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Luca Doria

Institut für Kernphysik Johannes-Gutenberg Universität Mainz

Lecture 2 Brief History of GR 1907: While working at the Bern’s Patent Office, A. Einstein realized how to start generalizing to generic reference frames. 1908: First paper about acceleration and relativity by A. Einstein. In this paper he states the and derives caused by gravitational fields. 1911: Second paper by A. Einstein on time dilation by gravitational fields where also light deviation by massive bodies was approximately derived. 1912: Einstein consults with M. Grossman about non-euclidean geometry. October 1915: First guess: Rij = Tij November 1915: Einstein publishes the General as we know it today. D. Hilbert obtained the same equations almost at the same time. 1919: Eddington confirms the deviation of light formula from GR using a solar eclipse in Brazil. 1959: Pound-Rebka Experiment (gravitational red shift) 1971: Hafele-Keating Experiment (time dilation) 1974: Hulse-Taylor binary pulsar. 2004: Probe-B and frame dragging (published in 2011) 2016: Direct detection of gravitational waves by LIGO

Sommersemester 2018 Luca Doria, JGU Mainz 2 The Equivalence Principle

Observer in an uniform gravitational field

Accelerated Observer

How do you tell the difference?

Sommersemester 2018 Luca Doria, JGU Mainz 3 The Equivalence Principle

What about this case?

Sommersemester 2018 Luca Doria, JGU Mainz 4 The Equivalence Principle

Strong Equivalence Principle: At every space-time point in a gravitational field, it is possible to choose a locally inertial coordinate system such that in a sufficiently small neighbourhood of that point, the laws of nature can be expressed in the same form as in an unaccelerated coordinate system.

Weak Equivalence Principle: Change “laws of nature” with “laws of motion of free-falling bodies” (gravity).

Sommersemester 2018 Luca Doria, JGU Mainz 5 CHAPTER 3. bodies. Clearly the strong version implies the weak on but not vice-versa. CHAPTER 3. GENERAL RELATIVITY bodies. Clearly the strong version implies the weak on but not vice-versa. 3.2 Free-FallingCHAPTER 3. GENERAL Bodies RELATIVITY bodies. Clearly the strongLet’s3.2 version tryFree-Falling implies to translate the weak in on Bodies mathematical but not vice-versa. formulas the ideas contained in the EP. Consider a free-falling body: according to the EP, there must exist lo- Let’s try to translate in mathematical formulas the ideas contained in the 3.2 Free-Fallingcally Bodies a coordinate system where the effect of gravitation vanishes (inside Einstein’sEP. Consider elevator) a free-falling and body: the equation according of to motion the EP, there in flat must space exist lo-time is cally a coordinate system where the effect of gravitation vanishes (inside Let’s try to translate in mathematical formulas the ideas contained in the Einstein’s elevator) and the equationd2 ofxµ motion in flat space time is EP. Consider a free-falling body: according to the EP, there must exist= lo-0 , (3.1) cally a coordinate system where the effect of gravitation vanishesd2xµ dt2 (insideFree-Falling Bodies = Einstein’s elevator) and the equation of motion in flat space2 time0 is , (3.1) µ n dt where dt = hµndx dx is the and hµn the Lorentz metric. 2 µ µ n whered xdt = hµndx dx is the proper time and hµn the Lorentz metric. µ Let’s get out= 0 from , theEquation elevator of motion and change (3.1)for a free-falling to new body coordinatesµ x , which canLet’s bed gett whatever2 out from we the elevator want (a and curvilinear change to system, new coordinates an acceleratedx , which or rotating can be whatever we want (a curvilinear system, an accelerated or rotating where dt = h dxµdxn is the proper time and h the Lorentz metric. µn system,system, etc..) etc..)Changing toµ na generic coordinate system: Let’s get out from the elevator and change to new coordinates xµ, which can be whatever we want (a curvilineardd system,∂x∂xa dxa anµdx acceleratedµ ∂xa d∂x2xa orµ d rotating2xµ∂2xa dx∂2µxdxa n dxµ dxn = = + + = 0 .= 0 (3.2) . (3.2) system, etc..) dt ∂xµ dµt ∂xµ dt2µ ∂2xµ∂xn dtµ dtn dt✓ ∂x d◆t ∂x dt ∂x ∂x dt dt a µ a 2 µ ✓2 a µ ◆n d ∂x dx Multiplying∂x d x the∂ lastx equationdx dx by ∂xl/∂xa and recognizing the presence µ = µ 2 + µ n = 0 .l (3.2) a dt ∂x dt Multiplyingof the∂x Christoffeldt the∂x symbol∂ lastx dt equationdt by ∂x /∂x and recognizing the presence ✓ ◆ Equation of motion: of the Christoffell a symbol Multiplying the last equation by ∂x /∂x and recognizing2 l theµ presencen d x l dx dx of the Christoffel symbol 2 2+lGµn µ= 0n . (3.3) dt d x dtl ddxt dx + Gµn = 0 . (3.3) d2Thexl last resultdxµ dxn is quite interesting:dt2 the presencedt dt of the gravitational field + Gl = 0 . (3.3) dtcan2 beµ seenn dt asd at of space-time where free-falling particles follow The last result is quite interesting: the presence of the gravitational field The last result is quite interesting:a geodetic the in presence that space. of From the gravitational the coordinate field transformation formula and candt = be seenh dx asµdx an curvature, it is clear that of space-time the metric tensor where of thefree-falling curved space particles is follow can be seen as a curvature of space-time µn where free-falling particles follow a geodetic in that space. Fromarelated geodeticSommersemester the to coordinate the2018 in Lorentzian that transformation space. flat From space formula theLuca by Doria, coordinate JGU and Mainz transformation formula6 and µ n dt = h dxµdxn, it is cleardt = that thehµn metricdx dx tensor, it is of clear the curved that the space metric is tensor of the curved space is µn ∂xa ∂xb related to the Lorentzianrelated flat space to by the Lorentziangµn = flat space= byh . (3.4) ∂xµ ∂xn ab ∂xa ∂xb a b gµn = = hab .∂x ∂x (3.4) ∂xµ ∂xn gµn = = h . (3.4) 3.3 Non-Relativistic Limit∂xµ ∂xn ab 3.3 Non-RelativisticSo far Limit our calculations were relativistic and now we would like to see if 3.3what weNon-Relativistic derived can be reduced to the Limit know non-relativistic result, which So far our calculations were relativistic and now we would like to see if 26 what we derived can be reducedSo far to our the calculations know non-relativistic were relativistic result, which and now we would like to see if what we derived26 can be reduced to the know non-relativistic result, which

26 CHAPTERCHAPTER 2. 2. GEOMETRICAL TOOLS TOOLS CHAPTER 2. GEOMETRICAL TOOLS AA general general mixed mixed tensor has has the the following following covariant covariant derivative: derivative:

a1..ar A general mixed tensor has the following covariant derivative: a1..ar ∂ ec T a1..ar ∂ ec Tbb..1b..bs ( e T)a1..ar = rr 1 s (2.37) (a ..eacT) b1..br = c (2.37) ∂ T rr1 cr b1..br ∂∂xxc a ..a ec b ..bs 1 r a 1da ..a a a1..ar 1d ( e T) = r a11 da22..arr ar r a1..ar (2.37)1d c b1..br ++GGdcTTc b ..b ++...... ++GGdcTTb ..b (2.38)(2.38) r ∂dcx b11..bss dc b1..1bs s a1 da2..ar ar da1..ara11..dar d a1..ar +G T + ... + G GTd Ta1..ar ... Gd Ta1..ar (2.38) (2.39) dc b1..bs dcGbb1c..Tbsdb ..b ... G bscT b ..b d (2.39) b11c db22..bss bsc b1..1bs s 1d1 Gd Ta1..ar ... Gd Ta1..ar (2.39) Notice b1c db that2..bs the derivation bsc b1..bs of1d covariant indices brings a negative sign in CHAPTER 2.Notice GEOMETRICAL that the derivation TOOLS of covariant indices brings a negative sign in Notice that the derivationfrontfront of of ofGG.. covariant indices brings a negative sign in front of G. A general mixed tensor has the following covariant derivative:

2.102.10 CalculationCalculation of of the thea1..ar Christoffel Symbols ∂ e T CHAPTERa1..ar 3.c b1 GENERAL..bs RELATIVITY 2.10 Calculation of the( e ChristoffelT) = r Symbols (2.37) HavingHaving a a procedure procedurer c b for for1..br calculating calculating∂xc derivatives derivatives of of tensors tensors along along a a direction direction a1 da2..ar ar a1..ar 1d Having a proceduregivengiven for calculating by by+ a aG tangent tangentT derivatives vector,+ ... + weG we of tensorsT need need now now along a a way a way direction to to calculate calculate(2.38) the the Christoffel Christoffel bodies. Clearly the strong versiondc b1..bs implies thedc b weak1..bs on but not vice-versa. given by a tangentsymbols. vector,symbols. weG needd Ta1 now..ar a... wayG tod T calculatea1..ar the Christoffel (2.39) b1c db2..bs bsc b1..bs 1d symbols. First,First, we we have have to know the the following following property property First,3.2 weNoticeFree-Falling have to that know the the derivation following Bodies of property covariant indices brings a negative sign in front of G. rrggµµnn==00 . . (2.40) (2.40) g = 0 .rr (2.40) Let’s try to translate in mathematicalrr µn formulas the ideas contained in the TheThe covariant covariant derivative of of the the metric metric tensor tensor is is zero. zero. This This property property is is EP. Consider a free-falling body: according to the EP, there must exist lo- The covariant derivativecalledcalled metric ofmetric the compatibility compatibility metric tensorandand is zero. it it is is always always This property assumed assumed is valid valid in in our our context, context, cally a2.10 coordinateCalculation system where of the the effect Christoffel of gravitation Symbols vanishes (inside called metric compatibilitybutbut nothing nothingand it prevents prevents is always to to assumed develop develop valid covariant covariant in our derivatives derivatives context, without without compati- compati- Einstein’s elevator) and the equation of motion in flat space time is but nothingHaving prevents a procedurebility.bility. to develop A A physicallyfor physically calculating covariant appealing appealing derivatives derivatives property property of without tensors of of compatibility compatibility along compati- a direction is is that that given given two two bility. Agiven physically by a tangentvectors, appealing vector, their property mutual we need of angle compatibility now a does way not to is calculate change that given the if they Christoffeltwo are parallely trans- vectors, their mutuald2xµ angleMetric does not and change Geodetics if they are parallely trans- vectors,symbols. their mutualportedported angle along along does a not curve change= by0 by a ,ifa compatible compatibletheyChristoffel are parallely derivative. derivative. trans-Symbols In In (3.1) other other words, words, the the ported along a curve by a compatibledt2 derivative. In other words, the First, we havescalarscalar to know product product the of following two vectors vectors property does does not not change change during during the the parallel parallel trans- trans- scalarMetric product of two vectorsµ n does not change during the parallel trans- where dt = hµport.port.ndx d From Fromx is Eq. Eq. the 2.40, proper we we timecan can write write and theh theµn following followingthe Lorentz three three metric. expressions, expressions, which which port. FromMetric Eq. compatibility 2.40, we can condition: write the followingrgµn = 0 three . expressions, which (2.40) Let’sGives getthe “distance” out fromareare thebetween the the elevator same same two except except points. andr change for for a a cyclic cyclic to new index index coordinates permutation permutationxµ (,r (r whichµµ n)n) are the same except for a cyclic index permutation (r µ n) ! ! can be whatever we want (a curvilinear system, an! accelerated! or rotating! ! Theimplies covariant cycling the derivative indices:2 of the metric2 tensor isl zero.2 Thisl property2 is In cartesian coordinates: d =(x rxgµn)=+(∂rgyµn Gyrnl)gln+(zGrnlgµzl =) 0 system,called etc..)metricg compatibility= ∂ g andG2lrgr itgµ is1n always=Gl ∂grgµ2 assumedn=0G1rngln valid 2Grn ing ourµ1l = context,0 r µn r µn rnrln rn µl l l but nothingr prevents to developg covariant= ∂ g derivativesG l g withoutG lg = compati-0 (2.41) 2 l µ gnr =l∂µ gnr Gµn glr Gµr gnl = 0 (2.41) In a general space:µganr =µ∂µgnr Ga rg2µ µnr G gµ2 nra=0 µµn lrn µr (2.41)nl bility.d A physically∂x dx d appealing= ∂xgijxµrndix propertylrxj µr∂ ofnlx compatibilitydxl dx isl that given two r = += ∂ G l =G0l .= 0. (3.2) vectors, theirµ mutual angleµl doesng2rµ not=l change∂nµggrµn ifGnr theygglµ areGn parallelyµggrl = 0. trans- dt ∂nxgrµ d=t∂ngrµ ∂xGnrrdgntlµrµ Gn∂µxgnrl∂rxµ=0.dtnrA.d tEinsteinlµ nµ rl ported along✓r a curve◆ by ar compatible derivative. In other words, the Calculating rgµn l µgnra ngrµ = 0 we obtain CalculatingMultiplyingGeodesicscalarand summing product theg Calculating last to of equationzero:g two vectorsrgr bygµn=∂ doesrx0 we/µ∂x notg obtainnrCHAPTER changerandn recognizinggr duringµ = 2.0 GEOMETRICAL we the obtain parallel the presence trans- TOOLS rr µn rµ nr rrn rµ r r ofShortest the Christoffelport. path From between Eq. symbol 2.40,two points we can write the following three expressions, which ∂ g ∂ g ∂ g + 2Gl = 0 , (2.42) arewe thehave same except for a cyclic∂ indexr g+µn permutationl ∂µ=gnr ∂n gr (µr + 2µGµln n=) 0 , (2.42) where we∂ usedrgµn the∂µg symmetrynr ∂ngrµr Gµkn2G=µn Gµk nr0(evident ,n rµ from! !µ Eq.n (2.42) 2.34). Multiply- d2xl dxµ ijdxn ji srl l l Geodesicing the equation last equationg = by+∂gGgµn weG cang solve= 0G g for .13=G: 0 (3.3) and solving for Gammarrdµtn2 multiplyingr13µn dt byrnd thetln metric rn tensorµl13 g = ∂ g Gl g Gl g = 0 (2.41) The last result is quiter interesting:µ nr 1µ nr theµ presencen lr µr ofnl the gravitational field s sr l l Christoffel Symbols Ggµn ==∂ gg ∂Gµggnr + ∂Gngrgµ =∂0.rgµn . (2.43) can be seen as a curvaturern r ofµ space-time2n rµ nr wherelµ free-fallingnµ rl particles follow a geodetic in that space. From the coordinate transformation formula and Calculatingµ n rgµn µgnr ngrµ = 0 we obtain dt = Thishµnd veryx dx nice,r it is result clearr tells thatr us the that metric given tensor the metric of the tensor curved we space can is calculate Sommersemester 2018 The metric is the onlyLuca Doria, information JGU Mainz needed! 9 relatedthe to the Christoffel Lorentzian symbols. flat space Said by in another way,l the metric tensor is all what ∂rgµn ∂µgnr ∂ngrµ + 2G = 0 , (2.42) Sommersemesteris 2018 needed to calculate covariantLuca Doria, JGU derivatives Mainz µn of tensors on a curved space.7 ∂xa ∂xb gµn = 13= h . (3.4) ∂xµ ∂xn ab 2.11 Geodesics 3.3 Non-Relativistic Limit As straight lines are the shortest lines in flat space, we can ask which So farcurve our calculations connecting were two relativisticpoints in a and curved now space we would is the shortest.like to see These if par- what weticular derived curves can are be reduced called geodesics to the knowand non-relativistic in this sense are result, the "straight which lines" in curved spaces. One way to see the problem of finding the geodesics is the following: the most "straight26 line" between two points is the one where a tangent vector T on the first point does not change (remains par- allel to itself) if transported in the direction pointed by the vector itself. Translating in mathematical language the previous statement:

n µ n µ T = T T = 0 , (2.44) rT rµ and substituting the definition of covariant derivative

µ dT µ + G TaTb = 0 . (2.45) dt ab The components of a tangent vector to a curve parameterized by a param- eter t are just Tµ = dxµ/dt, therefore we obtain the equation a geodesic curve must satisfy 2 µ a b d x µ dx dx + G = 0 . (2.46) dt2 ab dt dt The last result is a coupled system of second order linear differential equations and we know that there exists an unique solution given initial values for x and dx/dt. Another way to obtain the geodesic equation is to directly calculate what

14 Einstein Equations

Sommersemester 2018 Luca Doria, JGU Mainz 8 Einstein Equations

Requirements: 1) It should be a tensor equation (the same in any coordinate system). 2) In analogy to other situations known in physics, it should be of second order at most in the relevant variable (the gravitational potential, or in this case the metric tensor). 3) The equation must reduce to the Poisson equation in the non-relativistic limit. 4) The source of the gravitational field should be the energy-momentum tensor T. 5) If T=0, space-time must be flat

First time the Riemann tensor appears in Einstein’s notebooks.

Sommersemester 2018 Luca Doria, JGU Mainz 9 Einstein Equations R g R = T ↵ ↵ ↵

The Ricci tensor depends from the metric and its derivatives. Again, the metric is all what we need (together with T) for calculating the equations.

Einstein equations are non-linear! Unlike the Maxwell equations case, specifying the sources is not sufficient for calculating the fields. In this case, the energy-momentum tensor determines the geometry and the geometry modify the energy momentum tensor: very complex problem in the general case.

Properties: 1) It is a tensor equation (the same in any coordinate system). 2) In analogy to other situations known in physics, it is of second order at most in the relevant variable (the gravitational potential, or in this case the metric tensor). 3) The equation must reduce to Newtonian gravity in the non-relativistic limit. 4) The source of the gravitational field is the energy-momentum tensor T.

Sommersemester 2018 Luca Doria, JGU Mainz 10 The

The is spatially homogeneous and isotropic

Image from Cryhavoc

Sommersemester 2018 Luca Doria, JGU Mainz 11 CHAPTER 4. COSMOLOGICAL MODELS CHAPTER 4. COSMOLOGICAL MODELS

and time). We assume that all the observers on a spatial surface are able and time). We assume that all the observers on a spatial surface are able toto measure measure the the time time ("cosmic ("cosmic time") time") in the in the same same way, way, so we so can we choose can choose g00 = 1 (a constant not depending on the space-time point). It turns out g00 = 1 (a constant not depending on the space-time point). It turns out thatthat all all these these requirements requirements lead lead to to a space-time a space-time with with constant constant curvature curvature 3 (in(in particular particular such such space-times space-times are are called calledmaximallymaximally symmetric symmetric3). It). can It can bebe showed showed that that the the requirements requirements of of homogeneity and and isotropy in four in four dimensionsdimensions lead lead to to the the following following metric metric

2 2 2 2 2 Thek(udku(u) dFLRWu) Metric dsds2==g(gv()vdv)dv+2 +f (vf)(vd)u d+u2 + ,, (4.1) (4.1) 1 1ku2 ku2   wherewherevvisis a a coordinate coordinate and andu isu ais vector a vector of three of three coordinates coordinates and andf an f an From theunknownunknown Cosmological function function Principle of ofv vonly.only. This This metric metric is clearly is clearly rotationally rotationally invariant invariant as required. Introducing new "spherical coordinates" , q, f for the "space" as required. Introducing new "spherical coordinates"r r, q, f for the "space" we obtainvariables the Friedmann-Lemaitre-Robertson-Walkeru and a time coordinate t = g (metric:v)dv we obtain variables u and a time coordinate t = g(v)dv we obtain dr2 R p ds2 = dt2 R2(t) dr2+ r2dRq2 +pr2 sin2 qdf2 , (4.2) ds2 = dt2 R2(t)1 kr2 + r2dq2 + r2 sin2 qdf2 , (4.2)  1 kr2 where R(t) function of time to be determined, and k is a constant repre- sentingwhere R the(t) curvature.function Thisof time metric to be is invariant determined, under and thek rescalingis a constant repre- Notingsenting the rescaling the curvature. invariance This metric is invariant under the rescaling R R ! l R R r lr (4.3) !! l r k lr (4.3) k ! , ! l2 k k 2 , we canso always choosing choosel = k=+1,0,-1.pk, the curvarure !k lcan assume only the values k = 1, 0, 1. The function R(t) is usually normalized to its present value R can beso scaled choosing to e.g.l = 1 pfork ,t=today. the curvarure Usually kR(t)can is assumecalled a(t): only the the scale values factor.k = a(t)=R(t)/R(0). and a(t) is called cosmic . Eq1, 4.2 0, is 1. called The the functionFriedman-Lemaître-Robertson-Walker R(t) is usually normalized tometric its present (FLRW) value ( )= ( )/ (0). and ( ) is called . Sommersemesteranda 2018t representsR t R the mosta generalt Luca Doria, homogeneous JGUcosmic Mainz scale and isotropic factor space-time 12 inEq four 4.2 is dimensions. called the Friedman-Lemaître-Robertson-Walker metric (FLRW) Itand interesting represents to look the most at the general spatial geometry. homogeneous If k = and0, space isotropic is flat space-time and equivalentin four dimensions. to a three-dimensional "plane". If k = 1, space is equivalent It interesting to look at the spatial geometry. If k = 0, space is flat and 3Mathematically, a maximally symmetric space M of dimension D is a space with a metricequivalent admitting to D(D+1)/2 a three-dimensional Killing vectors. Killing "plane". vectors If formk = a vector1, space field describing is equivalent the infinitesimal isometric transformations in M. 3Mathematically, a maximally symmetric space M of dimension D is a space with a metric admitting D(D+1)/2 Killing vectors. Killing vectors form a vector field describing the infinitesimal isometric transformations35 in M.

35 FLRW

Georges Lemaître (1894-1966) Belgian Catholic Priest and Astronomer (1888-1925) Russian and Mathematician

Howard P. Robertson (1903-1961) Arthur G. Walker (1909-2001) USA physicist and mathematician UK Mathematician

Sommersemester 2018 Luca Doria, JGU Mainz 13 Comoving Coordinates

Sommersemester 2018 Luca Doria, JGU Mainz 14 CHAPTER 4. COSMOLOGICAL MODELS

Fig. 4.2: if the space-time is expanding (or contracting), the coordinates stretch according to the scale factor a(t) and an object (say, a galaxy) keeps CHAPTER 4. COSMOLOGICAL MODELS its position with respect to the axes. An extremely simplified case is the one where k = 0 and a(t)=constant. Fig. 4.2: if the space-time is expanding (or contracting),Substituting the the coordinates FLRW metric in the Einstein Equations gives Tµn = 0, stretch according to the scale factor a(t) and ansince object the (say, components a galaxy) keeps of the Ricci tensor Rµn are zero (and therefore also its position with respect to the axes. the curvature scalar R). In this case, the equations of General Relativity An extremely simplified case is the one wheredescribek = 0 and ana( emptyt)=constant. universe with a Lorentz (flat) metric. This case is Substituting the FLRW metric in the Einstein Equations gives T = 0, trivial but it is aµn check that the Einstein Equations have an additional since the components of the Ricci tensor Rµn are zero (and therefore also the curvature scalar R). In this case, the equationscorrect of limiting General Relativity case. describe an empty universe with a Lorentz (flat) metric. This case is trivial but it is a check that the Einstein Equations have an additional correct limiting case. 4.4

4.4 Friedmann Equations Now it is time to use the FLRW metric in the Einstein equations. This will allow us to extract the exact form of the still unknown scale factor Now it is time to use the FLRW metric in thea( Einsteint). The equations. left side of This the equation contains only the geometric quantities will allow us to extract theEinstein/Ricci exact form of thewhich still unknown can beTensor scale calculated factor from for the FLRW metric gµn which metric non-zero components a(t). The left side of the equation contains onlyare the geometric quantities which can be calculated from the metric gµn which non-zero components are g00 = 1 2 2 Non-zero componentsg00 = 1 of the FLRW metric tensor a (t)r g11 = a2(t)r2 1 kr2 (4.5) g = 11 1 2 2 2 kr (4.5) g22 = a (t)r CHAPTER 4. COSMOLOGICAL MODELS 2 2 g22 = a (t)r 2 CHAPTER2 2 4. COSMOLOGICAL MODELS g33 = a (t)r sin q . g = a2(t)r2 sin2 q . 33 and the Riemann tensor with Eq. 2.56. Contracting the Riemann tensor From the metric tensor, we can directly calculate the Christoffel symbols From the metricNon-zero tensor, components we can directly of calculate the the Christoffeland thewe symbols Riemann can calculate tensorNon-zero the with Ricci components Eq. tensor 2.56. which Contracting of the non-zero the components Riemann tensor are with Eq. 2.43. The non-zero Christoffel symbolswith are Eq.we 2.43. can The calculate non-zero the RicciChristoffel tensor symbols which non-zero are components are FLRW Christoffer symbols Ricci tensor a¨ R = 3 00a¨ a 0 aa˙ 0 2 0 2 2 aa˙ R = 3 G = ; G = aar˙ ; G = aar˙ sin 0q 00 0 2 02 2 2 11 1 kr2 22 33 G11 = 2 ; G22 = aaar˙ aa¨;+ 2Ga˙33+=2kaar˙ sin q 1 kr = 2 ˙ Raa¨11+ 2a˙ + 2k 2 (4.7) 1 1 2 2 3 3 a R = 1 kr ˙ G01 = G10 = G02 = G20 = G03 = G30 = 1 111 2 22 3 3 a (4.7) a G01 = G10 = G021==kr2G(20¨ =+ 2G˙032 +=2G30) = 1 2 1 2 2 (4.6) R22 r aa a kr a G = r(1 kr ) ; G = r(1 kr ) sin q R = r2(aa¨ + 2a˙2 + 2kr) 22 33 1 22 2 1 2 2 2 2 2 (4.6) G22 = r(1 kr ) R;33 =G33r (=aa¨ +r(21a˙ +kr2kr) )sinsinqq , 2 2 3 3 1 = 2( + 2 + ) 2 G12 = G21 = G13 = G31 = R33 r aa¨ 2a˙ 2kr sin q , r 2 2 3 3 1 2 3 3 where the "dot" representsG12 = theG21 = totalG13 derivative= G31 = with respect to the time G33 = sin q cos q ; G23 = G32 = cotwhereq , the "dot" represents the total derivative with respectr µn to the time (x˙ = dx/dt). Finally, contracting the Ricci tensor (g Rµn) we obtain the (x˙ = dx/dt).2 Finally, contracting the3 Ricci tensor3 (gµnR ) we obtain the curvatureG33 = scalarsin q cos q ; G23 = G32 = cot q µ,n 37 curvature scalar 6 scalar6 R = (aa¨ + 2a˙2 + k) . (4.8) R =37 (aa¨ +a2 a˙2 + k) . (4.8) a2 Now we have all the ingredients for calculating the left side of the Einstein Sommersemester 2018 LucaNow Doria, weJGU haveMainz all the ingredients for calculating the left15 side of the Einstein equations and we just need T . Again the Cosmological Principle can equations and we just need T . Againµn the Cosmological Principle can help us: an homogeneousµn and isotropic distribution of is the one help us: an homogeneous and isotropic distribution of matter is the one corresponding to a perfect fluid with r and P: corresponding to a perfect fluid with density r and pressure P:

T =(Trµ+n =(P)vr +v P)vPgµvn ,Pgµn , (4.9) (4.9) µn µ n µn wherewherev is thev four-velocityis the four-velocity vector withvector components with componentsv =(1, 0,v 0,=( 0)1,, since 0, 0, 0), since the "fluid"the "fluid" is at rest is at with rest respect with respect to the comoving to the comoving coordinates. coordinates. Again, Again, thinkingthinking at the at fluid the as fluid made as of made point-like of point-like galaxies, galaxies, they retain they their retain their placeplace with respect with respect to the to coordinate the coordinate axes: it axes: is just it the is just distance the distance among among themthem which which changes changes through througha(t). Densitya(t). Density and pressure and pressure can be time- can be time- dependent,dependent, but not but space-dependent, not space-dependent, otherwise otherwise this would this be would against be the against the CosmologicalCosmological Principle. Principle. ConsiderConsider now the now energy-momentum the energy-momentum conservation conservation

µ µ µ µ µ b µ b b µ b µ T = µ∂TnT =+∂GµTnT+ G G TnT =Gµ0nT ,= 0 , (4.10) (4.10) rµ n r µ n µb n µbµn b b and theand zero the component zero component (i.e. the (i.e. energy the energy component) component) of the above of the equa- above equa- tion tion a˙(t) a˙(t) ∂0r(t)+∂ 3r(t)+(r3(t)+(Pr((tt)))+ =P0(t)) . = 0 . (4.11) (4.11) 0 a(t) a(t) This equationThis equation expresses expressesenergyenergy conservation conservationin the FLRWin the Universe. FLRW Universe. No- No- tice thattice for that obtaining for obtaining Eq. 4.11 Eq. we 4.11 had we to had use the to use space-time the space-time geometry geometry

38 38 CHAPTER 3. GENERAL RELATIVITY

We have to use the weak-field and v c approximations together with ⌧ stationarity ∂gµn/∂t = 0. In this limit, the only non-zero component of the Ricci tensor is R00. The energy-momentum tensor reduces also to only the energy density component T00 which in the non-relativistic limit is just the matter density r. The approximate equation is therefore

1 R = 2h = kr , (3.15) 00 2r 00 which has to be compared to the Poisson equation for the gravitational potential 2f = 4pGr . (3.16) r where G is the Newton constant. Calculating the constant k, we can write the general relativistic Einstein equation (in natural units G = c = 1) CHAPTER 4. COSMOLOGICAL MODELS 1 R g R = 8pT . (3.17) µn 2 µn µn and the Riemann tensor with Eq. 2.56. Contracting the Riemann tensor we can calculate the Ricci tensor which non-zero components are In "normal units", 8p 8pG/c4. ! The equation has been checked against many astronomical data, anda¨ lab- R = 3 oratory and satellite experiments, always finding good agreement00 a up to now. aa¨ + 2a˙2 + 2k R = Eq. 3.17 is not the most general form allowed by our requirements.11 1 Sincekr2 (4.7) the covariant derivative of both sides of the equation vanishes and this R = r2(aa¨ + 2a˙2 + 2kr) happens also for the metric tensor, we can also addEnergy-Momentum a term22 which is pro- Tensor = 2( ¨ + 2 ˙2 + 2 ) sin2 q , portional to gµn R33 r aa a kr 1 R g R +whereLg = the8p "dot"T represents. the total (3.18) derivative with respect to the time µn µn µn µn µn 2 (x˙ = dx/dt). Finally, contracting the Ricci tensor (g Rµn) we obtain the curvature scalar The new constant L is called . 6 Given the symmetry of the tensors in the equation, there areR = only( 10aa¨ in-+ 2a˙2 + k) . (4.8) a2 dependent components.Calculated This meansin the previous that the slide Einstein equation represents Now we have all theStill ingredients missing…but for we calculating can use the left side of the Einstein a coupled system of 10 non-linear second-order partial differential equa- equations and weagain just needthe cosmologicalT . Again principle: the Cosmological Principle can tions and finding analytical general solutions is possible only in fewµn cases help us: an homogeneous and isotropic distribution of matter is the one characterized by high symmetry content. Besides the trivial flat solu- corresponding to a perfect fluid with density r and pressure P: tion, a particularly important space-time satisfyingE/m the tensor Einstein for an equations isotropic homogeneous “fluid”: is the Schwarzschild solution which finds wide applicationsTµn =( inr physics+ P)vµvn Pgµn , (4.9) problems involving a spherically symmetric gravitational field. where is the four-velocity vector with components =(1, 0, 0, 0), since v Density v the32 "fluid" is at rest withPressure respect4-velocity to the comoving coordinates. Again, thinking at the fluid as made of point-like galaxies, they retain their place with respect to the coordinateNote axes: this! it is just the distance among The “fluid” is at themrest wrt which the co-moving changes coordinates: through a(t). DensityGR is in general and “hard” pressure to solve: can be time- the metric shows up on BOTH dependent,v=(1,0,0,0) but not space-dependent,sides otherwise of the Einstein this Equations.. would be against the

Sommersemester 2018 Cosmological Principle.Luca Doria, JGU Mainz 16 Consider now the energy-momentum conservation

µ µ µ b b µ T = ∂ T + G T G T = 0 , (4.10) rµ n µ n µb n µn b and the zero component (i.e. the energy component) of the above equa- tion a˙(t) ∂ r(t)+3 (r(t)+P(t)) = 0 . (4.11) 0 a(t) This equation expresses energy conservation in the FLRW Universe. No- tice that for obtaining Eq. 4.11 we had to use the space-time geometry

38 CHAPTER 4. COSMOLOGICAL MODELS

through the covariant derivative and this brings us back to the complex- ity of the Einstein Equations, where curvature and energy/matter distri- bution are in general directly connected. At his point we are stuck: we have the scale factor and other two un- knowns in a single equation. The only way forward is to postulate a relationship between density and pressure, i.e. an equation of state for the energy/matter content of the universe. It can be showed that basi- cally all the cosmologically relevant perfect fluids have an equation of state like P = wr where w is a constant characteristic of the specific fluid. Substituting this generic equation of state in Eq. 4.11 we obtain

r˙ a˙ = 3(1 + w) , (4.12) r a

which has solutions like

3(1+w) r(t) µ a(t) . (4.13)

Now we have an equation that tells us how the density behaves while the universe expands or contracts. We still have to determine the dynamics of the scale factor. Using the expressions for the Ricci tensor, the curvature scalar and the energy-momentum tensor, we can substitute them into the Einstein equa- tions Eq. 3.17 finding (we leave L = 0 for the moment)

a¨ 3 = 4pG(r + 3P) for (µ, n)=(0, 0) a (4.14) a¨ a˙ 2 k + 2 + 2 = 4pG(r P) for (µ, n)=(i, j) . a a a2 ✓ ◆ For the spatial components iThe, j = 1, Friedmann 2, 3 there is only oneEquations equation as it should be, given the requirement of isotropy. Substituting the second Puttingderivative together of thea(t previous) from thecalculations, first of thewe can Eq. obtain 4.14 intothe Friedmann the second equations we obtain: the Friedmann Equations

a¨ 4pG Acceleration equation: = (r + 3P) a 3 (4.15) a˙ 2 8pG k Hubble par. equation: = r . a 3 a2 ✓ ◆ 39 The Friedmann equations are the Einstein equations for the FLRW metric and an isotropic homogenous fluid.

How to solve them for the scale factor a(t)? We need to know pressure and density of the “fluid”, or at least a relation between then: an equation of state.

Sommersemester 2018 Luca Doria, JGU Mainz 17 CHAPTER 4. COSMOLOGICAL MODELS CHAPTER 4. COSMOLOGICAL MODELS and the Riemann tensor with Eq. 2.56. Contracting the Riemann tensor weand can the calculate Riemann the tensor Ricci tensor with Eq. which 2.56. non-zero Contracting components the Riemann are tensor we can calculate the Ricci tensor which non-zero components are a¨ R00 = 3 a¨ R00 = a3 aa¨+ 2aa˙2 + 2k R = 2 11 a1a¨ + kr2a˙2 + 2k (4.7) R11 = (4.7) 2 1 kr2 2 R22 = r (aa¨ +2a˙ + 2kr) R = r2(aa¨ + 2a˙2 + 2kr) 22= 2( + 2 + ) 2 R33 r a2a¨ 2a˙ 2 2kr sin q2 , R33 = r (aa¨ + 2a˙ + 2kr) sin q , where the "dot" represents the total derivative with respect to the time where the "dot" represents the total derivative withµn respect to the time (x˙ = dx/dt). Finally, contracting the Ricci tensor (g Rµnµn) we obtain the curvature(x˙ = dx/ scalardt). Finally, contracting the Ricci tensor (g Rµn) we obtain the curvature scalar 6 R = (6aa¨ + 2a˙2 + k) . (4.8) R =a2 (aa¨ + 2a˙2 + k) . (4.8) a2 Now we have all the ingredients for calculating the left side of the Einstein Now we have all the ingredients for calculating the left side of the Einstein equations and we just need T . Again the Cosmological Principle can equations and we just needµTn . Again the Cosmological Principle can help us: an homogeneous and isotropicµn distribution of matter is the one help us: an homogeneous and isotropic distribution of matter is the one corresponding to a perfect fluid with density r and pressure P: corresponding to a perfect fluid with density r and pressure P:

Tµn =(r + P)vµvn Pgµn , (4.9) Tµn =(r + P)vµvn Pgµn , (4.9) where is the four-velocity vector with components =(1, 0, 0, 0), since wherev v is the four-velocity vector with componentsv v =(1, 0, 0, 0), since the "fluid" is at rest with respect to the comoving coordinates. Again, CHAPTER 4. COSMOLOGICALthe "fluid" is at rest with MODELS respect to the comoving coordinates. Again, thinkingthinking at at the the fluid fluid as as made made of of point-like point-like galaxies, galaxies, they they retain retain their their placeplace with with respect respect to to the the coordinate coordinate axes: axes: it it is is just just the the distance distance among among through the covariantthemthem which which derivativeCHAPTER changes changes through and4. through COSMOLOGICAL thisa(ta)( brings.t). Density Density us and back and MODELS pressure pressure to the can complex- can be be time- time- dependent,dependent, but but not not space-dependent, space-dependent, otherwise otherwise this this would would be be against against the the ity ofCHAPTER the EinsteinCosmological 4. COSMOLOGICAL Equations, Principle. where MODELS curvatureEquation and energy/matter of State distri- bution are in generalCosmological directly Principle. connected. ConsiderConsider nowthrough now the the energy-momentum energy-momentum the covariant derivative conservation conservation and this brings us back to the complex- At histhrough point the we covariant are stuck:ity derivative of the we Einstein have and this the Equations, brings scale us factor back where and to the curvature other complex- two and un- energy/matter distri- µ =µ µ +µ µ µ b b b b µ =µ knownsEnergy-momentumity of in the a Einstein single conservation: equation. Equations,µT whereµn TheTn =∂µ curvature only∂TµnTn +G wayµGbTn andTn forwardG energy/matterµGnTµnbT =0 is0 , to , postulate distri- (4.10) (4.10) a butionr arer in general directlyµb connected.b relationshipbution are between in general densityAt directly his point and connected. pressure, we are stuck: i.e. an weequation have the of scale state factorfor and other two un- At his pointandand we the the are zero zero stuck: component component we have (i.e. (i.e. the the the scale energy energy factor component) component) and other of of the two the above un- above equa- equa- the energy/mattertiontion contentknowns of the in a universe. single equation. It can be The showed only way that forward basi- is to postulate a knowns in a single equation. The onlya˙(t) way forward is to postulate a cally allFor the the cosmologically 0-th component:relationship relevant between perfecta˙(t) density fluids and have pressure, an equation i.e. an ofequation of state for relationship between density∂0 andr∂(0tr)+( pressure,t)+3 3 (r( i.e.tr)+(t)+ anP(Ptequation))(t)) = =00 . of . state for (4.11) (4.11) a(ta)(t) statethe like energy/matterP = wr where contentthew energy/matteris of a constant the universe. characteristic content It can of be the showed of universe. the that specific basi- It fluid. can be showed that basi- This equation expresses in the FLRW Universe. No- Substitutingcally all the thisThis cosmologically generic equationcally equation expresses all relevant theenergy of cosmologicallyenergy state perfect conservation conservation in fluids Eq. 4.11 have relevantin we an the obtain equation FLRW perfect Universe. of fluids No- have an equation of tice that for obtaining Eq. 4.11 we had to use the space-time geometry Choosestate the generic like P = ticeequationwr thatwhere state forof state: obtainingw likeis a constantP = Eq.wr 4.11 characteristicwhere we hadw is to a of use constant the the specific space-time characteristic fluid. geometry of the specific fluid. Substituting this genericSubstituting equationr˙ of this state generic ina˙ Eq. equation 4.11 we obtain of state in Eq. 4.11 we obtain = 3(1 + w) 3838, (4.12) r a r˙ a˙ r˙ a˙ = 3(1 + w) ,= 3(1 + w) (4.12), (4.12) which has solutions like r a r a

which has solutions like 3(1+w) whichr(t has) µ solutionsa(t) like . (4.13) 3(1+w) r(t) µ a(t) . (4.13) 3(1+w) Now we have an equation that tells us how the densityr(t) µ a( behavest) while. the (4.13) Now we have an equation that tells us how the density behaves while the Sommersemesteruniverse 2018 expands or contracts.Luca Doria, We JGU stillMainz have to determine the dynamics18 of theuniverse scale factor. expands orNow contracts. we have We still an equation have to determine that tells the us dynamicshow the density behaves while the of the scale factor. Using the expressionsuniverse for the Ricci expands tensor, or contracts. the curvature We still scalar have and to determine the the dynamics Using the expressionsof for the the scale Ricci factor. tensor, the curvature scalar and the energy-momentumenergy-momentum tensor, tensor, we we can can substitute them them into into the the Einstein Einstein equa- equa- Using the expressions for the Ricci tensor, the curvature scalar and the tionstions Eq. 3.17 Eq. 3.17 finding finding (we (we leave leaveLL == 00 for for the the moment) moment) energy-momentum tensor, we can substitute them into the Einstein equa- a¨tionsa¨ Eq. 3.17 finding (we leave L = 0 for the moment) 3 == 4pG((r++ 3P)) for (µ(, n)=()=(0, 0) ) 3 a 4pG r 3P for µ, n 0, 0 a (4.14) a¨ a2˙ 2 k a¨ (4.14) a¨ + 2a˙ +k2 = 4pG(r P) for3 (=µ,4np)=(G(ri,+j) 3P. ) for (µ, n)=(0, 0) 2 a + a2 a + 2 2a= 4pG(r P) for (µ, n)=(i, j) . a a ✓ ◆ a 2 (4.14) ✓ ◆ a¨ a˙ k For the spatial components i,+j 2= 1, 2, 3+ there2 is= only4pG one(r equationP) for as it(µ, n)=(i, j) . a a a2 For theshould spatial be, given components the requirementi, j =✓1, of 2,◆ isotropy. 3 there isSubstituting only one the equation second as it shouldderivative be, given of a(t the) from requirement the first of the of Eq. isotropy. 4.14 into Substituting the second we the obtain second For the spatial components i, j = 1, 2, 3 there is only one equation as it derivativethe Friedmann of a(t) from Equations the first of the Eq. 4.14 into the second we obtain should be, given the requirement of isotropy. Substituting the second the Friedmann Equations derivativea¨ of 4p(G) from the first of the Eq. 4.14 into the second we obtain = a t (r + 3P) the a 3 Friedmanna¨ 4p EquationsG (4.15) a˙ =2 8pG (r +k 3P) a = 3 r . a 3 a2 a¨ 4pG (4.15) ✓a˙ ◆2 8pG k = (r + 3P) = r . a 3 39 2 2 (4.15) a 3 a a˙ 8pG k ✓ ◆ = r . a 3 a2 39 ✓ ◆ 39 Cosmological Parameters CHAPTER 5. OBSERVATIONAL AND THE LCDM MODEL Defining the following expansion coefficients

1 da H(t)= Hubble Parameter a dt 1 d2a 1 da 2 q(t)= / Deceleration a dt2 a dt ✓ ◆ 1 d3a 1 da 3 j(t)= / "Jerk" a dt3 a dt ✓ ◆ 4 1 d4a 1 da (5.16) s(t)= / "Snap" (or Jounce) a dt4 a dt ✓ ◆ 1 d5a 1 da 5 c(t)= / "Crackle" a dt5 a dt ✓ ◆ 1 d6a 1 da 6 p(t)= / "Pop" a dt6 a dt ✓ ◆ ...

we can write the scale factor as

1 2 2 1 3 3 Sommersemester 2018 a(t)=a 1 + H (t t )Luca Doria,q H JGU( Mainzt t ) + j H (t t ) + ... (5.17) 19 0 0 0 2 0 0 0 6 0 0 0  The red-shift is (here we keep c = 1) 6 a(t ) 1 + z = 0 , (5.18) a(t D/c) 0 where D/c is the time at which the light signal was emitted and D is the distance travelled since the emission. Substituting the expansion in Eq. 5.17 in Eq. 5.18 and expanding it in the parameter x = H0D/c with

1 = 1 + x + ax2/2 + bx3/6 + cx4/24 + ... a b 1 x +(1 )x2 +(a 1)x3 (5.19) 2 6 1 + (6a2 36a + 8b c + 24)x4 + (x5) , 24 O 50 Hubble’s Law

Edwin Hubble (1889 – 1953)

a˙ d v = H a˙ = Ha / ) a )

Sommersemester 2018 Luca Doria, JGU Mainz 20 CHAPTER 4. COSMOLOGICAL MODELS CHAPTER 4. COSMOLOGICAL MODELS 4.6 Cosmological Models 4.6 Cosmological Models In order to treat all the possible cases, we will use the most general form Inof order the Einstein to treat all equations, the possible thus cases, including we will the use cosmological the most general constant formL. ofWhen the Einstein theSimplest cosmological equations, Cosmological constant thus including is taken the into cosmological account,Models the constant FriedmannL. Whenequations the cosmological become constant is taken into account, the Friedmann equations become CHAPTER 4. COSMOLOGICALa¨ MODELS 4pG L = H˙ + H2 = (r + 3P)+ Friedmann Equations a¨a 4pG3 L3 = ˙ + 2 = (r + )+ (4.22) with Cosmological Constant Term H H 8pG k3P L through the covariant derivativea and thisH2 brings= 3 usr back+ to the3. complex- (4.22) 8pG3 ka2 L3 ity of the Einstein Equations, where curvatureH2 = andr energy/matter+ . distri- 3 a2 3 bution are inRegarding general directly the equation connected. of state, the most relevant cases are At his pointRegarding we are stuck: the equation we have of state, the scale the most factor relevant and other cases are two un- 1 knowns in a single equation. Thew = only0 r way forward"Dust" is to postulate a ) ⇠1a3 relationship between density andw = pressure,0 r i.e. an"Dust"equation of state for 31 (4.23) the energy/matter content ofw the= 1/3 universe.) r⇠ a It can"Radiation" be showed that basi- ) ⇠1a4 (4.23) cally allEquations the cosmologically of state relevantw = 1/3 perfectr fluids"Radiation" have an equation of w = 1) r⇠ aconst4 . "Vacuum" . state like P = wr where w is a constant ) characteristic⇠ of the specific fluid. w = 1 r const. "Vacuum" . Substituting thisIn an generic universe equation filled with of state "dust",) ⇠ in i.e. Eq. point-like 4.11 we obtain massive particles, there is Innot an interaction universe filled between with them "dust", so i.e. the point-like pressure massive is zero and particles, therefore there w=0. is r˙ a˙ notIt is interaction a standard between physics= 3( them result1 + w so that) the for, pressure a volume is zero filled and with therefore (4.12) radiation w=0. only, ItP is= ar standard/3. The physicsr last case, result where thataP for= a volumer (a sort filled of negative with radiation pressure) only, can Sommersemester 2018 Parise= r/3. in different The last models.Luca case, Doria, JGU where ItMainz is calledP = "vacuumr (a sort energy" of negative since pressure) this21 equation can which has solutionsariseof state in different like can arise models. from quantum It is called field "vacuum theory. energy" We are since now in this the equation position to discuss some specific cosmological models. of state can arise from quantum3(1+w field) theory. We are now in the position to discuss somer( specifict) µ a( cosmologicalt) . models. (4.13) Now we have4.6.1 an equationThe Einstein that tells us Universe how the density behaves while the 4.6.1 The Einstein Universe universe expandsThis cosmological or contracts. model We still was have first to proposed determine by the Einstein, dynamics who tried to of the scale factor.Thisobtain cosmological a model without was first expansion proposed or by contraction. Einstein, who This tried goal can to Using the expressionsobtainbe achieved a static for only universe the if RicciL = without0. tensor, The expansion static the curvature conditions or contraction. scalar are a˙ = anda¨ This= the0 goal and fromcan 6 energy-momentumbeEq. achieved 4.22 tensor, we only obtain we if canL = substitute0. The static them conditions into the are Einsteina˙ = a¨ = equa-0 and from 6 tions Eq. 3.17Eq. finding 4.22 we (we obtain leave L = 0 for theL moment) a = k = 1 rLk ) (4.24) a¨ a = L k = 1 3 = 4pG(r + 3P)r =forr k ()µ, n. )=(0, 0) (4.24) a 4LpG 2 r = . (4.14) a¨ a˙ k 4pG + 2 + 2 = 4pG(r P) for 41(µ, n)=(i, j) . a a a2 ✓ ◆ 41 For the spatial components i, j = 1, 2, 3 there is only one equation as it should be, given the requirement of isotropy. Substituting the second derivative of a(t) from the first of the Eq. 4.14 into the second we obtain the Friedmann Equations

a¨ 4pG = (r + 3P) a 3 (4.15) a˙ 2 8pG k = r . a 3 a2 ✓ ◆ 39 CHAPTER 4. COSMOLOGICAL MODELS CHAPTER 4. COSMOLOGICAL MODELS

The previous equationsCHAPTER describe 4. a COSMOLOGICAL spherical universe MODELS with a constant ra- dius (scale factor), a constantThe density previous and equations a non-zero describe cosmological a spherical con- universe with a constant ra- The previousstant. equations An unpleasant describe characteristic a sphericaldius (scale universe of factor), this with universe a constant constant is ra- its density instability: and a a non-zero cosmological con- dius (scale factor), a constant density and a non-zero cosmological con- small perturbation wouldstant. make it An expand unpleasant or contract. characteristic In this sense of this this universe is its instability: a stant. An unpleasant characteristicsmall of this perturbation universe is its would instability: make a it expand or contract. In this sense this smallmodel perturbation is not would as static make as it its expand name or may contract. imply. In this sense this model is not as static as its name maymodel imply. is not as static as its name may imply. 4.6.2 The Matter-dominated Universe 4.6.2 The Matter-dominated4.6.2 UniverseThe Matter-dominated Universe Suppose that theMatter-Dominated universe is uniformly filled Universe only with non-interacting Supposebodies. that the In universe this case, is uniformly= 0Suppose (or filled= only that0), L with the= 0 non-interacting universe and from is Eq. uniformly 4.13 we filled have only with non-interacting = P = =w bodies. In this3 case, P 0 (or w 0),bodies.L 0 In and this from case, Eq. 4.13P = we0 have (or w = 0), L = 0 and from Eq. 4.13 we have r a(t)r3 =a(At)with= AA =withconstant.A =constant. The second The of the second Friedman of the equations Friedman equations · Eq.·“Dust”: 4.22 becomesw =0 r a(t)3 = A with A =constant. The second of the Friedman equations Eq. 4.22 becomes ) · 8pGEq.A 4.228p becomesG A a˙2 = a˙2 =k .k . (4.25) (4.25) 3 a 8pG A d⌘ 1 3 a a˙2 = k . (4.25) DefineIntroducing the ConformalIntroducing the Timeconformal the conformal time=h instead time ofh theinstead time t ofsuch the that timedh/tdtsuch= that3 dah/dt = ( ) dt a 1/a t 1/thea( lastt) the equation last equation becomes becomesIntroducing the conformal time h instead of the time t such that dh/dt = 8pG 2 = 1/a(t)2the last equation becomes a0 Aa2 ka8pG.2 (4.26) The 2nd Friedmann equation becomes 3 a0 = Aa ka . (4.26) 3 2 8pG 2 The apex in a0 indicates differentiation with respect to the conformala0 time.= Aa ka . (4.26) 3 The lastThe equation apex in cana0 indicates be easily differentiation integrated. For with example, respect choosing to the as conformal time. Withinitial solutions condition a(0)=0 we have The last equation can beThe easily apex integrated. in a0 indicates For differentiation example, choosing with respect as to the conformal time. initial condition4pGAa(0)=0 weThe have last equation4pGA can be easily integrated. For example, choosing as k = 1 a = (1 cos h) ; t = (h sin h) ) 3 4pGA initial condition3 a(0)=4p0GA we have k = 1 2pGAa =2 (1 cos h) 2pGA; 3 t = (h sin h) k = 0 a = ) h 3 ; t = h 4pGA3 4pGA ) 3 = 1 9 = (1 cos h) ; = (h sin h) 4pGA 2pGA k 4pGAa 2pGA t k = 1 k =a =0 a =(cosh h 1h)2 ; t = ) ;(sinht =h3 h) .h3 3 ) )3 3 3 2pGA 9 2 2pGA 3 4pGA k = 0 a = 4pGAh(4.27) ; t = h SommersemesterIn this2018 model,k = a closed1 a (k== 1)Luca universe Doria, (JGUcosh Mainz expandsh 1) and) eventually; t = 3 collapses(sinh22h h) . 9 again. Open ) (k <= 31) expand forever. In the boundary4pGA3 case 4pGA k = 1 a = (cosh h 1) (4.27); t = (sinh h h) . k = 0, the expansion continues forever, but the expansion) rate approaches3 3 zero forIn infinite this model, times ( aH closed0 when (k =t 1)+ universe•). expands and eventually collapses (4.27) ! ! again. Open universes (kIn<= this1) model, expand a forever. closed (k In= the1) universe boundary expands case and eventually collapses 4.6.3 k =The0, the Radiation-dominated expansion continuesagain. Universe forever, Open but universes the expansion (k <= rate1) expand approaches forever. In the boundary case zero for infinite times (H k =00, when the expansiont +•). continues forever, but the expansion rate approaches For radiation, we have already seen! that w = 1/3, so!r = 3P and ra4 = A with A=constant. Setting L = 0, thezero second for of infinite the Friedman times ( equationsH 0 when t +•). 4.6.3 The Radiation-dominated Universe ! ! 42 For radiation, we have already4.6.3 seenThe that Radiation-dominatedw = 1/3, so r = 3P and ra4 Universe= A with A=constant. Setting ForL = radiation,0, the second we have of the already Friedman seen that equationsw = 1/3, so r = 3P and ra4 = A with A=constant. Setting L = 0, the second of the Friedman equations 42 42 CHAPTER 4. COSMOLOGICAL MODELS CHAPTER 4. COSMOLOGICALRadiation-Dominated MODELS Universe Eq. 4.22 becomes Eq. 4.22 becomes 2 8pG A Radiation: w =1/3 a˙ =8pG A k . (4.28) a˙2 = 3 k2 . (4.28) ) 3 a2 a UsingUsing the the initial initial condition condition a(0)=0, a(0)=0, the the solutions solutions are are

8pGA With solutions k = 1 a = 2 8pt GAt2 2 k = 1) a s= 23a2 2 t t ) sr r 3a 8pGA (4.29) k = 0 a = 2 28pt GA k = 0) a s= r 23a t (4.29) ) s 3a2 8prGA k = 1 a = 2 t + t2 . ) s 3a2 r 8pGA 2 k = 1 a = 2 2 t + t . As in the matter-dominated ) universe,s ther closed3a solution expands and then recollapses, while the other cases expand forever. As in the matter-dominated universe, the closed solution expands and then4.6.4 recollapses,Vacuum-Dominated while the other Universe cases expand forever. In the vacuum-dominated model, there is no matter present, so P = r = 0. In this case, only the cosmological constant plays a role (L > 0). A Sommersemester4.6.4 2018 Vacuum-DominatedLuca Doria, JGU Universe Mainz 23 possible interpretation of this model comes from quantum field theory In the(QFT), vacuum-dominated where there are non-zero model, quantum there fluctuations is no matter even present, in the vac- so P = r = 0.uum In this where case, fields only have the zero cosmological average. QFT constant predicts a termplays analogous a role (L to > 0). A possiblethe cosmological interpretation constant. of In this this model case the comes second of from the Friedman quantum equa- field theory (QFT),tions where Eq. 4.22 there reduces are to non-zero quantum fluctuations even in the vac- uum where fields have zero average.La2 QFT predicts a term analogous to a˙2 = k , (4.30) the cosmological constant. In this3 case the second of the Friedman equa- tionsand Eq. the 4.22 solutions reduces are to 3La2 L k = 1 a a=˙2 = cosh k ,t (4.30) ) rL 3 r 3 ! 3 L and the solutions arek = 0 a = exp t (4.31) ) rL r 3 ! 3 L k = 1 a = 3 coshL t k = 1 )a = sinhL t 3. ) rLr r 3 !r ! 3 L k = 0 a =43 exp t (4.31) ) rL r 3 ! 3 L k = 1 a = sinh t . ) rL r 3 !

43 CHAPTER 4. COSMOLOGICAL MODELS

Eq. 4.22 becomes 8pG A a˙2 = k . (4.28) 3 a2 Using the initial condition a(0)=0, the solutions are CHAPTER 4. COSMOLOGICAL MODELS 8pGA 2 k = 1 a = 2 2 t t Eq. 4.22 becomes ) s r 3a 8pG A a˙2 = k . (4.28) 3 a2 8pGA = = (4.29) Using the initial conditionk a(0)=0,0 thea solutions2 are 2 t ) s r 3a 8pGA 8pGA k = 1= a = 2 = t t2 + 2 k ) 1 sa 3a2 2 t t . ) r s r 3a 8pGA As in the matter-dominatedk = 0 a = universe,2 t the closed solution(4.29) expands and ) s 3a2 then recollapses, while the otherr cases expand forever.

8pGA 2 k = 1 a = 2 2 t + t . 4.6.4 Vacuum-Dominated ) s r Universe3a As in the matter-dominated universe, the closed solution expands and thenIn the recollapses, vacuum-dominated while the other model, cases expand there forever. is no matter present, so P = r = 0. In this case, only the cosmological constant plays a role (L > 0). A 4.6.4possibleVacuum-Dominated interpretation of this Universe model comes from quantum field theory (QFT), where there are non-zero quantum fluctuations even in the vac- In the vacuum-dominated model, there is no matter present, so P = r = 0.uum In this where case, fields only the have cosmological zero average. constant QFT plays predicts a role (L a> term0). A analogous to possiblethe cosmological interpretation constant. of this model In this comes case from the quantum second of field the theory Friedman equa- (QFT),tions Eq. where 4.22 there reduces are non-zero to Vacuum-Dominated quantum fluctuations even in Universe the vac- uum where fields have zero average. QFT predicts a term analogous to the cosmological constant. In this case theL seconda2 of the Friedman equa- Vacuum: P = ⇢ =0 a˙2 = k , (4.30) tions Eq. 4.22 reduces to ) 3 La2 and the solutions are a˙2 = k , (4.30) 3 With solutions 3 L and the solutions are k = 1 a = cosh t L 3 ) 3 r L r ! k = 1 a = cosh t ) rL 3r 3 ! L k = 0 a = exp t (4.31) ) 3 L L 3 k = 0 a = expr t r !Note this(4.31) result ) L 3 ! r 3r L k = 1 a3= sinhL t . k = 1 a =) sinhrL t r. 3 ! ) rL r 3 !

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Sommersemester 2018 Luca Doria, JGU Mainz 24 CHAPTER 4. COSMOLOGICAL MODELS

The Friedmann equations are basically the Einstein equations for an ho- mogeneous and isotropic universe filled with a perfect fluid. For fully solve themCHAPTER and obtain 4. the COSMOLOGICAL function a(t), MODELS we need to specify the equation CHAPTER 4. COSMOLOGICALCHAPTER MODELS 4. COSMOLOGICAL MODELS of state (or - almost equivalently - the parameter w). The Friedmann equations are basically the Einstein equations for anCHAPTER ho- 4. COSMOLOGICAL MODELS The Friedmann equations are basically the Einstein equations for an ho- mogeneous and isotropicThe universe Friedmann filled equations with a are perfect basically fluid. the Einstein For fully equations for an ho- mogeneous and isotropicmogeneous universe and filled isotropic with a universe perfect fluid. filled For with fully a perfect fluid. For fully solve them and obtain the function a(t), weThe need Friedmann to specify equations the equation are basically the Einstein equations for an ho- 4.5 solveThe them Cosmological and obtainmogeneoussolve the them function and anda Parameters isotropic( obtaint), we need the universe function to specify filleda( thet with), we equation a need perfect to specifyfluid. For the fully equation of state (or - almost equivalentlyof state (or - the - almost parametersolveof equivalently state themw (or). and - - almost the obtain parameter equivalently the functionw). a -( thet), we parameter need to specifyw). the equation The expansion rate (howof state much (or - a(t) almost changes equivalently in units - the parameter of a(t) itself)w). is called 4.5 The Cosmologicalthe Hubble4.5 The Parameter Parameters Cosmological4.5 The CosmologicalParameters Parameters 4.5 The Cosmologicala˙(t) Parameters The expansion rate (how muchH( a(t)t)= changes in units. of a(t) itself) is called (4.16) The expansion rate (how much a(t) changesThe in units expansion of a(t) rate itself)a( (howt) is much called a(t) changes in units of a(t) itself) is called the Hubble ParameterThe expansion rate (how much a(t) changes in units of a(t) itself) is called the Hubble ParameterThe definition of H makesthe Hubble sense, Parameter sinceCHAPTERa˙(t) the scale 4. COSMOLOGICAL of a itself is not MODELS important: the HubbleH(t Parameter)= .a˙(t) (4.16) a˙(t) a(t) H(t)=a˙(t) . (4.16) in the ratio,H(t)= we obtain. a meaningful measurable (4.16)H(t)= quantity.a(t.) The today’s (4.16) TheThe definition Friedmanna( oft)H makes equations sense, are since basically the scale the of Einsteina itself isa( equationst not) important: for an ho- value of this parameterThe is definition called H of0. H makes sense, since the scale of a itself is not important: The definition of H makesin sense,mogeneous the ratio, since we the and obtain scaleThe isotropic a definition of meaningfula itself universe of isH not measurablemakes filled important: sense, with quantity. since a perfect the The scale fluid. today’s of a Foritself fully is not important: It is also useful to rewritein the the ratio, energy we obtain conservation a meaningful equation measurable in the quantity. FLRW The today’s in the ratio, we obtain avalue meaningfulsolve of them this parameter measurable and obtainin the is called ratio, the quantity. functionH we0. obtaina The(t) a, meaningful we today’s need to measurable specify the quantity. equation The today’s Universe (Eq. 4.11) usingvaluevalue the of of this Hubblethis parameter parameter parameter is is called calledH as.H0. value of this parameter isIt called isof also stateH useful (or. - to almost rewrite equivalently the energy conservation- the parameter equation0w). in the FLRW Universe (Eq.0 4.11) usingItIt is is also also the useful Hubble useful to to parameter rewrite rewrite the the as energy energy conservation conservation equation equation in the in FLRW the FLRW It is also useful to rewrite the energy conservationUniverseUniverser˙ + equation (Eq.3 (Eq.H( 4.11)r 4.11)+ using inP using)= the the0 FLRW the Hubble . Hubble parameter parameter as as (4.17) Universe (Eq. 4.11) using the Hubble parameterr as˙ + 3H(r + P)=0 . (4.17) 4.5 The Cosmological Parametersr˙ +r˙ 3+H3(Hr +(rP+)=P)=0 .0 . (4.17) (4.17) AnotherAnother relevant relevant quantity quantity is is the the decelerationdeceleration parameter parameter r˙ + 3H(r + P)=Another0 . relevantThe quantity Density is (4.17) the parameter The expansion rateAnother (how much relevant a(t)aa¨ quantitya changesa¨ is in the unitsdeceleration of a(t) itself) parameter is called q = , (4.18) Another relevant quantity isthe theHubbledeceleration Parameter parameterq = 2 ,aa¨ (4.18) a˙ a˙2 q = aa¨, (4.18) a˙(t) q =a˙2 , (4.18) Rememberwhich quantifies the aHubblea¨ the parameter: rate of changeH( oft)=H. Substituting. the Hubblea˙2 param- (4.16) which quantifiesq = the, ratewhich of quantifies change the of H ratea.(t Substituting) of change (4.18) of H. Substitutingthe Hubble the param- Hubble param- eter in the second2 Friedmannwhich quantifies equation the rate of change of H. Substituting the Hubble param- The definition a˙ ofeterH makes in the sense, second since Friedmann the scale equation of a itself is not important: eter in the second Friedmanneter8p in the equation second8p Friedmann equation which quantifies the rate of changein the ratio, of H. we2 Substituting obtainG a meaningfulk the HubbleG measurable param-k quantity. The today’s H = r 2 2 8rpG1 = k2 2 8.pG (4.19)k The 2nd Friedmann Equation8pG3 becomes ak )H32H8=pG8prG H ak k8pr G 1 = k. (4.19) eter in the second Friedmannvalue equation of this2 parameter is called H20. 3 a2 ) 3H2 H2a2 H = r 3HH2 = r r1 =2 292 r.13= 2 2 (4.19). (4.19) DefiningIt is also the usefulcritical to density rewriter thec 2= energy(today’s conservation2 3 value a ) equation221032H g/ incm the FLRWH a 8pG k 8pG 3 ka )8pG3H H⇡3H a ⇠ 29 3 2 4 3 Defining the critical density rc = 8pG (today’s2 value 10 g/cm H = 1.05r Universe10 eV (Eq./cm 4.11)r ) and1 using= the density the Hubble. parameter2 parameter (4.19)W = asr/rc 3weH have ⇡ 29 ⇠ 3 ⇥ 2 2 1.05Defining10 24 the2 /critical33H) and density the densityrc = parameter(today’sW =29 valuer/r we3 have10 g/cm Defining3 Introducing thea )critical the3 CriticalH density Density HraceV=cm (today’s value8pG 10 g/cmc ⇡ ⇠ 1.05⇥ 10r˙ 4+eV3H/cm(8rpk+3G)P and)= the0density . parameter W (4.17)= r/r we have 4 3 2 3 W 1 = .k ⇡ (4.20) c ⇠ Defining the critical1.05 density10r eV=/cmH (today’s) and the⇥ valuedensity H102 parametera229g/cm3 W = r/rc we have c 8pG W 1 = 2 2 . (4.20) 4 3 ⇥ Another relevant quantity is the⇡ deceleration parameter⇠ H a k 1.05 10 eV/cm ) and theTheAnd densitythe density Density parameter parameter Parameter is quiteW = importantr/rc wewe have sincehave: itW determines1 = the geome-. (4.20) ⇥ The density parameterkaa is¨ quite important H since2a2 it determines the geome- try (through the curvature kW) of the1q= universe:= ,. (4.18) (4.20) k tryThe (through density the parameter curvatureH2a˙a2 2 isk quite) of the important universe: since it determines the geome- W 1 = r2 <2 rc .W < 1 k = 1 (4.20) (Open) The densitywhich parameter quantifiesH a try the is() (through rate quite of change important() ther < curvaturer ofc H. since SubstitutingW k<)1 of it the determinesk universe: the= Hubble1 (Open)the param- geome- and therefore r = rc W = 1 k =()0 (Flat)() (4.21) eter in the second Friedmann() () equationr = r W = 1 k = 0 (Flat) (4.21) The density parametertry (through is quite important the curvature since itk) determines of ther universe: rc W > 1 k =()1() (Closed)()() try (through the curvature k) of the universe:()8pG ()rk>=rc 8pG W > 1= kk = 1= (Closed) 2 = r r()c W =()1 k 0 (Flat) (4.21) r < rcH W rc W > 140 k = 1 (Closed) () () 2() () r = rc W = 1 = 3kH = 0 (Flat) 29 3 (4.21) r = rc WDefining= 1 thek =critical0() density (Flat)r()c 8pG (today’s(4.21) value 10 g/cm () ()4 3 40 ⇡ ⇠ 1.05 10r>eVr/c cm ) andW > the1densityk parameter= 1W (Closed)= r/rc we have r > rc W > 1⇥ k = 1() (Closed)() () () k W 1 = . (4.20) 40 40 H2a2 The density parameter is quite important since it determines the geome- try (through the curvature k) of the universe: r < r W < 1 k = 1 (Open) c () () r = r W = 1 k = 0 (Flat) (4.21) c () () r > r W > 1 k = 1 (Closed) c () () 40 CHAPTERCHAPTER 4. COSMOLOGICAL 4. COSMOLOGICAL MODELS MODELS

The Thek = k1= case1 case is also is also known known as deSitter as deSitter Universe Universe. . The Thek = k0= case0 case predicts predicts an exponential an exponential growth growthCHAPTER of the of Universe the Universe 4. COSMOLOGICAL and asand we as we MODELS willwill see later see later on, this on, this solution solution has relevance has relevance in the in framework the framework of inflation- of inflation- ary modelsary models. If we. IfThe assume wek assume= 1L case

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