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Friedmann-Lemaitre cosmologies via roulettes and other analytic methods
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Please note that terms and conditions apply. JCAP10(2015)056 hysics P k energy theory ing the Friedmann le c,d ic t ersity - Shanghai, ar iversity, ich renders the integration rmer. Next the Friedmann for the Randall-Sundrum II ics, Henan University, 10.1088/1475-7516/2015/10/056 . First it will be demonstrated sentation as a roulette such that f the solutions of the Friedmann eresting phenomena. Finally an doi: h law. [email protected] , strop A and Yisong Yang b [email protected] , osmology and and osmology C Gary W. Gibbons a 1508.06750 dark matter theory, cosmology of theories beyond the SM, dar In this work a series of methods are developed for understand
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ou An IOP and SISSA journal An IOP and 2015 IOP Publishing Ltd and Sissa Medialab srl Kaifeng, Henan 475004, P.R. China D.A.M.T.P., University of Cambridge, Cambridge CB3 0WA, U.K. Department of Mathematics, PolytechnicBrooklyn, School, New New York York Un 11201, U.S.A. NYU-ECNU Institute of Mathematical3663 Sciences, North New Zhongshan York Road, Univ Shanghai 200062,E-mail: P.R. China Institute of Contemporary Mathematics, School of Mathemat b c d a c ArXiv ePrint: Received September 10, 2015 Accepted September 24, 2015 Published October 27, 2015 Abstract. equation when it is beyondthat the every reach of solution the of Chebyshev the theoreminformation Friedmann equation on admits the a repre latterequation may is be integrated useduniverse, for to leading a obtain to that a quadraticanalytic harvest for method equation of the is of a used fo richequation, to state collection isolate when and the of the asymptotic new equation behaviorimpossible, int o of and to state establish is a of universal an exponential extended growt Keywords: form wh Shouxin Chen, Friedmann-Lemaitre cosmologies via roulettes and other analytic methods J
JCAP10(2015)056 5 5 6 6 9 2 3 4 28 29 29 30 30 30 30 31 31 31 32 32 32 11 12 13 16 16 17 19 21 23 24 25 17 23 26 27 11 ns – 1 – A.4.5 Sinusoidal spirals A.4.6 Black holes A.4.7 Other examples A.4.8 Some hyperbolicA.4.9 spirals Further examples A.4.1 Circles A.4.2 Conics A.4.3 Algebraic spirals A.4.4 Logarithmic or equiangular spiral 3.2.1 Steiner’s theorem A.3 Chebyshev’s theorem A.4 Examples A.1 Affine andA.2 rescaling of Non-linear angle transformations 5.3 Chaplygin fluid case 6.1 Extended6.2 Chaplygin fluids Universal6.3 growth law Examples 4.3 Degenerate4.4 situation Polytropic equation of state case 5.1 Linear5.2 equation of state Solutions case of finite lifespans 3.4 A3.5 two-component fluid in ΛCDM a cosmology closed universe 4.1 Restrictions4.2 to parameters in Global non-degenerate solutions situatio 3.1 Rolling3.2 without slipping The Friedmann equation 3.3 A single-component fluid in a closed universe 7 Conclusions A Pedal equations and their inversion 6 Universal growth law for the extended Chaplygin universe 5 The Randall-Sundrum II cosmology 4 Quadratic equation of state 3 Friedmann’s equations and roulettes Contents 1 Introduction 2 Friedmann’s equations JCAP10(2015)056 , t 1 , that , that is century, th are related Friedmann which is the P roulette ) against time t . In the light of and 18 ( scussion, showing a th g one of Huyghens’ catenary Tolman Universe lette construction of a rals of binomial differen- directrix ded into three categories: ]. In this way we obtain a applied to the d a e special solutions. Recall and Pressure 6 cle commences by recalling t it is a mathematical fluke le factor lutions when the barotropic , ρ iately to the motion of light, resented as a slipping along a straight line. othesis is taken, the Einstein 5 ical constant. The purpose of ] and the integrability of the id whose pressure is negligible ssible and how in principle to y g. More recently such models ose of Richard Chace Tolman wide variety of iconic problems 4 the recent systematic works [ Stoics. r spacetime which contains just s [ ch may be used to obtain either e wheel of time steadily rolling pping” the straight line interval ic and conformal times, and for ed in the 17 own as the Friedmann equations, , century, we can also use the same 3 s of mechanics. More surprisingly ear ordinary differential equations, of the Friedmann equations beyond th tions. ) is a semi-circle which of course may t ( a – 2 – moving in a vertical stratified medium with variable rolled without slipping along its familiar in cosmology it will become apparent that the ideas Snell’s Law parabola . In its closed form, the scale factor has the form of a in the literature, it is here proposed to provide a general di . In this case, for a closed universes, P 3 1 = The idea of roulettes offers opportunities for obtaining som Evidence that this may be so is provided by what is often calle Recall also that the Chebyshev theorem applies only to integ Few of those textbooks give details or a derivation of the rou lacuna ρ ], we explored a link between the Chebyshev theorem [ description to describe the shapes of axisymmetric soap film perhaps we may, as Delaunay first pointed out in the 19 refractive index, and to the familiar central orbit problem be obtained as a sort of limiting case by “rolling without sli given by it radius.Dark Further Energy evidence is provided by De-Sitte among other things thatobtain a the roulette form construction ofsome is of the always the rolling po classical curve. theory of In plane order curves,and to mainly Raychaudhuri do develop equations sohave the a arti much wideror relevance. sound In rays particular governed they by extend immed this locus of the focus of a 2 1 Introduction When a homogeneous andequations isotropic of general perfect-fluid relativity universe aregoverning the hyp reduced Hubble into parameter a in terms setwhich of of are a lin scale of factor, basic kn importance in evolutionary cosmology. In framework for understanding in ain unified physics. geometric fashion a Friedmann equations and obtainedequation a of wealth state of isflat either new and linear explicit non-flat or spatial so the nonlinear, sections present with in paper and both is withoutexact cosm to expressions a present or cosmolog several insightful analytic knowledgethe methods of reach the whi solutions of theroulettes, Chebyshev explicit theorem. integrations, and These analytic methods approxima may bethat divi many textbooks mention the fact that the graph of the sca although its roots go back to the Greeks. As the theory is then as the locus of a point on or inside a curve which rolls without for a closed Friedmann-Lemaitre cosmologyfollows supported is by a cycloid, flu from thus providing what an Fred attractiveisochronous Hoyle image dubbed pendulum of and a th about inviting Big cosmologies speculations, Bang which to such arehave as a been cyclic called th Big or Ekpyrotic Crunch cyclic following up remindin an old to tradition acycloid of rescalin the and none, asor far whether as any plot we of are the aware, scale explain factor whether against or time no may be rep is one in which the fluid is a radiation gas whose energy densit by JCAP10(2015)056 ] ], 0. 0, 9 – 16 7 ≥ > (2.1) – when m 14 hen the ,...,α , the solution necessary and 1 1 n ,α 0 1+ finite lifespan > − -dimensional and the -hyperboloid, the flat not be converted into n of constant scalar cur- m n ,...,n, st duration for the scale plygin fluid form [ we review the Friedmann M solutions. In section 4 we lettes and the Friedmann ) we collect some relevant tributing to the right-hand
) and , then θ p ) s. If the point 3 u D t is smooth and − d ]. Suppose a closed convex curve d k ( ) and hence determine the equation ). The relation ( r γ γ t (1 ∂D πG ( 24 )= − ( 2 ′ – . We also have a relation ( and f γ f ∈ . Using | O . If 17 r = y,y ) and if we are fortunate we may be able | ) we may determine the relation ( ′ O ( L p )+ x )+(1 ( = a defines a (single-valued) bounded (and hence y y,y lies in a plane Π. Physically we think of fa ( lies outside ⇐⇒ r ( ρ – 5 – ρ trochoid , D 2 O 2 to the point a ) over ∂D a . ∈ x 2 2 ( y O x ′ = , = at the point f d d y y O 2 2 γ ) γ = = = a r θ 2 -axis of a Cartesian coordinate system ( ′ 1+ d d y ) ) by the same constant factor y x : a q ′ ). We define 2 3.3 1+(˙ r 1 t γ r ( r y,y 1+(˙ in ( t = 1+ , where . Otherwise to a r γ with respect to the origin r to be the is distance from )= with boundary and γ L r of a fixed point = a ′ D p of p,r cycloid ( ). Conversely given the graph . Thus x will be a smooth simple curve. If to a ( which rolls without slipping along the a straight line γ ′ y -axis. If, by some contrivance, γ γ x by its equation to the tangent of the curve ) are plane polar coordinates for the plane Π with origin γ rolls. the arc length along the rolling curve equals the arc length a O lamina We conclude this section by recalling Steiner’s theorem, wh Simple geometry gives , then ˜ pedal equation • r, θ D . The locus in Π ′ to solve for of the pedaldetermining curve whether in the in pedalbooks, curve polar is and coordinates among if the by so known quadratures. what plane are Th its properties. 3.1 Rolling withoutIn slipping what follows, extensive use has been made of [ if we are fortunate we may be able to solve for Thus given a pedal curve we get a relation ( enclosing a domain The area under the roulette is twice the area swept out by the r 3.2 The FriedmannIt equation is convenient in what follows to use units in which 8 to the form If ( Now we may rescale rigid Friedmann equation as no-monotonic) periodic graph Π may suppose the line of meets the the associated to the graph be a single-valued graph and it will have self-intersection normal of of from curve one refers JCAP10(2015)056 ) ), so η ( 3.7 f (3.7) (3.9) (3.5) (3.6) (3.8) , the a (3.10) (3.12) (3.11) η d = a a = t d . byshev theorem include he rolling curve. , defined by ainst time has no obvious η of conformal time, 4 2 lling curve would be equal to r p s that if we are considering the , obtained by differentiating ( 4 p . ) )= , = 1 gives our basic equation: ) p k ), and we deduce the pedal equation, ( a , f a ( we may choose the scaling factor ρ b 1 σ ( − ρ ρ 2 3 4 swept out would be equal to cosmic time βa (1 a , ρ 2 . 6= η 2 + f , = d γ − = 2 γ ) 2 ), the arc-length along the rolling curve equal 3 γ 1 1 a 2 t 3 2 η = 1 and a − ( ( )+ a ⇐⇒ a = ka k a kb = – 6 – αa b fp = R = + + = 1) as ( ρ ). In the sequel we shall proceed in both directions. g ρ 2 2 ] that in this case, if one defines a k a 2 )= ( = 1 and ) swept out is twice the area swept out on the rolling a f 26 ρ a η p ( b η 4 k t ( d d d ρ d p d ρ ) 4 t = p ( 2 a r = R 2 r ] the inverse scale factor, or redshift factor by 25 ) that the pedal equation of the roulette we seek is given by , which is equivalent to the familiar Raychaudhuri equation 3.1 η are arbitrary constants. may be chosen as we wish. Setting f α,β,σ Before doing so we recall that in terms of conformal time It is readily seen that the integrable cases in view of the Che If we introduce [ and as a consequence this would be3.3 twice the area swept A out single-componentIn by fluid the t in simplest a barotropic closed case with universe which we represented ascosmic a time and roulette, the then area arc-length under the of graph the ro curve. The areameaning under but if the instead graph we of consider the scale scale factor factor as plotted a ag function Friedmann equation may be written (if to cosmic time and the area which is a first integral of a non-linear oscillator equation It follows from ( 3.2.1 Steiner’s theorem With this preparation wescale note factor that as a Steiner’s function theorem of implie cosmic time, that we find or given, a pedal equation, we deduce with respect to where which may be read in two directions. Either we know In fact John Barrow has pointed out [ the model where JCAP10(2015)056 (3.20) (3.21) (3.19) (3.17) (3.22) (3.15) (3.13) (3.16) (3.18) (3.14) ) with ) whose 3.8 3.14 onformal time ere the origin is at a straight horizontal edal equation is seen , 2 , radiation. Both require ose Cartesian equation is 3 4 2 huri equation ( above discussion, the same , is available and cosmic time re general model ( 2 = − 2 η γ γ η. 2 3 s . − 2 2 . , cos y γ β 3 − , where the origin is on its circumfer- θ − , = = 0 2 1 ) 2 2 γ 2 θ g 3 θ, x g , = , = 1 2 − γ 2 2 − = 3 γ g 2 4 3 = 1 − 2 2 2 4 ) r = cos p 2 − 2 = cos(2 − y 2 r 2 γ – 7 – = r γ η, t 3 + − 2 3 2 2 r = cos γ x 3 ( ) + 2 γ 3 = sin − α 2 g γ − 2 3 4 η a − + d d r , we have 2 k 3 1 as a function of conformal time = g η +( a d d 2 P , g = 0, we have η 4 3 d d , we have P ρ = is a critical case separating solutions which oscillate in c 1 9 γ , which leads to the updated equation 3 2 3 2 = = 1, = P 6= γ , γ γ 9 10 = and γ γ 3 ence. which is whichits is centre. the polar Howeverline. equation the This of locus is a not of consistent circle such with a of the second-order rolling radius Raychaud curve 1 would , be wh which is the polar equation of the Lemniscate of Bernoulli wh For radiation, solution is If which is the polar equation of a circle of radius For dust, Not surprisingly, in view of the previous subsection and the However we wish to construct the solution as a roulette. The p Some special cases are as follows. − • • • = may then be obtained in principle by quadratures. t reduction may be carried out in the context of the slightly mo and so a general solution for then the Friedmann equation becomes Note that the σ and may be solved as before. and those that blow up exponentially. Another special case i whence we obtain the Raychaudhuri equation special treatment. which is the pedal equation of the curve to be JCAP10(2015)056 (3.27) (3.26) (3.33) (3.29) (3.25) (3.31) (3.30) (3.28) (3.32) (3.23) (3.24) in which 2 π = α . π = θ sp artesian equation is an equation is . m 2 and origin at its focus. Catalan’s trisectrix. , in which case it is a circle 2 . To begin with we assume 2 π ρ 2 1 3 y = . − with time unless 2 + α 2 = , , x ) , , θ + 8) P , θ, θ 3 x , θ 3 , )( = 1 α α, ) = 1 = 27 αθ x 3 2 2 θ 2 3 1 3 − y − cot sin (1 + cos e x − 0). 2 , that is 1 2 0 , = 1+cos cos(2 2 p 3 2 = cot a − – 8 – = cos x 2 = cos = (1 = r 2 2 ) = r r ) = 2 = r 2 r a 2 y y r (˙ γ r )=(0 + 27 1, the pedal equation becomes , we have 2 x,y ρ x ≥ = 0, 4( 0 = ρ γ P ) 3.3 =2, , we have , we have , we have ρ ρ ρ γ 3 9 6 2 1 1 ) or ( which in polar coordinates becomes − − − 3.6 α = = = P P P , , , 1 8 5 3 9 6 . Since Friedmann’s equation reads in this case 0 = = = is a constant. This is a logarithmic spiral unless a γ γ γ 0 a which is the polar equation of Cayley’s sextic whose Cartesi If Tschirhausen’s cubic is also known as cubic L’Hˆopital’s or which is the polar equation of a cardioid with origin at the cu If If For a cosmological term, which is the polar equation of a parabola of semi-latus rectu which is the polar equation of Tschirhausen’s cubic, whose C For stiff matter, which is the rectangular hyperbola with the origin at its centre ( We now turn to the special case • • • • • = 1. Now from ( of radius it is constant. we see that the scale factor increases or decreases linearly for some constant k where JCAP10(2015)056 ) we (3.41) (3.42) (3.40) (3.36) (3.37) (3.39) (3.44) (3.43) (3.34) (3.38) (3.35) . The 3.6 ) is not 3.34 ty. The model ) with ( 3.37 curtate cycloid - or hypoycloid from the centre of a circle , nsformation ( . paring ( A ) by virtue of the invertible β β ce B B 3.35 ]. se to a bounce and hence oscilla- , B + 2 . B + 2 , we have a . A A A , η Ra 2 + , Rp, ) η η, , R 2 2 η. cos + 2 ) 2 C a 4 cos B A < R sin 2 . − 2 a A A sin sin B + 2 t a B + 2 B = cos R 2 R A − + A 2 A − A R − + 2 − . If 2 a ( − R C β p 2 β + − − A B – 9 – 1) ( A 4 2 = 3 = R Rη R − a A 2 a 2 = = )cos )sin k = = r 2 ], we have carried out a systematic study which shows = = ˙ B B = C t a 2 a 2 +( t ρ a + + r but was untouched in [ d d ρ ) is integrable, in view of the Chebyshev theorem, to yield 4 A A 1+˙ a prolate cycloid is conformal time. Moreover 3.37 η increases to a maximum, and the model ends with a big crunch. R > A )=( )=( a β β ), we see that ( ( y x 3.6 . However, in [ ) when and thus η is a mixture of dust and radiation with positive energy densi d 3.34 A > R a = t . One has d R A > R Geometrically we have a A more general example is given by the pedal equation of an epi Comparing with ( energy density of thetory radiation behaviour. is negative This and periodic this behaviour gives ri is clearly seen in ( which may also be verified by a simple geometric argument. Com starts out from a big bang, whose parametric equation in Cartesian coordinates is 3.4 A two-componentFor fluid our next in example, a consider closed the locus universe of a point at a distan This feature may not be so obvious since the time variable tra with invertible when that the Friedmann equation ( a big-bang type periodic solution in cosmic time transformation ( find that One thus has The pedal equation works out to be which, if of radius Note that JCAP10(2015)056 (3.45) (3.50) (3.48) (3.46) (3.52) (3.47) (3.49) (3.51) . The first 3 4 with positive = and negative if 2 3 2 γ = ) certain values of γ >B 3.41 2 A ive energy density. = 2 with positive energy tenary. = 2 with negative energy if γ γ . . 2 2 ative energy density and positive . 2 2 4 − 2 B a ) B A 2 + 2 , , . a and hence by ( − + 2 1 A 2 2 A 1 = 1 = 1 = 1 A a + 4 + (1 + 2 2 2 2 2 2 A/B + 2 4 2 ) y y y B B B ) 2 k B 2 B 2 2 2 B 2 p − + + 1 p − 2 p AB A A 2 2 2 2 2 2 A 4 – 10 – 4. 3. (1 x x x (2 A A A − − − = − = = 2 = = = 2. = 2 1 ρ r C = 1 this has stiff matter = 1 this has stiff matter 2 = 1. r r k k = A/B A/B A/B A/B ρ . If . For particular ratios 3 2 = ) corresponds to radiation with a positive pressure, γ = 1 , the first two cases correspond to a fluid with 3.45 k . , the Deltoid, , the Astroid, 2 1 1 8 3 , the Cardoid, , the Nephroid, 8 3 9 4 − − = 1, the energy density is = 0 or (0) = 0, a xplicit cases of the 1, may be integrated for atenary. time but finding the curve − es of roulettes giving rise to eat deal less inhibited (see to solve a quartic curve for growing observational support es and/or pressures exceeding . A> -bang solutions with 2 = 0 and olved “exotic equations of state” . om traditional texts on cosmology − 2 k ) . 2 2 1 2 p . a 1 A + 4 − . = 1 , 2 3 − Λ p (1 3 2 Λ 2 2 4 y + B + ) + B Aρ, 4 2 3 2 1 − 3 p 1 p p 1 a AB = A 3 2 Λ 2 4 – 11 – x (2 ) grows following a power law, when Λ A P 2 = = 0) Friedmann equation in the linear equation of t + p ( ρ k = p a = s ρ r p 2 r = = r but now with origin at a focus cosmology ]), and we have therefore felt justified in including as many e = 1, the energy density is is a constant satisfying the non-phantom condition 33 k – A ΛCDM This is an hyperbola The curve pursued by the scale factor is called a hyperbolic c If There is a simple formula for the scale factor as a function of The pedal equation is therefore The reader will have noticed that many of the explicit exampl 27 • solutions of Friedmann’s equationswith descibed above components have having inv forenergy densities example in magnitude. negative These energyon were densiti the often excluded grounds fr offor lack of the physical presence interest.e.g. of However [ with dark the energy the literature has become a gr so that when Λ = 0 the scale factor arbitrary values of the cosmological constant Λ to allow big roulette construction as we could find. 4 Quadratic equation ofIt state is well known that the flat-universe ( state case corresponding to dust which is known to be integrable. 3.5 The presently favoured, so-called concordance model has of which it is the roulette appears to be difficult since one has where JCAP10(2015)056 0, (4.9) (4.6) (4.4) (4.5) (4.7) (4.3) (4.2) (4.8) B < her more ) oscillates t ( 0. Using this a ρ> 0, which prohibits − ]. For convenience, > 36 ) – A 34 , 0, then ) ion of the general equation ] when the equation of state n (1 + are not allowed. If . 9 − 2 1 ) – . a n 7 B > ) , ier in [ . − Ca − 0 the scale factor n 2 , ρ D − = 0 is ruled out. 2 | < Ca ρ √ Bρ B , D , D | Ca D 2 = ln( √ ± + √ n 2 ) n ρ Bρ = 0 B . Hence, absorbing a possible integrating − = ln( ) − A 2 A ˙ = ln( ) Bρ a a A + B 1, throughout our study. Ca ) 2 ) Ca A 1+ + − 0 P ± Aρ (1 + ρ − ± ρ ) 1 + 1 − + − 1 = ρ – 12 – (1 + A ρ ρ +(1+ A> 0 d 1 1 2 ( ρ 0 = initiating from sufficiently small values and evolving ( 0 P ρ n = 2 P − a , B ρ 1 = + ρ ρ | ˙ − ρ . Hence P +(1+ )=
B | 0 ρ 1 2 where ( ρ ρ P ) − − Q 0 ρ ρ ρ . Z
), we see that small values of 0 to denote the discriminant of the quadratic − 1 ρ ln ≡ ( 4.6 D BP B 1 2 J 4 − − ) has two roots: − D ρ = ( 2 0 in ( ) = Q J A ). We note that cosmologies governed by quadratic and some ot J ≥ to be small, we obtain the restriction ρ ( ρ a ) assumes the form P ) into the law of energy conservation, 4.6 6= 0, there are three subcases to consider. = 0 is a constant. = (1+ 0. Then 4.2 B P D = 0. Then to assume large values. In conclusion, the case C > then ( fact and D> We are interested in a scale factor a To allow into arbitrarily large values (‘big bang cosmology’). If resulting in constant, we have D For For convenience, use In this section, we consider the more general situation [ (i) (ii) exponential law (a dark energy regime), and, when Λ where 4.1 Restrictions toInserting parameters ( in non-degenerate situations That is, we obtain follows the quadratic law we again assume non-phantom condition, which may be viewedof state, as a second-order truncation approximat periodically. extended forms of equations of state were investigated earl JCAP10(2015)056 (4.18) (4.16) (4.10) (4.13) (4.17) (4.15) (4.11) (4.14) (4.12) are not ), we get a 4.9 ), we get . 4.9 . into ( 0 . ) to get ) ≥ ∞ n . + Λ) ) − 0 2 0 4.15 ρ P )= | n Ca , ) =0 into ( ∞ B ) to get | ( D πG a a √ 4.4 (8 = ln( , B< n ) into ( . + 4 0 a 2 1) 1) ] > . = A 4.13 A − , − C 2 u 2 2 2Λ B n D n ( ( L ( Cρ C C 1+ a √ n a n [1 + | d + . Inserting . + + u + + + u = 0) Friedmann equation B 0 2 p = + d B D | D D ρ D L 2 2 A k ρ √ 2 ≤ − ) is made precise: 2 C √ √ √ n ] n n L + 0 n 1) n a C + a a a A 4.9 1 P 1+ 1 D + − , L | D L D + ρ πG √ – 13 – u n √ = − B n u 1 | ( n = 16 a = 2 √ L n a ([1 + 1 2 ρ + Λ) | ρ ) stays bounded, we see that small values of L 1 r = B = 1 ˙ | ρ a a s 2 2 0 n Z 4.14 0 is also ruled out. ρ arctan Z ˙ = a a D πG | = 1 1 ≡ √ (8 ), we see that ( B ρ | D< t n I ) gives us =0 ∀ 1) B a D | ( 2 = = ρ )= − − 2 2 4.16 t √ ∞ n ) has no real roots. We can integrate ( ( ( ρ <ρ ρ n ( ) Q t = ( 0 is an updated constant. Inserting the epoch 1 0 only. For this purpose, we may insert ( L <ρ 1 C > ρ D> 0. Then allowed. Thus the case In other words we find a sign restriction for where D< Therefore we must have Since the right-hand side of ( Using (iii) 4.2 Global solutions Hence we now aim at integrating the flat-space ( for which the Chebyshev theorem is not applicable in general where Thus, an integration of ( in the case JCAP10(2015)056 0. > = 0. 1 (4.24) (4.25) (4.26) (4.20) (4.19) (4.21) (4.22) (4.23) 1 >L >L 2 2 L L . , . More precisely, 1 2 1 L L C −∞ 0. Thus + − = > 1 ,
t 1 1 C 2 2 − C t L C C L L v − 2 d v + 1 2 q q 1 )
L L . L 2 2 C − + C C . L L = 0, or v v − −
a q q 2 , 0. Hence = v ln 2 . − + → −∞ → ∞ 2 = 0 which is false. Thus ≥ 2 L 1 C v v L v C C 2 C 2 L
D d + v − , − + + 1)( , t , t 2 2 ,L ln 2 r C 2 D 1 C v D D L L v 2 2 √ − 1 and + L 2 t − C √ √ n v L D 2
L n n r 2 2 2 2 L 2 a √ v C 1 1 L a a 1 ρ v n L 1 r – 14 – − L L 2 = 1 1 t L q L s = √ √ L D ( e v = ), we may also deduce the asymptotic behavior 1 s 1 √ = − + ρ u Z Z n to get = v v v ) we have ) 4.20
D 2 v + C )=O 2 or t L ln √ v ( )=O − 4.16 n t 1 a D − ( D 2 ) give us the general solution of the Friedmann equation when 2 L 1 a v 2 1 √ = √ ) and ( 2 L √ n C n I L 4.20 + CL + 4.19 = = = u 2( u 1 6= 0 otherwise D t u 1 L 2 = √ ), we may deduce the property 6= 0. From ( L n ) and ( I r 1 L = = = 4.20 4.19 t v = 0, then ) and ( is an integration constant. Since is an integration constant and 1 1 1 L C C 4.19 If From this solution we see that the initial condition Furthermore, from ( Now assume = 0. 1 cannot be achieved at any finite time but can only be realized a from ( Consequently, we obtain where In this case we can set In other words, ( L where In this case we can set JCAP10(2015)056 . ∞ tical = (4.30) (4.33) (4.32) (4.31) (4.27) (4.28) (4.29) a ). In par- 4.30 , ) that the scale , 1 )–( , L , , 0 4.26 0 +Λ 4.29 ≥ i > | 0 = 0 to the epoch , a 0 +Λ) = +Λ ) and ( +Λ=0 +Λ BP onential asymptotic growth 1 | > | ρ 4 | | 0 n 0 nt Λ is permitted to assume a , 0 0 4.25 − P . | 2 ) dictates the necessary and suffi- BP πG | ) BP BP B | | | 4 (8 . More precisely, we have A 4 4 4.2 − → ∞ 1) → −∞ − − + 4 2 2 2 −∞ ) 2 − 2 ) ) (1 + ) A is given by the explicit formula n = , t A A A , t ( p t 1 n t L − t 1 (1 + ) 2 > √ (1 + (1 + L (1 + C L A like before. p √ – 15 – p p e q ) may never allow a finite initial time at which the 2 , − e C 0. to evolve from the epoch L − − 0 + Λ) ) 4.2 ) ) (1 + a < 2 A q h A A ρ 0 . n n = | P 0 )=O obeys the exponential growth law ( t v B )=O (1 + ( | ≤ πG πG t (1 + (1 + a , B< a 4 ( 0 0 (8 a n | n n ≤ | | 1) B 1) 0 | πG B B ) and may hardly be surprising. Similarly, we have , P − | | πG πG 4 2 P 0 4 4 − . Consequently we see in view of ( 2 n grows following a power law. On the other hand, in the non-cri 1 ( n L 4.22 1 ( n a √ n B < = = 2 ), we have C 1 L v > L = 0 corresponds to a 4.17 cannot vanish at any finite time but at a ticular, in this lattersuitable situation, negative value the when cosmological consta cient condition scale factor vanishes. in order to allow the scale factor situation however, the scale factor the scale factor From ( In summary, we can conclude: (i) A nontrivial quadratic equation of state given in ( (ii) The quadratic equation of state ( (iii) In the critical situation This is an importantlaw. conclusion It is since worth we noting have that achieved the an growth exp rate we see that factor which leads to which is the same as ( JCAP10(2015)056 0, 99 ∞ and = (4.34) (4.37) (4.36) (4.41) (4.35) (4.40) (4.38) (4.39) 0 ρ D > ition just → ∞ t of order 10 ), i.e., P ] ρ 4.5 39 , and linear case. 38 ] for which the equation 24 , 37 − 0 the cosmological density. It ≥ , . , . ) Λ 2 n into a slightly more general P ) γ iven by [ | ρ ρ ρ A criminant of ( − 0 0 and the limiting value . , γ P | )(1 (1+ γ A n 1 + 1) − + 6= 1 1 κ< − ) α a (1+ ( A is of order 10 n A = 0 gives rise to the initial value A − − | Λ a (1+ 1+ , , γ . κ ρ n | 1 1 4 0 C Ca αρ − 1+ − a < ≤ = = + A | 0 -fluid model studied in [ κ Λ Λ P ρ – 16 – = P may assume arbitrarily large values as 1+ ) ρ α ρ ρ + C a A ∞ ) gives us the relation γ ρ , A> − + 1) γ 1 4.4 A ρ α 1 ) ( κρ simply and unsurprisingly adds a shift to the cosmological is the Planck density, and A +(1+ − 1+ 0 + 0 P P =0, ( = P = ρ | Aρ ) is well observed too. (1 + B ρ P gives rise to the condition = , in particular. Since ∞ 4.35 α P ) indicates that the epoch = a ) into the equation of energy conservation, we have 4.40 ) is consistent with a part of the necessary and sufficient cond 4.39 4.34 0 is a parameter, 0 is an integration constant. 1. Then ( ≥ 6= 0 otherwise the equation of state becomes the well-studied ]), the condition ( α κ C > and the epoch γ < Inserting ( We may compare our results with the 37 0, we again arrive at the necessary condition (i) ≥ Hence where (cf. [ translates itself into the form where ρ derived. Furthermore, the positivity condition for the dis 4.3 Degenerate situation In the degenerate situation, of state is Since we are interested in the case when which does not involve where In other words, the presence of is seen that ( constant. 4.4 Polytropic equationIt of will state case alsosituation be where interesting the to equation extend of state our is study of in a polytropic this one sectio g JCAP10(2015)056 0 γ (5.2) (5.3) (5.4) (5.1) y, we and (4.42) (4.43) B < 0 again A a. κ< ] d ) 10 γ − 1 2(1 − 1) | κ − | 2Λ n + ( ) n . is elementary when γ 0 = 0. − I = solutions. ∞ )(1 edmann equation [ ρ A λ , , λ γ + (1+ 1 2 − , C> n 2 1 0 ρ − 2 +1 α A Ca = 0 gives rise to the condition | n A> κ + a n | 1 Aρ ρ πG 1+ − 1 1+ a = α 0 is analogous to the necessary condition 16 – 17 – ] where P = Z , = ) 13 0 2 = κ< A = ), we have – ρ k a 2 I 11 (1+ 4.3 + n = 2 − t , α H ) γ Ca 1) n − 1 = − 2(1 gives rise to the limiting value πG ρ n − ( ) 16 ∞ n A ) implies that the epoch = = a 1 ) in the flat-space Friedmann equation with Λ = 0 for simplicit 4.40 (1 + α 1 2 4.40 ) for the quadratic equation of state problem. 1) n − 4.31 1. Then ( n πG ( n and the epoch γ > and the initial value Note that the necessary condition Applying ( 4 (ii) in the Randall-Sundrum II universe [ stated in ( are specific physical parameters. 5.1 Linear equation ofIn state view case of the linear equation of state In view of the Chebyshev theorem, we know that the integral arrive at the integral and the law of energy conservation ( are arbitrary rational numbers. Here we omit presenting the 5 The Randall-Sundrum II cosmology In this section we consider the braneworld cosmological Fri JCAP10(2015)056 ann (5.7) (5.9) (5.5) (5.6) (5.8) (5.11) (5.10) , , , 0 0 = 0 , . 0 ,λ> 1 (0) = 0. Indeed, is an integration β ]. a 2 . 1 , λ β 40 0 0 C |− , , t> + λ = 0, we easily obtain 1 | ) 2 k , . , ) A and β , λ< 2 0 0 2 2 u , , t> u (1+ , λ> ! ) | = 0 α is away from the phantom 1 + 4 ) n 2 − 2 λ ) 2 u | 2 a 1 u ) β 1 C 1 A λt β √ ,λ< tial condition A λ + u β p + ) situation [ √ ( ) = q e A u 1 2 (1+ + 2 A 2 n β ) )( | (1+ β , λ> | | a 1 A λ n u 1 λ λ | λ | β a 1 β | 2 1 = (1 + √ 2 2 − + β (1+ λ 0, the solution will be of a finite lifespan − 2 β ( and n )=O u ) 1 n ) | t ( 2 A = 1 λ u A , the growth laws 1 ( | q a ( = 2 | a + λ< (1+ ) λ + arctan + 2 (1+ q n Cα | A n a β 2 1 λ 2 ! → ∞ 2 ) + β 2 = , u , α − ) (1+ , λ + β arctan ) t A – 18 – ) n 2 2 1 a | ) A 2 2 ) = 0; 2 u β | λ β + a A | 2 u ln β λ − a u | u , λ + (1+ + √ λ d 2 1 + (1 + ) a 2 n + (1+ q ) 1 u 2 − β β √ β d u A a n ) u A , λ ) ) 2 1 d n
1 a )( A β A A 1 + )( − 1 + u u ) (1+ √ ) + 4 (1+ ( u = β 2 1 2 n λ A A n (1+ 1 2 2 (1+ (1+ 1 − β 2 2 2 n a − + n n − β λu 1 1 (1+ − − ) (1+ 1 u t arctan n ) and restricting to the flat-space case β u a n q A p u ( d t u | d 1 ln ( a λ p 1. To see this fact more transparently, we rewrite the Friedm | β + 5.1 λ p = (0) = 0, we solve 1 (1+ Z p 1 − β √ √ n u 1 p )
) ) ) α 2 β 1 a C A A A 1 A = 2 = λa )=O | Z 1 A t (1+ (1+ (1+ t λ u ( ) into ( p 1 | n n n d d ], a = (1 + 2 . Using n 5.4 Z I ) 45 = arctan = – A = = = = 1 41 αt u 1 (1+ n C ), we have a + = 5.5 t However, it may be a surprise to see, when Inserting ( From the above results we deduce, as u or periodic, instead, depending on how far the parameter divide line [ from ( These two situations are analogous to those in the classical and constant which may always be chosen to allow the big-bang ini equation as the integration to get since it is of the Chebyshev form, where where JCAP10(2015)056 0 ]. t 0 t 2 , of the (5.12) (5.17) (5.16) (5.13) (5.18) (5.14) (5.15) 0 t 0 when a [ . 1 n ) through ∈ ≥ t ) t ( 5.10 1+ a − with , = 0 t ot exist beyond 2 − ] obtained above and ) are autonomous, the 0 0 t t , A . Thus 2 0 1 , 5.13 , t>t u | gative, the solution < , we find λ . ) = | 2 ) by 2 t u u ( u p ˙ 1 u . ) ) and ( u , we should have ˙ ) 0 5.10 when ) 0 A 2 2 √ t (0) = 0 has either a finite lifespan A u ) would suffer from a discontinuity 0 < u a α t 1 t reaches a ‘stagnation’ point, where 5.10 1 ( 2 (1+ − − u a n u (1 + u 1 1 √ n − u , . u 1 ( )= 1 1 n n = t √ 2 1 . Thus we may obtain the solution of the − ( α ˙ 0 (say) when − a )) ) − t 1+ 1+ − 0 > ( ) and the solution t , α 0 − u 0 . Such a procedure can be repeated so that a t ) a ( (2 t – 19 – 0 2 )= (2 ) t ≥− t u ( = → A → ˙ t A< A may simply be obtained from that of ( u t + arctan + t 1 = 2 ] for the solution of ( − 1 ) indicates that ˙ u ) 2 π 0 ; lim u 0 t t (1 + )( ,t 1 n n u (2 ] by shifting the solution over [0 lim 5.17 [0 0 as 1 0 α )= satisfies → t − t 0 ) at is constructed. ∈ 4 1+ t ) is invalid. Instead, 1 t = )= → A , ( t ( 0 t u 0 − 0 t u ( ) a ( t t t ˙ a , and starts to descend. Hence it must evolve through another ( 5.10 1 p a u α with t − = 0. Thus ( , A> = ) = )=0. a 0 ), we have 0 t t u t d ( d −∞ ) satisfying u (2 5.13 u )= where 5.13 t the equation ( ( 0 ) implies that ˙ ˙ t . In other words, the solution to the Friedmann equation cann a 0 0 t − t ) 0 5.16 t If the coupling constant Thus, we conclude that, when the cosmological constant is ne = 2 (2 ) = 0 and lim 0 t → t t passes 2 ( ˙ which may well be called the lifespan of the solution. at u The above relation remains valid until Beyond On the other hand, if the solution would exist beyond 2 so that Friedmann equation over [2 solution of ( replacing the variable branch of the Friedmann equation, which may be integrated similarly. Actually, since both ( In particular, then ( In fact, from ( 5.2 Solutions ofWe finite now lifespans show that the solution ceases to exist beyond 2 From the relation between the scale factor Friedmann equation evolving from the initial condition maintain the continuity of ˙ t periodic solution of period 2 JCAP10(2015)056 ] 47 , grows (5.21) (5.20) (5.23) (5.19) (5.24) (5.22) 46 ∞ 0 so that 0 that the fulfills the 0 as in the . < )= A . 0 A> 0 ∞ ( < a A < ] indeed as in the Λ 45 – , , t> 2 41 . ) t | here explicitly in terms of A subject to the equation of . 1 Λ 2 α | T ) α (1+ 1) n , when the solution has only a A 2 n 1 n . − − classical case with Λ a ny solution with n 2 (0) + ) (1 + 1+ α 2( A n 4 2 → ∞ γ 2 1 a r C (1+ ≥− + n + a + +1 n a , t d 2 2 ) n A , as before, subject to the equation of state α G A G (0) = 0 is permitted when 1+ ) βρ 1 2 a A C α (1+ 2 + 0 are constants. Although this equation cannot n → ∞ (0) = 0 is not permitted and the Big Rip [ (1+ 1. – 20 – − n (0) + t a ) αρ r − a t 1 A 1 as 0 is a constant. Inserting this into the Friedmann = α | 2 6= 0, the energy conservation law gives us the relation (1+ Cα ) A> A n A H 2 α,β,γ > a A p + 2arctan 2 )=O a C > t (1+ π 1+ , according to whether the coupling constant ( 1 n | 0 a t C Z α t n ) is implicitly given by | 1 t = = 0, we have the integration Λ ( = | a − r = 2 I 0 n t ,λ n 2 = T ). = )=O t t = a s ( = 0 2 1 still corresponds to the phantom divide line [ a Λ = 0), where A 5.18 k , α − 1 + as before where = 1+ = 0 ) 1). ) or ( ) A k A − = A ) with (1+ T 5.14 ], the following slightly more general Friedmann equation n (1+ 0, however, the initial condition 5.1 ), the phantom world is still spelled out by the condition 1 + a n A> or is of period 48 ( 1 − C ) grows following the law 0 α 5.3 t t A< ( Ca To end this subsection, it will be instructive to check that, For later convenience, we document the lifespan or period Thus we have unveiled a peculiar situation, In [ Aρ a = 2 = = = a state ( big-bang solution is periodic for any condition ( T Consequently we see that the initial condition classical situation. Inρ fact, for 1 + Hence the scale factor In particular, P classical situation, as anticipated. the original physical parameters: happens at finite lifespan. This situation is in sharp contrast with the be integrated, some asymptotic analysis establishes that a according to the law is considered ( If 1+ equation ( JCAP10(2015)056 (5.34) (5.33) (5.32) (5.29) (5.25) (5.30) (5.26) (5.27) (5.28) (5.31) , . b 1 A α √ B > 1+ 2 = , α 1 , b ! C u, u s: a a d d 1 2 + ρ, 2 1 2 d ) 2 − , b , 1 ρ > b. u 0 b 2 ) into the law of energy conser- !) 1 α , √ 2 i 2 α u α √ 2 1 ) into the braneworld Friedmann α b ρ 2 1 + + 5.25 < − + 2 1 , ρ A ) 5.26 α b 2 2 1 α ) (1 A . u α B A √ 2 , B> , ρ r 1 − 1
1 2 1+ 2 (1+ ) (1+ − , α − u α n b n 2 bα 2 b 2 2 2 + 1 ρ u + α − − α ) α 1 − √ 1 − ) A 1 2 α 4 b − ρ α Ca = ]. Inserting ( u
( 1 arctanh h (1+ − – 21 – 2 r n 2 , A> 51 1 2 2 2 Z Z – = α ρ − ρ B − = ) ) ( ρ 49 ) + , α A A u b − A 1 Ca b 1 2 1 arctanh ) gives us α 1 √ 1 i α √ (1+ 1 2 b 2 Aρ n (1 + 2 + (1 + − = 0 and integrating, we arrive at = > + 5.26 ) into a rational one, n = n 2 C ρ λ ) 1 2 α b 1 − A α P α = α √ 5.29 ), we have − = a ( (1+ − n I b = 2 2 5.28 − I √ α ) = 0 and A Ca k 1 and h − 1 2 ) and ( α α 1 (1 + √ n ) with 5.27 Z 0 is an integration constant. Substituting ( ), we obtain = 5.1 u< ≡ I 4.3 ≤ I C > We begin by displaying the result in the first regime as follow To proceed, we note that ( = t Combining ( respectively. 5.3 Chaplygin fluidConsider case the equation of state which can be integrated separately in the parameter regimes which is not of the Chebyshev type. Nevertheless, set of a generalized Chaplyin fluid model [ Then 0 where vation, ( equation ( which renders the integral ( JCAP10(2015)056 ) ). 5.37 (5.38) (5.39) (5.40) (5.35) (5.36) (5.41) (5.37) and the 5.37 +1 . ) and ( n n , ] G , 1 ) dominates as 2 1 2 5.36 C G 5.35 1 n 2 1+ 1 ρ, → ∞ 5.37 b 1 − 1) n +1 α 2 1 π 1 √ )–( . On the other hand, the growth condition n 2 − , t arctan πG n α + 1 . n π 1 2 1 πG arctanh √ 1) 2 5.36 ( 16 σt or 16 n − α 1 2 α n 16 e 16 − – 22 – − = ) by setting 2 < n = πG b 1 ( 2 α 2 2 n (0) = α n √ )= α u − G bα A 5.37 H )=O < ∞ b + 1 √ B t . Consequently, we arrive at ( 2 ( α b b √ 1 u 1+ a or α = 4 α √ √ r σ b 1 b α b = √ 1 )= √ (0) = 0 or 2 α 1 2 √ where − a ∞ ( = ρ b 1 <α α I √ 1 ) α make their contributions to the dark energy, + A 2 B α ), we have the same asymptotic state ( (1 + b n and 1 5.39 √ gives rise to α A is an integration constant. It is clear that we may choose = ∞ 1 σ , which leads to the following explicit asymptotic estimate C It is interesting to note the different ways the Newton consta Likewise, in the third regime, we have With ( It will also be enlightening to compare ( We omit here the critical case )= ∞ → ∞ ( where under the condition constants where obtained for the big-bang solution of the classical Friedma as anticipated. In other words, the first term on the right-ha big-bang initial condition a t still hold. which may simply be obtained from ( JCAP10(2015)056 ] = 2 l = 0 ( (6.6) (6.2) (6.3) (6.1) (6.5) (6.7) (6.4) (6.8) ] k l 1 51 – = state 49 l A . . , we derived in [ +1) α 1 quation with ) and α → ∞ 2( , ρ k, here we consider the is an analytic function 1 A as a universal growth law in f ≤ B ,...,M . α 1+ equation of state: . e equation of state [ ≤ = 1 , . A 0 2 1 0 → ∞ → ∞ ion of the Friedmann equation. α l B B ρ ) to dark energy problem, has been ρ , 1+ 0 ( ( 1) 0 − , t n f α + B − ρ , ρ> + 1 +1 Aρ n πG α ( − ρ m A > +1) m + l α n α B ρ ρ ρ )( ) that there holds the scale factor and energy l = , B> A A 1 0; A l = =1 B 0 are constants and N 6.1 − A X ρ m – 23 – (1+ = 4 1+ =1 M l n X ≥ + σ − − ) = P N = ρ , ρ> ; Ca ( P 1 f ∞ , A> is rational. Nevertheless, for any real = − )= ρ α ρ = B ρ α ( > → ∞ +1 ,...,α → h P ) 1 α − ) ρ ρ α t ( ), we have ( ′ , t ρ Aρ f 6.2 σt ] when e = 2 is the only positive root of the function 0 0 and P C ∞ ρ ∼ > ) into ( ], the extended Chaplygin fluid model given by the equation of ) t (0) = 0; N ( 16 6.3 f a ,...,B ] and [ ), respectively, by numerical means. Motivated by their wor 1 15 B , In view of the Chebyshev theorem, we know that the Friedmann e Inserting ( Suggested by the above results, we establish in this section Unsuprisingly, 14 ,...,M much extended situations, without resorting to an integrat 6.1 Extended ChaplyginIn fluids [ extended Chaplygin model governed by the following general and Λ = 0 is integrable [ is studied in the specific situations when 6 Universal growth lawRecall for that the the extended generalized Chaplygin Chaplygin fluid universe is governed by th whose cosmological implication,studied especially extensively. its It relevance is direct to see from ( 1 where density relation the following explicit universal growth law: satisfying JCAP10(2015)056 (6.9) (6.11) (6.16) (6.10) (6.14) (6.15) (6.17) (6.12) (6.13) ) with t ( . a ∞ = 0. , ,a ] and discussed ρ>ρ 0 ) 2 t ρ ) ( ≥ ρ> ρ ∞ ≤ 0 , ρ 1 ρ ( = 1 1) n t h <ρ ρ ), we arrive at − πG 0. Inserting this into the − ∞ n ( ) 16 , 6.11 n r , ( , 1 . ρ > 0 and 0 n 1 0 n = . r, ρ − h ) d > a > ∞ ) ) ) m ), we have ρ is that the function m 0 r ) into ( ( 0 α t ( . a ∞ 1 t 1) , α B ( ) ρ f q ( ) ρ , r> a ρ ρ nh a 1 0 ) − 6.14 ( 0 for all ∞ ρ 6.12 r − =1 − 1 )+ln 2Λ ρ N ) strictly increases in ( n Z ( a ]. X ρ h r m > ( 1 q ), we get ρ ) ( ) ( ρ n − ) 16 ( nh − Q h + + ∞ ρ = ln – Q ), we rewrite the integrand of it as − ) ρ ( )) + ) ) ) ) and ( 4.15 1 0 = a ρ ) 14 ) t ( ∞ − ∞ ∞ h | ∞ ( ρ ρ ∞ f ρ ρ ( ( ρ 6.11 ρ a – 24 – ρ ∞ 1 6.13 , motivated from the results in [ Q ρ h ) + − − 1) n f − 0, since ln( ∞ r ρ r r 0. Inserting ( − 1 1 r ρ d − )=( +e ( n )( )( > πG ρ 1 ≡ )( ρ | n h ( r ∞ ∞ ( ∞ + ) 16 e ∞ r> ( h ρ ρ ln | ρ ρ n 1 ρ ( ( ( α ) 1 1 h ∞ h = = h h ρ ∞ ρ + 0 ), we have ρ 0 ρ ρ ) in the form 1 is an initial time when 0 − ( Z λ ≡ = λ 1 4.3 0 0 t 6.9 h ) ρ | = ∞ 2 ρ ln 0 small, we see that in ( ) − ˙ h a a 0 and ∞ 1 r ρ > ρ 1 ( )( > 1 r ) ( h 0 1 ) into the Friedmann equation ( t h 0 for ( ρ )= < ρ 6.15 = ( ) is a smooth function in ) r ρ Q 0 ( ( ρ q h . Note that we are dealing with an expanding universe such tha In order to extract information from ( A direct consequence of the above assumption on It is clear that our assumption on 0 t ≥ where where has exactly one positive root, say t where where we set 6.2 Universal growthRewrite law the function above, covers all the examples considered in [ Since which is a bounded function. Combining ( Inserting ( law of energy conservation ( JCAP10(2015)056 ), ). 6.4 6.19 (6.19) (6.21) (6.24) (6.23) (6.22) (6.18) (6.20) , ) and ( 1] , a . d [0 6.18 1) is given by ( ! ∈ . − a ∞ ) 1 2 2Λ 0 ρ a n 1 ( ( λ i n , √ ) a 1 d + 1) − A ]. a, ξ 1 2 uation of state ! ) − 2 d 2Λ a ! ∞ n 3 ρ 0 ( ( ch allow explicit calculations of (1 + B 1 λ 1 n ) . , 3 − ∞ as follows: √ nh A ρ B + ρ ), we see that ( − a B 1 ) − a 3 − = 1. 1 +1 ) + 4 → ∞ ∞ ) 6.1 nh ρ α 3 A ρ ( 2 α ( ∞ − ρ ) 1 Q ρ 3 a 1 ) ( ) ) 1 + 4 A , t A nh ρ A ∞ 1 ∞ ( 2 ρ − ρ nh ( B ) ) for Q 1 ( + 0. Consequently, in view of ( t 1 ) 1 − 1 B 0 − h 1+ ρ a λ 2 ∞ > A a e 1 (1 + ) nh ρ ) √ ρ ( 6.21 ) – 25 – ρ − A α ( e 1 ( p a h Q ∞ 1) n ) Q ) + = ρ ) ρ (1 + )e ( ( ∞ 0 1 − ∞ )+ 1 a ρ P Q πG λ ρ p 1 ( ( ) h ( n 1 h 1 ( a A h ∞ 16 p )=O h 3 αξ d ρ n e t a e ( , ( 1 A 1 ) α + α a h 2 ) recovers ( = a since 0 e (1 + ( + 0 λ Z α R λ 0 ∞ 6.24 = λ p + ), we obtain the integration + + 1) n ). 0 p a ∞ a 0, ( a λ ρ − 0 near a Z πG 6.17 ln λ d n 6.17 p a → 1 2 ( 0 16 √ a 3 λ n 1 Z − . A √ Z Z 0 = = λ ) and ( 0 = = = ) = λ a t ( 6.16 R is given in ( 0 λ which generalizes the corresponding formula obtained in [ Hence In the limit Consequently we have It is direct to see that , and hence, (i) For the classical generalized Chaplygin model ( (ii) Consider the extended Chaplygin fluid governed by the eq ∞ which is bounded for where where In view of ( we deduce the universal growth law for the scale factor 6.3 Examples It will be enlighteningρ to display a few special examples whi JCAP10(2015)056 (6.26) (6.25) (6.32) (6.31) (6.29) (6.30) (6.28) (6.27) 0. → 0 B , = 0. In other words, we verse containing pressure α 1 hod is given for finding the 1 point on the circumference . 1+ Adding radiation, one finds , α . 0 is allowed to be arbitrary, l. For example, we consider α ocus of a point in the plane of ) ! 0 1 , e ) . 1+ A 0 ≥ 1 0 ≥ 1 ! ≥ A 1 , n of state . 2 α ≥ 1 (1 + B ρ 1 ) B = 0 ) >α (1 + A B , α A 2 2 2 − , α α α − B 0 B α ρ ρ + 4 ) 2 B B . Hence 2 0 = 1 again in the limit A 2 , α − − 1 B 1+2 2 2(1 + 2 − B 2 α A ρ 1 α α 1 α A +4(1+ α B ρ 2 p B ρ 2+ − 2 1 Aρ + 4 ρ – 26 – 2(1 + + 1 − B 2 α − We have shown that every solution of Friedmann 2 = 0 ) ρ B 1 A 1 ρ p 1 ) with α ) B =1+2 P B ρ A − + + A 2 ) explicitly: = ρ 1 0 − α 6.21 1 λ Aρ B ∞ A (1 + ρ (1 + Aρ = = p ) or 1 = = P P α (hence P ∞ = − ρ ∞ 2 = 1. ∞ ρ α ρ α = 2( 2 α ), we obtain 6.29 which allows us to return to ( Then we have Friedmann equation and roulettes. equation admits a representation as aa roulette, curve that which is rolls the without l curve. slipping on A a well-known straight example line.free is A matter. the met The scale scale factorof factor of a is closed circle, a uni which cycloid, rolls i.e. without the slipping locus on of a a straight line. Similarly, with There are other explicitly solvable cases of interest as wel (i) (iii) As another simple example, let the equation of state be we require 1 + the equation of state which contains the case (iii) above as a special example wher To solve the associated equation find an explicit solvable general situation with the equatio 7 Conclusions Our results in this work are summarized as follows. which covers (ii) with we have With ( JCAP10(2015)056 (A.1) o whether . In the general (0) = 0 has only a a small amount of a −∞ lytic function and a xplicitly given regime big-bang solution and = s to extract all the key necessary and sufficient nce of dark energy near stic is that the presence t given. ear, it is shown that the th rate formula. rcumference. Many other nential growth rate of the ssical situation, but when lly supported by National ed in concrete situations. isfying ble in general. To tackle coupling parameters, when to vanish at a finite initial her contexts in physics not blems and geometric optics. tate to allow the scale factor nce and Frontier Technology 0. ergy and the pre-inflationary situation; when the equation ion, although near the phan- a ns of state which have been of ) by r, φ , ) φ ( u = r 1 – 27 – = 0 may only happen at = a u In the Randall-Sundrum II universe, there are two in- Although the flat-space Friedmann equation is not in a For the extended Chaplygin fluid universe for which the grows exponentially even when the cosmological constant Λ a grows following a power law or an exponential law according t a to evolve from zero to infinity. A distinguishing characteri condition is unveiled for thea coefficients of the equation ofof s the quadratic term in the equation of state prohibits binomial form so that the Chebyshev theorem is applicable, a Quadratic equation of state. explicit examples are given, somecurrent involving interest exotic to equatio cosmologists inuniverse. connection with dark en The Friedmann equation, ordirectly its connected equivalent, with also cosmology arises suchExamples in as of ot central the orbit pro use of roulettes are in this wider setting are that curve is still a circle but the point moves outside its ci time. In other words, the epoch assumes a negative value above a specific critical level. non-critical situations, Randall-Sundrum II universe. teresting flat-space cases:scale When factor the equation of state is lin negative inverse powerthe function, problem, an an integration analyticparameters method is that give is impossi rise introducescale to which factor. a enables universal Such u formula a for formula the covers expo all known formulas deriv Universal growth formula. equation of state is decomposed into the sum of a positive ana finite lifespan, which nevertom happens divide in line the the classical solutionof situat is state periodic, is asdescribe in that its the of classical exponential a growthΛ pattern Chaplygin = in fluid, 0. terms of weexotic We various explicitly see nonlinear matter as derive would in the the lead the phantom to classical divide a line, Chaplygin large as fluid amount realized model of by case prese the that exponential grow the cosmological constant ΛΛ is is zero negative, or however, a positive,away from new as the phenomenon in phantom occurs divide the so line, cla that the in big-bang solution an sat e (ii) (iv) (iii) Acknowledgments The research of Chen wasProgram supported Funds in under part by Grant HenanNatural No. Basic Science Scie 142300410110. Foundation of Yang China was under partia Grant No.A 1147110 Pedal equations andIf a their curve inversion in the plane is given in polar coordinates ( JCAP10(2015)056 h ), r, φ (A.6) (A.7) (A.2) (A.3) (A.4) (A.8) (A.9) (A.5) (A.11) (A.10) ) (includ- a ( ρ . 2 B 2 at the point ( , and A t 2 d . λ a 1 B − ) per unit mass and has = s to solving the differential r 1 2 2 η ( A u d . ) , V 2 2 + ) e form . a = λ k 2 ) 2 φ k B − − a . , ) . , − 1 a = 1 u u ) ) (1 ( 1 2 3 . = + (1 u u P 2 ( ( 2 u )+ u V 1 2 = a )+(1 , P P ( + ) 2 a ( u 2 u = = E − , B ( ABu ρ 1 2 2 1 2 2 2 + P πGρ λ ) with λ u u u φ 2 2 2 d d 2 = h λ + + – 28 – = 4 πGa + 2 A.4 2 2 2 u φ 1 = p and conserved angular momentum per unit mass 2 d d = 1 )= a 3 1 2 B E 2 u φ φ u u 1 ) = 4 ( p d d + d d − = u P ( 2 2 2 2 ˙ a a 1 A P u p ) with , λ 1 1 A A.4 2 P 2 A . Then λ 2 ) of Friedmann-Lemaitre universe with energy density B = t A ( 3 a )+ A )= ) arises in many other contexts. Here are two examples. λφ 2 ) satisfies ( u φ ( A.4 ( 2 Af f P = = 2 1 is the perpendicular distance from the origin to the tangent u u p which may be brought to the form ( ing any contribution due to a cosmological term) satisfies then its orbit satisfies ( If a particle moves in a spherically symmetric potential conserved energy per unit mass The scale factor Equation ( Let We may compose two such tranformations to get a third: • • one has and if A.1 Affine andSuppose rescaling of angle Conversely, if we are given the pedal equation of a curve in th the problem ofequation finding the polar equation of the curve amounts with JCAP10(2015)056 . 2 (A.12) (A.21) (A.16) (A.18) (A.17) (A.19) (A.13) (A.20) (A.14) (A.15) λ 2 A y times the semi-direct and nonzero real num- . x λ 2 d 1 u q he second transformation . + r 2 6= 0) + rameter group. The first two u q p ) 2 r ), we have 2 ( u n , βx ( , B r − P + + distance from the origin to the tangent. + q, . 1 u φ . q p B q p p,q,r 2 u + A u + βu + (1 αx 1 u , corresponding to − u A + r n − 0 + 1 B ) 1 n ) u 1 q p × 1 u d p Z u 0 λ ( P R u Q B 4 = 2 αu P 0 0 1 0 . Acting on 1 u A + A x =( P + – 29 – p B 2 d 1 2 2 u λ A n − )= q u u Z n , q, ) → u r 2 ( and → = 2 P u ) . r 2 )= + 1 βx P A 1 1 φ u u φ n u ( to a power: If p 2 ( and so integrability in finite terms is possible if + ,λ Q A u P 1 2 1 α 2 )= ( λ − u p ,B ( 1 x 2 = → A P ) q Z u ( , then = 2 P I u ) transforms under pull-back composed with mutiplication b u − corresponding to ( ) = 1 is particularly simple: Q + u ( n P ⋉ R is an integer or half integer. follows the first ) we deduce that × )= , the integral 2 R u r +1 ( A.4 p ,λ Q 2 A matrix representation may be given The case in this section should not be confused for the perpendicular α,β p ,B 1 2 which reveals the three-dimensional group as where Thus the set offormulas central are potentials commutative is butA acted the upon second by a is three-pa not. In the above t Chebyshev’s theorem states that for rational numbers A.3 Chebyshev’s theorem From ( is elementary if and only if at least one of the quantities bers Or if is an integer. In our case which says that product A.2 Non-linear transformations The simplest of these is to raise then JCAP10(2015)056 (A.32) (A.27) (A.30) (A.31) (A.22) (A.24) (A.28) (A.25) (A.23) (A.26) (A.29) . 2 ) 2 u ) . 2 ) 4 2 A u . A 2 − +2 R 2 R 4 + 4 m R 2 , , 2 u . . p 2 2 2 1 1 )( 1 1 u u 2 m , , (1+( 1 Au − A + − 2 + , = 2 2 = 1 = 1 2 u A 1 − 2 2 + 4 2 Ax, Au u B B 2 2 1 2 2 p 2 2 1 B B y y B B = r = + − 2 2 ( = 4 2 2 2 2 A A – 30 – 2 + − 2 2 1 1 2 p p p 1 1 A A y 2 2 B B 2 2 = x x A A = = , = = 2 Rp, 2 2 ) m 1 1 2 2 2 p p from a circle of radius 1 1 − p p p + 2 φ A 2 − = 2 R r u ( − 2 2 A A = 2 r : Cotes’s Lituus. 1 2 : Fermat’s Spiral. 2 1 − = 1: Archimedes Spiral. = 1: Hyperbolic or Reciprocal Spiral. = 2: Galilei’s Spiral. = = m m m m m • • • • • and with respect to the centre A.4 Examples A.4.1 Circles With respect to a point distance with rsepect to its vertex A.4.3 Algebraic spirals A.4.2 Conics An ellipse and with respect to the centre A parabola A hyperbola with respect to the focus with respect to the focus with respect to the focus JCAP10(2015)056 ] 53 , (A.33) 52 . . 2 2 . . u u 2 2 3 4 ) . u . u 2 2 2 8 4 5 u + u m 3 − 2 8 9 4 1 ). These are particular null − . These are particular null k 4 6 4 3 − 2 3 u + Arnold-Bohlin duality [ = u u 3 − u . 3 1 1 9 2 u . 2 1 p 2 u . geodesics of spherically symmetric + = 9 u = , = 4 3 8 3 3 = 4 2 + (1 φ 2 3 2 . These are particular null geodesics 2 2 1 u 1 1 p 3 p − ) p 2 − 1 p u 1 , = 2(1+ , m , u = = 3 2 . φ , 3 φ φ − ) 2 2 1 2 1 3 φ p k 1 p + 3 2 = 4 = cos p = 4 2 2(1 1 3 , 2 u − 1 u 2 p 3 1 φ p 2 2(1+ = = cos 3 , is the angle beween the curve and the radius . 1 u . = cos , m = cos φ 2 1 u φ 2 p cos α u kφ = – 31 – kφ k u 1 = cos 2 we have 2 1 = cos we have u p = u 1 2 2 = 1 φ = cos 2 p − = cos √ = cos +2 φ u and φ u u , where u we have 1 k , α 2 − cosh − +1 m cosh 2 ]. ) ) φ φ sin 54 r kφ = = coth kφ cosh cosh = u u = p or or u = (cos = (cos 2 . Cayley’s Sextic 2 u or − +1 u 1 3 φ ⇐⇒ . The Trisectrix of De Longe . Maclaurin’s Trisectrix φ φ √ 1 1 1 2 3 α +2 − = = 1. The Oeuf Double φ φ = 1. The Witch of Agnesi = 1. The Cubic Duplicatrix = 1. The Cruciform = = 1. The Kampyle = cot cosh 2 cosh 2 ,k ,k φ 3 2 ,k ,k ,k ,k ,k ,k cosh cosh − we have 3 2 = − − = tanh 1 k = 1 we obtain the Rhodenea 2 = = = 3 = 2 = 1 = = 2 = 1 = e u u − − u is the pedal of the Cardioid. vector. m m m m m m m u m If Some particlular sinusoidal spirals are: If If = 1 we obtain the Epi-spirals = = 3 1 • • • • • • • • • m m m = 3 gives the Trifolium. = 2 the Quadrifolium and = A.4.5 Sinusoidal spirals If A.4.4 Logarithmic or equiangular spiral A.4.6 Black holes As central orbits, the first two examples below are related by of a four-dimensional black hole. geodesics of a seven-dimensional black hole. while the third is self-dual.Ricci They flat arise black in hole the theory metrics of [ null k k k If geodesics of a five-dimensional black hole. If JCAP10(2015)056 (A.36) (A.39) (A.34) (A.37) (A.35) (A.41) (A.43) (A.38) (A.40) (A.42) . 2 u ) 2 k . 2 2 u m 2 , 2 . 2 − u + 1) ) u 2 ) 2 2 k k 2 k . 2 + (1 2 4 m m u m +1 m + 4 m + (2 2 2 . u u 4 +(1+ ) . +(1+ . u ) 1 , 1 . + m , . 2 2 2 1 − 1 m 1 − + u 1 u k − m m − 2 2 2 m 2 m 2 − 2 φ 2(1 m + − 16 ∓ u u u u 2 u 2 2 =4+9 u 3 3 8 2 2 2 u u u − u 2 u 2 2 k 1 9 + p 4 2 k ) 2 2 − = + 1) m = 1 k m = m 2 2 2 − (7 =1+3 2 1 – 32 – 2 p 1 k 1+2sin = 2, 2 p 1 = m 2 p 4 1 = u p 2(1+ 2 = u ,k 1 =( 2 p 1 = = p 2 2 1 1 r 2 p 2 p = 1 1 = 1, k , p , , 2 m m m ,k m m ) ) , ) 1 φ = kφ − kφ kφ 2 1 = p m , = cot 2 , = (sinh φ = (coth u m = (cosh ) u u ,u kφ m = cot ) u kφ = (tan u The deltoid is Moulin ´avent The astroid is Kappa The nehproid is = 1 we get the two types of Poinsot Spirals for which = (tanh • • • • • u m respectivley. A.4.9 Further examples A.4.7 Other examples Freeth’s Nephroid is whence A.4.8 Some hyperbolic spirals If JCAP10(2015)056 , , (1853) ]. 18 (2009) 531 Adv. High J. 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