Cosmology from the two-dimensional renormalization group acting as the Ricci flow
CCosmology from the two-dimensional renormalization groupacting as the Ricci flow
Daniel Friedan ∗ New High Energy Theory Center and Department of Physics and Astronomy,Rutgers, The State University of New Jersey,Piscataway, New Jersey 08854-8019 U.S.A. andScience Institute, The University of Iceland, Reykjavik, Iceland (Dated: September 3, 2019)The two-dimensional renormalization group acting as the Ricci flow Λ ∂∂ Λ g µν = R µν produces a specific 1+3 dimensional space-time metric which describes an expandinguniverse that starts with a big bang a ∼ t / √ then decelerates until z = 0 . t max = 1 . t H with a big blowup a ∼ ( t max − t ) − / √ . Theonly free parameters are the overall time scale and the value of the present time t . These are fixed by the Hubble constant H = t − H and the present decelerationparameter q ( t ). This crude calculation of cosmology omits all but the gravitationalfield. The only energy-momentum is purely gravitational dark matter and energy.This is a preliminary exploration towards a specific, comprehensive, testable calcu-lation of cosmology from a fundamental theory in which physics is produced by aquantum version of the two-dimensional renormalization group. The renormalization group of the two-dimensional general nonlinear model acts at leadingorder as the Ricci flow [1–3] Λ ∂∂ Λ g µν = R µν (1)The field of the nonlinear model takes its values in a Riemannian manifold M . The Rie-mannian metric g µν encodes the couplings of the model. The beta-function at leading orderis the Ricci tensor R µν . The renormalization group drives g µν to a fixed point (modulo atmost a finite number of relevant parameters). The fixed points are the solutions of R µν = ∇ µ v ν + ∇ µ v µ (2)where v µ ( x ) ∂ µ is a vector field on M . A vector field is an infinitesimal reparametrization˙ x µ = v µ ( x ) of M . This is a field redefinition in the nonlinear model equivalent to theredundant perturbation ˙ g µν = ∇ µ v ν + ∇ ν v µ . Thus the fixed point equation (2) expressesphysical 2-d scale invariance .The 2-d quantum field theory is manifestly well defined when g µν has euclidean signature.Assume that M is a four-dimensional Riemannian manifold of the form I × S where I isa real interval and S is the unit 3-sphere. Assume SO(4) invariance. Parametrize I × S as a spherical shell in R . The flat euclidean metric is δ µν dx µ dx ν = dr + r ds S where ds S is the round metric on the unit 3-sphere. The general SO(4)-invariant metric has the form The general nonlinear model (also called the nonlinear sigma model ) was constructed in [1–3] in 2+ (cid:15) dimensions with fixed point equation R µν − (cid:15)g µν = ∇ µ v ν + ∇ µ v µ . The solutions were called quasi-Einsteinmetrics. The Ricci flow was introduced in Mathematics in [4]. The solutions of R µν − (cid:15)g µν = ∇ µ v ν + ∇ µ v µ have been called Ricci solitons or Ricci flow solitons in the Mathematics literature. a r X i v : . [ a s t r o - ph . C O ] M a y g µν dx µ dx ν = F ( r ) dr + F ( r ) ds S . The metric can be made conformally flat g µν dx µ dx ν = a (cid:0) dτ + ds S (cid:1) a = e f ( τ ) (3)by reparametrizing r → τ ( r ) with dr/dτ = F /F . The general SO(4)-invariant vector fieldis v τ ( τ ) ∂ τ . Then [5] R µν dx µ dx ν = ( − ∂ τ f τ ) dτ + (cid:0) − ∂ τ f τ + 2 − f τ (cid:1) ds S ( ∇ µ v ν + ∇ ν v µ ) dx µ dx ν = (2 ∂ τ v τ − v τ f τ ) dτ + (2 v τ f τ ) ds S f τ = ∂ τ f v τ = g ττ v τ = a v τ (4)The fixed point equation (2) becomes the ordinary differential equation ∂ τ f τ = − f τ v τ − f τ + 2 ∂ τ v τ = 4 f τ v τ + 3 f τ − τ to real time T = i − τds = a (cid:0) − dT + ds S (cid:1) (6)The real-time ode is ∂ T f T = − f T v T − f T − ∂ T v T = 4 f T v T + 3 f T + 3 ∂ T = i∂ τ f T = if τ = ∂ T a/a v T = iv τ (7)The solution is f T = cos 2 T + √ T v T = −√ T a = t (cid:48) sin ν T cos − ν T T ∈ (0 , π/
2) (8)with ν = √ / − / . . . . . The only free parameter is the overall time scale t (cid:48) . Inco-moving time t ds = − dt + a ds S dt = adTtt (cid:48) = (cid:90) T sin ν T (cid:48) cos − ν T (cid:48) dT (cid:48) = 12 B sin T (cid:18) ν , − ν (cid:19) t ∈ (0 , t max ) t max t (cid:48) = 12 B (cid:18) ν , − ν (cid:19) = 1 . . . . (9) B ( p, q ) is the Euler beta function, B x ( p, q ) the incomplete beta function. At the limits T → tt (cid:48) → T ν ν a → t (cid:48) (2 + ν ) / √ (cid:18) tt (cid:48) (cid:19) / √ T → π t max − tt (cid:48) → (cid:0) π − T (cid:1) − ν − ν a → t (cid:48) (1 − ν ) − / √ (cid:18) t max − tt (cid:48) (cid:19) − / √ (10)This is an expanding universe which begins with a big bang a ∼ t . ... and ends with a bigblowup a ∼ ( t max − t ) − . ... . The Hubble parameter H = ∂ t a/a = f T /a is H = 1 t (cid:48) (cos T + ν ) sin − − ν T cos − ν T (11)The deceleration parameter q = − a∂ t a/ ( ∂ t a ) = − ∂ T f T /f T is q = 2 (cid:0) √ T + 1 (cid:1)(cid:0) cos 2 T + √ (cid:1) T q =0 = 12 arccos( − / √
3) = (0 . . . . ) π q >
0) until T = T q =0 then accelerates ( q <
0) until the end.For a first estimate of the time scale t (cid:48) and the present time t use the decelerationparameter at the present time q = q ( T ) ≈ − . T = 0 . π/ H to H ( T ) in (11) to obtain t (cid:48) = 1 . t H where t H =1 /H ≈ . × y . Then (9) gives t max = 1 . t H , t = 0 . t H and (8) gives a = a ( t ) =1 . t H .Einstein’s equation R µν − Rg µν = 8 πG T µν (13)is satisfied with energy-momentum tensor T µν = 18 πG ( ∇ µ v ν + ∇ ν v µ − ∇ σ v σ g µν ) (14)which is that of a perfect fluid of density ρ ( t ) and pressure p ( t ) T µν dx µ dx ν = ρ ( t ) dt + p ( t ) a ds S πG ρ ( t ) = a − (cid:0) f T + 3 (cid:1) πG p ( t ) = a − (cid:0) f T v T + 3 f T + 3 (cid:1) (15)The energy-momentum is purely gravitational, tautologically dark. The equation-of-stateparameter w ( t ) = p/ρ and the density parameter Ω( t ) = 8 πGρ/ H = 1 + 1 /f T are w = cos 2 T T + 2 √ (cid:18) sin 2 T cos 2 T + √ (cid:19) (16)The estimate T = 0 . π/ w = − .
6, Ω = 1 .
5. The cosmologicalparameters are graphed as functions of t in Figure 1. Past values for a selection of redshifts z = a /a − T, f T , v T ) → ( − T, − f T , − v T ) and constant of motion C = 1 a (cid:18) v T + 2 f T v T + f T + 1 (cid:19) ∂ T C = 0 (17)Changing variables to h ± = f T + (1 ± / √ v T the constant of motion and ode become h + h − + 1 = Ca ∂ T h ± = − (1 + h ± ) + λ ± ( h + h − + 1) λ ± = 2 ± √ tt H -2-112 zqw Ω HH t FIG. 1. The cosmological parameters as functions of the co-moving time t . The present is t .TABLE I. Cosmological parameters at selected values of zz H/H q w Ω z (cid:29) (cid:29) z . .
73 0 .
16 1 . . × .
73 0 .
16 1 . . × .
73 0 .
15 1 .
010 48 0 .
74 0 .
15 1 . T = T . .
74 0 .
08 1 . . .
69 0 .
02 1 . q = 0 0.2 1 . − . . t = t − . − . . The cosmological solution (8) is C = 0, h − = cot T , h + = − tan T .The phase portrait of the ode is shown in Figure 2. Every C (cid:54) = 0 trajectory asymptotesto a C = 0 trajectory. The separatrix S and its time-reflection S (cid:48) each does so at one end;all the other C (cid:54) = 0 trajectories do so at both ends. The asymptotic behavior is one of T → ± h − → T h + → − T − T + cT ( T ) ν cC > h + → T h − → c (cid:48) T ( T ) − ν c (cid:48) C > T , namely ( T ) ν or ( T ) − ν , implies that such asolution cannot be the analytic continuation of a solution in imaginary time τ = iT . -6 -4 -2 2 4 6 f T -6-4-2246 v T C = 0 C = 0 C < C < C > SS -4 -2 2 4 h − -10-5510 h + C = 0 C = 0 C < C < C > S S FIG. 2. Phase portrait of the real-time ode (7,18) in terms of ( f T , v T ) on the left, ( h − , h + ) on theright. The trajectories are the solutions. The arrows point in the direction of increasing T . Theinterval between points in the trajectories is ∆ T = 0 .
01. The universe is expanding when f T = ∂ t a is positive. The expansion is accelerating when f T is increasing. The two red trajectories are the C = 0 solutions. The one in the lower right quadrant labeled C =0 is the cosmological solution (8).Its time reflection is labeled C =0 (cid:48) . The separatrix is labeled S . Its time-reflection is labeled S (cid:48) . The trajectories that start from h − = ∞ , h + = 0 at T = 0 are characterized by thenumerical invariant c = ( h + h − + 1)( h − ) ν e (cid:82) T dT (cid:48) h − ( T (cid:48) ) = lim T → ( h + h − + 1)( h − ) ν (20)The separatrix S has c S = 0 . . . . . The c < c S trajectories — the trajectories below S — are all roughly realistic as cosmologies. They all start with a big bang a ∼ t / √ thendecelerate than accelerate to a big blow-up a ∼ t − / √ , all driven by purely gravitationaldark energy-momentum. The gap between the C =0 cosmological solution and the separatrix S means that the cosmological solution (8) is stable against real-time perturbations. Thezone of stability on the side C > T → SO (4) space-time symmetry will need to be justified byproperties of the 2-d renormalization group flow — by stability properties of the fixed pointand/or as the dynamical result of a distinguished trajectory of the rg flow.The analyticity assumption needs precise characterization and justification. The assump-tion here that f T ( T ) and v T ( T ) should be analytic in T singles out the C = 0 cosmologicalsolution. Analyticity serves as a selection principle providing specificity. It should expressa physical principle.The next exploratory steps will add standard model fields to the calculation, then spatialfluctuations and quantum effects, hoping to find analytic solutions of 2-d renormalizationgroup fixed point equations that capture more and more qualitative features of the cosmologyof the real world, aiming to find a solution that can be tested in quantitative detail. Thiswould provide support for the fundamental theory [8] in which a quantum version of the two-dimensional renormalization group acts mechanically to produce physics and would provideguidance for deriving real world cosmology from that fundamental theory.This work was supported by the Rutgers New High Energy Theory Center and by thegenerosity of B. Weeks. I am grateful to the Mathematics Division of the Science Instituteof the University of Iceland for its hospitality. ∗ [email protected]; [1] D. Friedan, Nonlinear Models in 2 + (cid:15) Dimensions, Phys. Rev. Lett. , 1057 (1980).[2] Daniel Harry Friedan, Nonlinear Models in (cid:15)
Dimensions , Tech. Rep. (Lawrence BerkeleyLaboratory LBL-11517, 1980) U.C. Berkeley PhD thesis (1980).[3] Daniel Harry Friedan, Nonlinear Models in 2 + (cid:15)
Dimensions, Ann. Phys. , 318 (1985),republication of [2].[4] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. ,255 (1982).[5] Calculations are shown in the accompanying supplemental material which consists ofa note Calculations for ”Cosmology . . . ” and three SageMath [6] notebooks of numer-ical calculations. The supplemental material can be found at or at https://share.cocalc.com/share/6a7035ba-9879-4b05-b3d0-688b1309a21c/Cosmology_I_supplementary_material/?viewer=share/ .[6] The Sage Developers,