvture perturbtion only involves sclrs i, δ[ (3) R] = 4 2 (k2 3K)(H (0) L H(0) T )Q(0)
18 Covrint Sclr Equtions DOF counting exercise Vribles (4 metric; 4 mtter) Einstein equtions Conservtion equtions +2 Binchi identities 2 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom without loss of generlity choose sclr components of the stress tensor δp, Π.
19 Covrint Sclr Equtions Einstein equtions (suppressing 0) superscripts (k 2 3K)[H L H T ] 3(ȧ )2 A + 3ȧ + ȧ ḢL kb = = 4πG 2 δρ, 00 Poisson Eqution k 2 (A + H L + 1 ( ) d 3 H T ) + dη + 2ȧ (kb ḢT ) = 8πG 2 pπ, ij Anisotropy Eqution ȧ A ḢL 1 K 3ḢT k 2 (kb ḢT ) = 4πG 2 (ρ + p)(v B)/k, 0i Momentum Eqution [ (ȧ ) ] 2 2ä 2 + ȧ [ d dη k2 d A 3 dη + ȧ ] (ḢL kb) = 4πG 2 (δp + 1 δρ), ii Accelertion Eqution 3
20 Covrint Sclr Equtions Conservtion equtions: continuity nd Nvier Stokes [ d dη + 4ȧ [ ] d dη + 3ȧ δρ + 3ȧ δp = (ρ + p)(kv + 3ḢL), ] [ ] (ρ + p) (v B) k = δp 2 3 (1 3 K )pπ + (ρ + p)a, k2 Equtions re not independent since µ G µν = 0 vi the Binchi identities. Relted to the bility to choose coordinte system or guge to represent the perturbtions.
21 Seprte Universes For perturbtions lrger thn the horizon, locl observer should just see different (seprte) FRW universe Sclr equtions should be equivlent to n ppropritely rempped Friedmnn eqution Unit norml vector N µ to constnt time hypersurfces N 0 = (1 + AQ), N i = 0 N 0 = 1 (1 AQ), N i = BQ i Expnsion of sptil volume per proper time is given by 4-divergence µ N µ θ = 3H(1 AQ) + k 3 BQ + ḢLQ
22 Sher nd Accelertion Other pieces of ν N µ give the vorticity, sher nd ccelertion with ν N µ ω µν + σ µν θp µν µ V ν P µν = g µν + N µ N ν ω µν = P α µ P β ν ( β N α α N β ) σ µν = 1 2 P α µ P β ν ( β N α + α N β ) 1 3 θp µν µ = ( ν V µ )V ν projection, trce free ntisymmetric, symmetric nd ccelertion
23 Sher nd Accelertion Vorticity ω µν = 0, σ 00 = σ 0i = 0 = 0 Remining perturbed quntities re the sptil sher nd ccelertion σ ij = (ḢT kb)q ij i = kaq i A convenient choice of coordintes might be sher free Ḣ T kb = 0 A lone is relted to the perturbed ccelertion
24 Seprte Universes So the e-foldings of the expnsion re given by dτ = (1 + AQ)dη N = = dτ 1 3 θ (ȧ dη + ḢLQ + 1 ) 3 kbq Thus if kb cn be ignored s k 0 then H L plys the role of locl chnge in the scle fctor, more generlly B plys the role of Eulerin Lgrngin coordintes. Chnge in H L between seprte universes relted to chnge in number of e-folds: so clled δn pproch, simplifying equtions by using N s time vrible to bsorb locl scle fctor effects We shll see tht for dibtic perturbtions p(ρ) tht Ḣ L = 0 for n pproprite choice of slicing. This conservtion lw plys n importnt role in simplifying clcultions
25 Seprte Universes Dropping the ḢL nd B terms, we get tht the 00 Einstein equtions t k 0 re (ȧ ) 2 A = 4πG 3 2 δρ which is to be compred to the Friedmnn eqution H 2 + K 2 = 8πG 3 ρ Noting tht H = H(1 AQ) we get 2δH H = 8πG 3 δρq = 2AQ H 2 = 2AQ 2 (ȧ ) 2
26 And the spce-spce piece [ 2ä 2 Seprte Universes (ȧ ) 2 + ȧ ] d A = 4πG dη 3 2 (δp + 3δρ) which is to be compred with the ccelertion eqution d (H) = 4πG dη 3 2 (p + 3ρ) gin expnding H = H(1 AQ) nd lso dη = (1 + AQ)d η d d (H) = (1 AQ) dη d η d d (H) 2AQ d η d η (H)[1 AQ] ȧ + ȧ d d η AQ
27 Finlly the continuity eqution is to be compred to Seprte Universes δρ + 3ȧ (δρ + δp) = 0 d ρ = 3(H)(ρ + p) dη which gin with the substitutions becomes (1 AQ) d ( ρ + δρq) = 3(H)(1 AQ)[ ρ + p] 3(H)[δρ + δp]q d η d d η δρ = 3ȧ(δρ + δp)
28 Covrint Vector Equtions Einstein equtions (1 2K/k 2 )(kb (±1) Ḣ(±1) T ) [ d dη + 2ȧ Conservtion Equtions = 16πG 2 (ρ + p)(v (±1) B (±1) )/k, ] (kb (±1) Ḣ(±1) T ) = 8πG 2 pπ (±1). [ ] d dη + 4ȧ [(ρ + p)(v (±1) B (±1) )/k] = 1 2 (1 2K/k2 )pπ (±1), Grvity provides no source to vorticity decy
29 Covrint Vector Equtions DOF counting exercise Vribles (4 metric; 4 mtter) Einstein equtions Conservtion equtions +2 Binchi identities 2 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom without loss of generlity choose vector components of the stress tensor Π (±1).
30 Einstein eqution [ d 2 dη 2 + 2ȧ Covrint Tensor Eqution DOF counting exercise ] d dη + (k2 + 2K) H (±2) T = 8πG 2 pπ (±2). 4 Vribles (2 metric; 2 mtter) 2 0 Einstein equtions Conservtion equtions +0 Binchi identities 0 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom wlog choose tensor components of the stress tensor Π (±2).
31 Arbitrry Drk Components Totl stress energy tensor cn be broken up into individul pieces Drk components interct only through grvity nd so stisfy seprte conservtion equtions Einstein eqution source remins the sum of components. To specify n rbitrry drk component, give the behvior of the stress tensor: 6 components: δp, Π (i), where i = 2,..., 2. Mny types of drk components (drk mtter, sclr fields, mssive neutrinos,..) hve simple forms for their stress tensor in terms of the energy density, i.e. described by equtions of stte. An eqution of stte for the bckground w = p/ρ is not sufficient to determine the behvior of the perturbtions.
32 Guge Metric nd mtter fluctutions tke on different vlues in different coordinte system No such thing s guge invrint density perturbtion! Generl coordinte trnsformtion: η = η + T x i = x i + L i free to choose (T, L i ) to simplify equtions or physics - corresponds to choice of slicing nd threding in ADM. Decompose these into sclr T, L (0) nd vector hrmonics L (±1).
33 Guge g µν nd T µν trnsform s tensors, so components in different frmes cn be relted g µν ( η, x i ) = xα x µ x β x ν g µν(η, x i ) = xα x µ x β x ν g µν( η T Q, x i LQ i ) Fluctutions re compred t the sme coordinte positions (not sme spce time positions) between the two guges For exmple with T Q perturbtion, n event lbeled with η =const. nd x =const. represents different time with respect to the underlying homogeneous nd isotropic bckground
34 Sclr Metric: Guge Trnsformtion à = A T ȧ T, B = B + L + kt, H L = H L k 3 L ȧ T, H T = H T + kl, HL H T = H L H T ȧ T curvture perturbtion depends on slicing not threding Sclr Mtter (Jth component): δ ρ J = δρ J ρ J T, δ p J = δp J ṗ J T, ṽ J = v J + L, density nd pressure likewise depend on slicing only
35 Guge Trnsformtion Vector: B (±1) = B (±1) + L (±1), H (±1) T = H (±1) T + kl (±1), ṽ (±1) J = v (±1) J + L (±1), Sptil vector hs no bckground component hence no dependence on slicing t first order Tensor: no dependence on slicing or threding t first order Guge trnsformtions nd covrint representtion cn be extended to higher orders A coordinte system is fully specified if there is n explicit prescription for (T, L i ) or for sclrs (T, L)
36 Slicing Common choices for slicing T : set something geometric to zero Proper time slicing A = 0: proper time between slices corresponds to coordinte time T llows c/ freedom Comoving (velocity orthogonl) slicing: v B = 0, mtter 4 velocity is relted to N ν nd orthogonl to slicing - T fixed Newtonin (sher free) slicing: Ḣ T kb = 0, expnsion rte is isotropic, sher free, T fixed Uniform expnsion slicing: (ȧ/)a + ḢL + kb/3 = 0, perturbtion to the volume expnsion rte θ vnishes, T fixed Flt (constnt curvture) slicing, δ (3) R = 0, (H L + H T /3 = 0), T fixed Constnt density slicing, δρ I = 0, T fixed
37 Threding Threding specifies the reltionship between constnt sptil coordintes between slices nd is determined by L Typiclly involves condition on v, B, H T Orthogonl threding B = 0, constnt sptil coordintes orthogonl to slicing (zero shift), llows L = c trnsltionl freedom Comoving threding v = 0, llows L = c trnsltionl freedom. Isotropic threding H T = 0, fully fixes L
38 Newtonin (Longitudinl) Guge Newtonin (sher free slicing, isotropic threding): B = H T = 0 Ψ Ã (Newtonin potentil) Φ H L (Newtonin curvture) L = H T /k T = B/k + ḢT /k 2 Good: intuitive Newtonin like grvity; mtter nd metric lgebriclly relted; commonly chosen for nlytic CMB nd lensing work Bd: numericlly unstble
39 Newtonin (Longitudinl) Guge Newtonin (sher free) slicing, isotropic threding B = H T = 0 : [ (k 2 3K)Φ = 4πG 2 δρ + 3ȧ ] (ρ + p)v/k k 2 (Ψ + Φ) = 8πG 2 pπ Anisotropy so Ψ = Φ if nisotropic stress Π = 0 nd Poisson + Momentum [ ] d dη + 3ȧ δρ + 3ȧ δp = (ρ + p)(kv + 3 Φ), [ ] d dη + 4ȧ (ρ + p)v = kδp 2 3 (1 3 K )p kπ + (ρ + p) kψ, k2 Newtonin competition between stress (pressure nd viscosity) nd potentil grdients Note: Poisson source is the density perturbtion on comoving slicing
40 Totl Mtter Guge Totl mtter: (comoving threding, isotropic slicing) B = ṽ (T 0 i = 0) H T = 0 ξ = Ã R = H L (comoving curvture) = δ (totl density pert) T = (v B)/k L = H T /k Good: Algebric reltions between mtter nd metric; comoving curvture perturbtion obeys conservtion lw Bd: Non-intuitive threding involving v
41 Totl Mtter Guge Euler eqution becomes n lgebric reltion between stress nd potentil (ρ + p)ξ = δp ( 1 3K ) pπ k 2 Einstein eqution lcks momentum density source ȧ ξ Ṙ K k 2 kv = 0 Combine: R is conserved if stress fluctutions negligible, e.g. bove the horizon if K H 2 Ṙ + Kv/k = ȧ [ δp ρ + p + 2 ( 1 3K ) ] p 3 k 2 ρ + p Π 0
42 Guge Invrint Approch Guge trnsformtion rules llow vribles which tke on geometric mening in one choice of slicing nd threding to be ccessed from vribles on nother choice Functionl form of the reltionship between the vribles is guge invrint (not the vrible vlues themselves! i.e. eqution is covrint) E.g. comoving curvture nd density perturbtions R = H L H T ȧ (v B)/k ρ = δρ + 3(ρ + p)ȧ (v B)/k
43 Newtonin-Totl Mtter Hybrid With the guge in(or co)vrint pproch, express vribles of one guge in terms of those in nother llows mixture in the equtions of motion Exmple: Newtonin curvture nd comoving density (k 2 3K)Φ = 4πG 2 ρ ordinry Poisson eqution then implies Φ pproximtely constnt if stresses negligible. Exmple: Exct Newtonin curvture bove the horizon derived through comoving curvture conservtion Guge trnsformtion Φ = R + ȧ v k
44 Hybrid Guge Invrint Approch Einstein eqution to eliminte velocity ȧ Ψ Φ = 4πG 2 (ρ + p)v/k Friedmnn eqution with no sptil curvture (ȧ With Φ = 0 nd Ψ Φ ȧ ) 2 = 8πG 3 2 ρ v k = 2 3(1 + w) Φ
45 Newtonin-Totl Mtter Hybrid Combining guge trnsformtion with velocity reltion Φ = 3 + 3w 5 + 3w R Usge: clculte R from infltion determines Φ for ny choice of mtter content or cusl evolution. Exmple: Sclr field ( quintessence drk energy) equtions in totl mtter guge imply sound speed δp/δρ = 1 independent of potentil V (φ). Solve in synchronous guge.
46 Synchronous Guge Synchronous: (proper time slicing, orthogonl threding ) Ã = B = 0 η T H L 1 3 H T h L 6H L T = 1 L = dηa + c 1 1 dη(b + kt ) + c 2 Good: stble, the choice of numericl codes Bd: residul guge freedom in constnts c 1, c 2 must be specified s n initil condition, intrinsiclly reltivistic, threding conditions breks down beyond liner regime if c 1 is fixed to CDM comoving.
47 The Einstein equtions give Synchronous Guge η T K 2k 2 (ḣl + 6 η T ) = 4πG 2 (ρ + p) v k, ḧ L + ȧ while the conservtion equtions give ḣl = 8πG 2 (δρ + 3δp), [ ] d dη + 3ȧ δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 1 2ḣL), [ ] d dη + 4ȧ (ρ J + p J ) v J k = δp J 2 3 (1 3 K k 2 )p JΠ J. Lck of lpse A implies no grvittionl forces in Nvier-Stokes eqution. Hence for stress free mtter like cold drk mtter, zero velocity initilly implies zero velocity lwys.
48 The Einstein equtions give Synchronous Guge η T K 2k 2 (ḣl + 6 η T ) = 4πG 2 (ρ + p) v k, ḧ L + ȧ = 8πG 2 (δρ + 3δp), ḣl (k 2 3K)η T + 1 ȧ = 4πG 2 δρ 2 ḣl [choose (1 & 2) or (1 & 3)] while the conservtion equtions give [ ] d dη + 3ȧ δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 1 2ḣL), [ ] d dη + 4ȧ (ρ J + p J ) v J k = δp J 2 3 (1 3 K k 2 )p JΠ J.
49 Synchronous Guge Lck of lpse A implies no grvittionl forces in Nvier-Stokes eqution. Hence for stress free mtter like cold drk mtter, zero velocity initilly implies zero velocity lwys. Choosing the momentum nd ccelertion Einstein equtions is good since for CDM domintion, curvture η T is conserved nd ḣl is simple to solve for. Choosing the momentum nd Poisson equtions is good when the eqution of stte of the mtter is complicted since δp is not involved. This is the choice of CAMB. Cution: since the curvture η T ppers nd it hs zero CDM source, subtle effects like drk energy perturbtions re importnt everywhere
50 Sptilly Flt Guge Sptilly Flt (flt slicing, isotropic threding): H L = H T = 0 L = H T /k Ã, B = metric perturbtions (ȧ ) 1 ( T = H L + 1 ) 3 H T Good: elimintes sptil metric in evolution equtions; useful in infltionry clcultions (Mukhnov et l) Bd: non-intuitive slicing (no curvture!) nd threding Cution: perturbtion evolution is governed by the behvior of stress fluctutions nd n isotropic stress fluctution δp is guge dependent.
51 Uniform Density Guge Uniform density: (constnt density slicing, isotropic threding) H T = 0, ζ I H L B I B A I A T = δρ I ρ I L = H T /k Good: Curvture conserved involves only stress energy conservtion; simplifies isocurvture tretment Bd: non intuitive slicing (no density pert! problems beyond liner regime) nd threding
52 Uniform Density Guge Einstein equtions with I s the totl or dominnt species (ȧ (k 2 3K)ζ I 3 ) 2 A I + 3ȧ ζ I + ȧ kb I = 0, ȧ A I ζ I K k B I = 4πG 2 (ρ + p) v B I k The conservtion equtions (if J = I then δρ J = 0) [ d dη + 4ȧ [ ] d dη + 3ȧ ] δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 3 ζ I ), (ρ J + p J ) v J B I k = δp J 2 3 (1 3 K k 2 )p JΠ J + (ρ J + p J )A I.,
53 Uniform Density Guge Conservtion of curvture - single component I ζ I = ȧ δp I ρ I + p I 1 3 kv I. Since δρ I = 0, δp I is the non-dibtic stress nd curvture is constnt s k 0 for dibtic fluctutions p I (ρ I ). Note tht this conservtion lw does not involve the Einstein equtions t ll: just locl energy momentum conservtion so it is vlid for lternte theories of grvity Curvture on comoving slices R nd ζ I relted by ζ I = R ρ I. (ρ I + p I ) comoving nd coincide bove the horizon for dibtic fluctutions
54 Uniform Density Guge Simple reltionship to density fluctutions in the sptilly flt guge ζ I = 1 3 δ ρ I. (ρ I + p I ) flt For ech prticle species δρ/(ρ + p) = δn/n, the number density fluctution Multiple ζ J crry informtion bout number density fluctutions between species ζ J constnt component by component outside horizon if ech component is dibtic p J (ρ J ).
55 Vector Guges Vector guge depends only on threding L Poisson guge: orthogonl threding B (±1) = 0, leves constnt L trnsltionl freedom Isotropic guge: isotropic threding H (±1) T = 0, fixes L To first order sclr nd vector guge conditions cn be chosen seprtely More cre required for second nd higher order where sclrs nd vectors mix
4 The dynamical FRW universe
4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which
More informationAstro 321 Set 3: Relativistic Perturbation Theory. Wayne Hu
Astro 321 Set 3: Relativistic Perturbation Theory Wayne Hu Covariant Perturbation Theory Covariant = takes same form in all coordinate systems Invariant = takes the same value in all coordinate systems
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Prt III Mondy 12 June, 2006 9 to 11 PAPER 55 ADVANCED COSMOLOGY Attempt TWO questions. There re THREE questions in totl. The questions crry equl weight. STATIONERY REQUIREMENTS Cover
More information+ x 2 dω 2 = c 2 dt 2 +a(t) [ 2 dr 2 + S 1 κx 2 /R0
Notes for Cosmology course, fll 2005 Cosmic Dynmics Prelude [ ds 2 = c 2 dt 2 +(t) 2 dx 2 ] + x 2 dω 2 = c 2 dt 2 +(t) [ 2 dr 2 + S 1 κx 2 /R0 2 κ (r) 2 dω 2] nd x = S κ (r) = r, R 0 sin(r/r 0 ), R 0 sinh(r/r
More informationA5682: Introduction to Cosmology Course Notes. 4. Cosmic Dynamics: The Friedmann Equation. = GM s
4. Cosmic Dynmics: The Friedmnn Eqution Reding: Chpter 4 Newtonin Derivtion of the Friedmnn Eqution Consider n isolted sphere of rdius R s nd mss M s, in uniform, isotropic expnsion (Hubble flow). The
More informationNonlocal Gravity and Structure in the Universe
Nonlocl rvity nd Structure in the Universe Sohyun Prk Penn Stte University Co-uthor: Scott Dodelson Bsed on PRD 87 (013) 04003, 109.0836, PRD 90 (014) 000000, 1310.439 August 5, 014 Chicgo, IL Cosmo 014
More informationInflation Cosmology. Ch 06 - Initial Conditions. Korea University Eunil Won. Korea U/Dept. Physics, Prof. Eunil Won (All rights are reserved)
Infltion Cosmology Ch 6 - Initil Conditions Kore University Eunil Won 1 The Einstein-Boltzmnn Equtions t Erly Times For nine first-order differentil equtions for the nine perturbtion vribles, we need initil
More informationOutline. I. The Why and What of Inflation II. Gauge fields and inflation, generic setup III. Models within Isotropic BG
Outline I. The Why nd Wht of Infltion II. Guge fields nd infltion, generic setup III. Models within Isotropic BG Guge-fltion model Chromo-nturl model IV. Model within Anisotropic BG Infltion with nisotropic
More informationA Vectors and Tensors in General Relativity
1 A Vectors nd Tensors in Generl Reltivity A.1 Vectors, tensors, nd the volume element The metric of spcetime cn lwys be written s ds 2 = g µν dx µ dx ν µ=0 ν=0 g µν dx µ dx ν. (1) We introduce Einstein
More information-S634- Journl of the Koren Physicl Society, Vol. 35, August 999 structure with two degrees of freedom. The three types of structures re relted to the
Journl of the Koren Physicl Society, Vol. 35, August 999, pp. S633S637 Conserved Quntities in the Perturbed riedmnn World Model Ji-chn Hwng Deprtment of Astronomy nd Atmospheric Sciences, Kyungpook Ntionl
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationarxiv: v1 [gr-qc] 8 Apr 2009
On the Stbility of Sttic Ghost Cosmologies rxiv:0904.1340v1 [gr-qc] 8 Apr 2009 John D. Brrow 1 nd Christos G. Tsgs 2 1 DAMTP, Centre for Mthemticl Sciences University of Cmbridge, Wilberforce Rod, Cmbridge
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationPhysics Graduate Prelim exam
Physics Grdute Prelim exm Fll 2008 Instructions: This exm hs 3 sections: Mechnics, EM nd Quntum. There re 3 problems in ech section You re required to solve 2 from ech section. Show ll work. This exm is
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationThe Active Universe. 1 Active Motion
The Active Universe Alexnder Glück, Helmuth Hüffel, Sš Ilijić, Gerld Kelnhofer Fculty of Physics, University of Vienn helmuth.hueffel@univie.c.t Deprtment of Physics, FER, University of Zgreb ss.ilijic@fer.hr
More informationBypassing no-go theorems for consistent interactions in gauge theories
Bypssing no-go theorems for consistent interctions in guge theories Simon Lykhovich Tomsk Stte University Suzdl, 4 June 2014 The tlk is bsed on the rticles D.S. Kprulin, S.L.Lykhovich nd A.A.Shrpov, Consistent
More information4- Cosmology - II. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1
4- Cosmology - II introduc)on to Astrophysics, C. Bertulni, Texs A&M-Commerce 1 4.1 - Solutions of Friedmnn Eqution As shown in Lecture 3, Friedmnn eqution is given by! H 2 = # " & % 2 = 8πG 3 ρ k 2 +
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationSummer School on Cosmology July Inflation - Lecture 1. M. Sasaki Yukawa Institute, Kyoto
354-1 Summer School on Cosmology 16-7 July 01 Infltion - Lecture 1 M. Ssi Yuw Institute, Kyoto Summer school on cosmology ICT, 16-18 July 01 Miso Ssi Yuw Institute for Theoreticl hysics Kyoto University
More informationToday in Astronomy 142: general relativity and the Universe
Tody in Astronomy 14: generl reltivity nd the Universe The Robertson- Wlker metric nd its use. The Friedmnn eqution nd its solutions. The ges nd ftes of flt universes The cosmologicl constnt. Glxy cluster
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 2015-16 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationQuantum Mechanics Qualifying Exam - August 2016 Notes and Instructions
Quntum Mechnics Qulifying Exm - August 016 Notes nd Instructions There re 6 problems. Attempt them ll s prtil credit will be given. Write on only one side of the pper for your solutions. Write your lis
More informationPhys 6321 Final Exam - Solutions May 3, 2013
Phys 6321 Finl Exm - Solutions My 3, 2013 You my NOT use ny book or notes other thn tht supplied with this test. You will hve 3 hours to finish. DO YOUR OWN WORK. Express your nswers clerly nd concisely
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
More informationFrame-like gauge invariant formulation for mixed symmetry fermionic fields
Frme-like guge invrint formultion for mixed symmetry fermionic fields rxiv:0904.0549v1 [hep-th] 3 Apr 2009 Yu. M. Zinoviev Institute for High Energy Physics Protvino, Moscow Region, 142280, Russi Abstrct
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationGAUGE THEORY ON A SPACE-TIME WITH TORSION
GAUGE THEORY ON A SPACE-TIME WITH TORSION C. D. OPRISAN, G. ZET Fculty of Physics, Al. I. Cuz University, Isi, Romni Deprtment of Physics, Gh. Aschi Technicl University, Isi 700050, Romni Received September
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationCovariant Energy-Momentum Conservation in General Relativity with Cosmological Constant
Energy Momentum in GR with Cosmologicl Constnt Covrint Energy-Momentum Conservtion in Generl Reltivity with Cosmologicl Constnt by Philip E. Gibbs Abstrct A covrint formul for conserved currents of energy,
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationarxiv:gr-qc/ v1 14 Mar 2000
The binry blck-hole dynmics t the third post-newtonin order in the orbitl motion Piotr Jrnowski Institute of Theoreticl Physics, University of Bi lystok, Lipow 1, 15-2 Bi lystok, Polnd Gerhrd Schäfer Theoretisch-Physiklisches
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationInflation Cosmology. Ch 02 - The smooth, expanding universe. Korea University Eunil Won
Infltion Cosmology Ch 02 - The smooth, expnding universe Kore University Eunil Won From now on, we use = c = k B = The metric Generl Reltivity - in 2-dimensionl plne, the invrint distnce squred is (dx)
More informationBig Bang/Inflationary Picture
Big Bng/Infltionry Picture big bng infltionry epoch rdition epoch mtter epoch drk energy epoch DISJOINT Gret explntory power: horizon fltness monopoles entropy Gret predictive power: Ω totl = 1 nerly scle-invrint
More informationd 2 Area i K i0 ν 0 (S.2) d 3 x t 0ν
PHY 396 K. Solutions for prolem set #. Prolem 1: Let T µν = λ K λµ ν. Regrdless of the specific form of the K λµ ν φ, φ tensor, its ntisymmetry with respect to its first two indices K λµ ν K µλ ν implies
More informationMath 124A October 04, 2011
Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model
More informationPHYS 601 HW3 Solution
3.1 Norl force using Lgrnge ultiplier Using the center of the hoop s origin, we will describe the position of the prticle with conventionl polr coordintes. The Lgrngin is therefore L = 1 2 ṙ2 + 1 2 r2
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More information221A Lecture Notes WKB Method
A Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using ψ x, t = e
More informationarxiv:astro-ph/ v1 5 Jul 2005
Cosmologicl nonliner hydrodynmics with post-newtonin corrections Ji-chn Hwng, Hyerim Noh b, nd Dirk Puetzfeld c Deprtment of Astronomy nd Atmospheric Sciences, Kyungpook Ntionl University, Tegu, Kore b
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationThe Cosmology of the Nonsymmetric Theory of Gravitation (NGT)
The Cosmology of the Nonsymmetric Theory of Grvittion () By Tomislv Proopec Bsed on stro-ph/050389 nd on unpublished wor with Wessel Vlenburg (mster s student) Bonn, 8 Aug 005 Introduction: Historicl remrs
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More information1.3 The Lemma of DuBois-Reymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBois-Reymond We needed extr regulrity to integrte by prts nd obtin the Euler- Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationImperial College QFFF, Cosmology Lecture notes. Toby Wiseman
Imperil College QFFF, 07-8 Cosmology Lecture notes Toby Wisemn Toby Wisemn; Huxley 507, emil: t.wisemn@imperil.c.uk Books This course is not bsed directly on ny one book. Approprite reding for the course
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationHomework # 4 Solution Key
PHYSICS 631: Generl Reltivity 1. 6.30 Homework # 4 Solution Key The metric for the surfce of cylindr of rdius, R (fixed), for coordintes z, φ ( ) 1 0 g µν = 0 R 2 In these coordintes ll derivtives with
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationpotentials A z, F z TE z Modes We use the e j z z =0 we can simply say that the x dependence of E y (1)
3e. Introduction Lecture 3e Rectngulr wveguide So fr in rectngulr coordintes we hve delt with plne wves propgting in simple nd inhomogeneous medi. The power density of plne wve extends over ll spce. Therefore
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationGauge Invariance and. Frame Independence in Cosmology
Guge Invrince nd Frme Independence in Cosmology Cover design by Mrten vn Gent ISBN: 978-90-8891-693-9 Printed by: Proefschriftmken.nl Uitgeverij BOXPress Published by: Uitgeverij BOXPress, s-hertogenbosch
More informationLecture 10 :Kac-Moody algebras
Lecture 10 :Kc-Moody lgebrs 1 Non-liner sigm model The ction: where: S 0 = 1 4 d xtr ( µ g 1 µ g) - positive dimensionless coupling constnt g(x) - mtrix bosonic field living on the group mnifold G ssocited
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationSome basic concepts of fluid dynamics derived from ECE theory
Some sic concepts of fluid dynmics 363 Journl of Foundtions of Physics nd Chemistry, 2, vol. (4) 363 374 Some sic concepts of fluid dynmics derived from ECE theory M.W. Evns Alph Institute for Advnced
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationModelling non-linear evolution using Lagrangian perturbation theory re-expansions
MNRAS 431, 799 823 2013) Advnce Access publiction 2013 Mrch 07 doi:10.1093/mnrs/stt217 Modelling non-liner evolution using Lgrngin perturbtion theory re-expnsions Shrvri Ndkrni-Ghosh 1,2 nd Dvid F. Chernoff
More informationWMAP satellite. 16 Feb Feb Feb 2012
16 Feb 2012 21 Feb 2012 23 Feb 2012 è Announcements è Problem 5 (Hrtle 18.3). Assume V * is nonreltivistic. The reltivistic cse requires more complicted functions. è Outline è WMAP stellite è Dipole nisotropy
More informationLyapunov function for cosmological dynamical system
Demonstr. Mth. 207; 50: 5 55 Demonstrtio Mthemtic Open Access Reserch Article Mrek Szydłowski* nd Adm Krwiec Lypunov function for cosmologicl dynmicl system DOI 0.55/dem-207-0005 Received My 8, 206; ccepted
More informationVorticity. curvature: shear: fluid elements moving in a straight line but at different speeds. t 1 t 2. ATM60, Shu-Hua Chen
Vorticity We hve previously discussed the ngulr velocity s mesure of rottion of body. This is suitble quntity for body tht retins its shpe but fluid cn distort nd we must consider two components to rottion:
More informationThis final is a three hour open book, open notes exam. Do all four problems.
Physics 55 Fll 27 Finl Exm Solutions This finl is three hour open book, open notes exm. Do ll four problems. [25 pts] 1. A point electric dipole with dipole moment p is locted in vcuum pointing wy from
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationPHY4605 Introduction to Quantum Mechanics II Spring 2005 Final exam SOLUTIONS April 22, 2005
. Short Answer. PHY4605 Introduction to Quntum Mechnics II Spring 005 Finl exm SOLUTIONS April, 005 () Write the expression ψ ψ = s n explicit integrl eqution in three dimensions, ssuming tht ψ represents
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationM.Gasperini. Dipartimento di Fisica Teorica, Via P.Giuria 1, Turin, Italy, and INFN, Sezione di Torino, Turin, Italy. and. G.
CERN-TH.778/94 Dilton Production in String Cosmology M.Gsperini Diprtimento di Fisic Teoric, Vi P.Giuri, 05 Turin, Itly, nd INFN, Sezione di Torino, Turin, Itly nd G.Venezino Theory Division, CERN, Genev,
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationPhysics 712 Electricity and Magnetism Solutions to Final Exam, Spring 2016
Physics 7 Electricity nd Mgnetism Solutions to Finl Em, Spring 6 Plese note tht some possibly helpful formuls pper on the second pge The number of points on ech problem nd prt is mrked in squre brckets
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More informationMassachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 6
Msschusetts Institute of Technology Quntum Mechnics I (8.) Spring 5 Solutions to Problem Set 6 By Kit Mtn. Prctice with delt functions ( points) The Dirc delt function my be defined s such tht () (b) 3
More informationTHREE-DIMENSIONAL KINEMATICS OF RIGID BODIES
THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationLecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c
Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion
More informationTorsion free biconformal spaces: Reducing the torsion field equations
Uth Stte University From the SelectedWorks of Jmes Thoms Wheeler Winter Jnury 20, 2015 Torsion free iconforml spces: Reducing the torsion field equtions Jmes Thoms Wheeler, Uth Stte University Aville t:
More informationFierz transformations
Fierz trnsformtions Fierz identities re often useful in quntum field theory clcultions. They re connected to reordering of field opertors in contct four-prticle interction. The bsic tsk is: given four
More informationTravelling Profile Solutions For Nonlinear Degenerate Parabolic Equation And Contour Enhancement In Image Processing
Applied Mthemtics E-Notes 8(8) - c IN 67-5 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ Trvelling Profile olutions For Nonliner Degenerte Prbolic Eqution And Contour Enhncement In Imge
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More information