Quantum eld theory in curved spacetime. Introduction on ination. Dennis Sauter. Seminar coordinator. Dr. Javier Rubio

Transcription

1 Quantum eld theory n curved spacetme Introducton on naton Denns Sauter Semnar coordnator Dr. Javer Rubo

2 CONTENTS Contents 1 Introducton 1 2 Problems n standard cosmology Horzon problem Flatness problem General dea of naton Slow roll condtons Inaton scalar eld Flat potental m 2 Φ 2 - potental

3 1 INTRODUCTION 1 1 Introducton To dscuss the state of n aton n the early unverse t s essental to ask why such a theory s even necessary to descrbe a proper evoluton of the unverse whch we see today. The framework whch we are dealng wth s a unverse whch evolved accordng to the Fredmann equatons and therefore s homogeneous and sotropc on large scales. Hence the metrc we use wll be the Fredmann Lemaître Robertson Walker metrc n sphercal coordnates. ds2 = a(t)2 dτ 2 dr2 + r2 dω2 (1) Here, τ s the proper tme, a(t) s the tme dependend scale factor and c = ~ = 1 and therefore were left out. In standard cosmology and the Bg Bang theory, what one actually descrbes s only the outcome of such a "Bg Bang" and how the unverse evolves afterwards. Through the cosmc mcrowave background radaton ( g. 1) we can see how the unverse looked at early stages. From ths we get the nformaton that the unverse s extremely sotropc and homogeneous wth hgh accuracy up to Because of lmtatons n expermental setups we can only test theores up to > s after the Bg Bang, so anythng earler would just be an assumpton or an extrapolaton of the known evoluton backwards n tme. Fgure 1: Cosmc mcrowave background radaton

4 2 PROBLEMS IN STANDARD COSMOLOGY 2 2 Problems n standard cosmology In every physcal process one s nterested n the way a certan system evolves further n tme. Therefore two thngs are necessary, namely startng condtons such as spacal dstrbuton and momentum dstrbuton and equatons whch descrbe the evoluton wth gven startng condtons. In case of the Fredmann unverse such equatons are gven by the Fredmann equatons (ȧ ) 2 H 2 = = 8 a 3 πgρ E k (2) a 2 Ḣ + H 2 = ä = 4 3 πg (ρ E + 3p) (3) wth the Hubble Parameter H, the gravtatonal constant G, the curvature parameter k, the energy densty ρ E and the pressure p. The queston whch now arses s: What startng condtons lead to the unverse we see today? To gve us a rough estmate of the startng condtons the followng approxmatons can be made. 2.1 Horzon problem Frst of all we look at lenght scales and compare them between early tmes (denoted wth an ) and today (doneted wth a 0). The rato of both scales s n order of the rato of the correspondng scale factors l l 0 a a 0 = ct 0 a a 0. (4) The rato of the scale factors can be expresed by the rato of the temperatures whch drves the expanson as some form of energy. We wll choose Planck values for T = T P l so that a T If ths s then compared a 0 T P l to the lenght scale of a causal regon at that tme whch s the Planck tme t P l s, one arrves at the expresson l t 0 a 10 28, (5) l c t P l a 0 0

5 2 PROBLEMS IN STANDARD COSMOLOGY 3 where we also used that for a power dependence of t for a(t) we can wrte ȧ = da dt a. From ths result we conclude that n very early tmes the unverse t conssted of around causally dsconnected regons of space. Nevertheless we can observe a homogenety up to 10 5 accuracy wch evolved from multple regons wthout any nteracton among each other. Such an event would be very unlkely and therefore s called the Horzon problem of standard cosmology. 2.2 Flatness problem Lets assume that somehow all the matter was dstrbuted homogeneously and gnore the horzon problem. The second problem arses f we look at the energy dstrbuton at prmordal tmes to see how the knetc energy has to be set for the unverse to nether nstantly recollapse under the gravtatonal forces nor emptyng out due to too fast expanson. Snce total energy needs to be conserved the followng relaton holds: E p + Ek = E tot = E tot 0 = E p 0 + E k 0 (6) wth the total energy beng the sum of potental and knetc energy. knetc energy depends on the velocty squared and t follows the relaton The ( ) 2 E k E0 k a. (7) 0 Now we dvde eq. (6) by the knetc energy of prmordal tmes to get a relatve accuracy for the quantty E tot E k = Ep + Ek E k Ep 0 + E k 0 E k 0 ( ) 2 a0. (8) For todays energy dstrbuton E p 0 E k 0 ths then leads to E tot E k (9)

6 2 PROBLEMS IN STANDARD COSMOLOGY 4 We conclude that the knetc energy of prmordal matter would have necessarly been determned wth an accuracy of up to To rewrte ths problem n terms of atness we look at the densty parameter Ω whch tells us whether the unverse s hyperbolc/open (Ω < 1), at (Ω = 1) or ellptc/closed (Ω > 1) Ω(t) 1 = Ω 1 = (Ω 0 1) (ah)2 0 (ah) 2 k (a(t)h(t)) 2 (10) ( a0 = (Ω 0 1) ) (11) Flatness s an unstable state n a Fredmann evolvng unverse. As a consequence of the atness whch s observed today, we would need an even atter unverse n early tmes, meanng Ω = 1 + O(10 56 ). Therefore ths problem of extreme netunng n startng condtons s called the atness problem of standard cosmology.

7 3 GENERAL IDEA OF INFLATION 5 3 General dea of naton The tme dervatves of a enters ndependently from each other nto both problems. In a matter or radaton domnated deceleratng Fredmann unverse where gravty slows down the expanson ȧ shows a monotone fallng behavour. Therefore the rato 0 wll always be a very small number. A small value can only be avoded f the gravtatonal force acted as a repulsve force at some pont n the past for a sucently long tme. Ths characterstc behavour denes naton: "Inaton s a stage of accelerated expanson of the unverse when gravty acts as a repulsve force." Fgure 2: Introducng a state of naton n early tmes to avod dvergng ȧ for t 0. As gure 2 ndcates, ȧ shows growng behavour up to some pont where then a transton occurs to the decelerated Fredmann expanson. The atness problem would mmedately be solved snce t follows from (10) Ω 1 1 ȧ that naton pushes Ω 1 to zero and therefore Ω to 1. Ths means that atness s an attractor state for repulsve gravty.

8 3 GENERAL IDEA OF INFLATION 6 Now we need to clarfy how gravty can become repulsve. Accelerated expanson means that ä > 0. From the Fredmann equatons (3) follows ä = 4 3 πg (ρ E + 3p) > 0. (12) From ths condton then follows that ρ E + 3p < 0. (13) There are two possble solutons for fulllng ths condton. The rst soluton would be a negatve energy densty ρ E < 0 ; 3p < ρ E. We wll later show that ths result wll not lead to the unverse whch we observe toady. A more promsng soluton s the second one where we assume a negatve pressure p < 0 ; ρ E < 3 p. Then gravty acts as a repulsve force and f p = ρ E ths leads to a so called de Stter spacetme wth the characterstc exponental grow of the scale factor a(t) e Ht. 3.1 Slow roll condtons A purely de Stter spacetme would never end to accelerate the expanson of the unverse and s only vald n rst order approxmaton. For a succesful naton model t s necessary to end at some pont. It s therefore crucal for ä to change the sgn after some tme (ȧ ) 2 (ȧ ) 2 H 2 + Ḣ = + ä a a a = ä a. (14) For ä to change sgn, the Hubble parameter needs a tme dependence wth a negatve tme dervatve such that ä s postve n the begnnng and then becomes negatve ( ) ä a = H2 1 Ḣ H 2 (1 ɛ). (15) H 2

9 3 GENERAL IDEA OF INFLATION 7 The rst slow roll parameter ɛ < 1 gves a condton for accelerated expanson. It wll later become clear why t s called "slow roll" parameter. The parameter ɛ has to grow wth tme and at some pont exceed the value 1 to end naton. Ths gves rse to the second slow roll parameter, as H 2 has to change faster than Ḣ Ḧ 2 Ḣ H (16) η Ḧ 1. (17) 2 Ḣ H As long as these condtons are fullled our natonary model s successful n descrbng a stage of naton whch then goes over n the already known decelerated Fredmann unverse. In contrast to prevous calculatons we now have to consder three ponts n tme. The pont n tme today t 0 stays the same, but we now have ntroduced naton such that the prevous ntal tme translates to the nal tme after naton t t f and t wll be the ntal tme before naton happened. Now we want to estmate how long naton should at least persst to explan the homogenety of the cosmc mcrowave background 10 5 > = 0 a f a f = H a f. (18) a 0 H f a f 0 We have shown earler that f a f a > H H f (19) We used that H has a weak tme dependence and only decreases slowly. Wth the rst order de Stter approxmaton a(t) e Ht we arrve at a f a e H f t f H t e H t > 10 33, (20)

10 3 GENERAL IDEA OF INFLATION 8 where we agan used that the Hubble parameter does not change much n tme. t > 76H 1 (21) As result we get that naton should at least hold for 76 Hubble tmes or so called "e-folds" to explan n a reasonable way the measured concept of the unverse.

11 4 INFLATON SCALAR FIELD 9 4 Inaton scalar eld 4.1 Flat potental We calculated condtons for naton and showed that the model contans possble solutons for the ntal problems. Now t s natural to ask what leads to repulsve gravty and accelerated expanson. To answer ths we ntroduce a scalar eld Φ, the so called Inaton eld wth the acton S = d 4 x [ 1 g 2 R + 1 ] 2 gµν µ Φ ν Φ + V (Φ). (22) From the energy-momentum tensor, gven by T µν = 2 ( ) δs Φ 1 g δg = µφ µν ν Φ g µν 2 σ Φ σ Φ + V (Φ) (23) we can calculate the energy densty and the pressure under the assumpton of a homogeneous scalar eld only dependend on tme ρ E = T 00 = 1 2 Φ 2 + V (Φ) (24) p = 1 3 T = 1 2 Φ 2 V (Φ) (25) The negatve pressure condton p ρ E can then be rewrtten to V (Φ) Φ 2. The evoluton of such a scalar eld s determned by the equatons Φ + 3H Φ + V Φ = 0 (26) H 2 = 8π 3 ( ) 1 2 Φ 2 + V (Φ), (27) whch are derved from the Fredmann equatons and the Klen-Gordon equaton. Now t becomes evdent why the parameters are called "slow roll" parameters. They descrbe a scalar eld n a potental that s ntally at wth neglgble knetc energy, but "roll" slowly down the potental to a mnmum,

12 4 INFLATON SCALAR FIELD 10 where the knetc energy term domnates over the potental term. The prncple structure of such a potental s shown n gure 3. The duraton of naton s determned by the length of the at regon of the potental. Fgure 3: Example of a at potental that suces the condtons gven for accelerated expanson. Whle the eld s large n ths example, the knetc energy contrbuton compared to the potental contrbuton s neglgble and p ρ E holds. If both contrbutons become comparable naton wll end and we get a transton to a decelerated Fredmann unverse. Now we can argue why the negatve energy densty s not a vald soluton to descrbe naton. Eq. (24) requres a negatve potental whch s agan much larger than the knetc energy. A negatve potental however leads to another type of accelerated evoluton, namely deaton or contracton that wll eventually end n a sngularty. Snce we stll exst today, ths possble soluton can be excluded. 4.2 m 2 Φ 2 - potental As a smple example of how ths mechansm works we consder a quadratc potental V (Φ) = 1 2 m2 Φ 2. If one nserts eq. (27) n eq. (26), the result s a

13 4 INFLATON SCALAR FIELD 11 coupled derental equaton for Φ and Φ Φ + ( ) 1 12π Φ2 + m 2 Φ 2 2 Φ + m 2 Φ = 0, (28) whch can be rewrtten by usng Φ = Φ d Φ dφ d Φ ( ) 1 12π Φ2 + m 2 Φ 2 2 Φ + m 2 Φ dφ =. (29) Φ To make solvng ths equaton easer we look at two derent approxmatons. Frstly lets assume the case Φ Φ where the knetc energy term domnates over the potental term. Then eq. (29) smples to The derental equaton s solved by d Φ dφ = 12π Φ. (30) Φ e 12πΦ. (31) We conclude that any gven large knetc energy wll decrease exponentally and decay quckly compared to the potental. Secondly we assume the case wth domnant potental Φ Φ and addtonally to ensure atness the knetc energy should not vary n approxmaton d Φ = 0. In ths case eq. (29) then dφ smples to 0 = 12πmΦ + m 2 Φ Φ (32) Φ = m (33) 12π Wth these to specal case solutons we can understand the behavour of the general solutons whch are shown n gure 4.

14 4 INFLATON SCALAR FIELD 12 Fgure 4: phase dagramm of solutons for the coupled derental equaton (eq. (29) ) for Φ and Φ. The phase dagramm shows that almost every startng condton for Φ and Φ wll jon the attractor soluton at some pont and then fulll the condtons necessaary for accelerated expanson. The only crtera on the startng condtons s the mnmum tme of 76 e-folds between jonng the attractor and leavng the at regon. At rst t seems surprsng that the clearly not at parabola potental fulls ths requrement. However, eq. (26) has the form of a harmonc oscllator equaton wth a dampng term proportonal to the Hubble parameter. Then agan the Hubble parameter s proportonal to the square root of the potental. Ths leads to the fact that a hgh potental value tself causes the slow roll by actng as a hgh frcton term. In fact, every potental of hgher order lke λφ 4 etc. leads to such a soluton. After the eld exts the at regon t begns to oscllate and starts the state of reheatng whch wll be the next topc to dscuss.

15 REFERENCES 13 References [1] Physcal Foundaton of Cosmology, Vatcheslav Muhkanov, Secs [2] TASI Lectures on Inaton, Danel Baumann, arxv: Secs ,4,5, [3] Cosmology wth Negatve Potentals, Gary Felder, Andre Frolov, Lev Kofman, Andre Lnde, arxv:hep-th/ [4] Lecture about Inatonalry Cosmology at MIT, Alan Guth, URL: