Precision cosmology made more precise
PPrecision cosmology made more precise
Giorgio Galanti ∗ and Marco Roncadelli
2, 3, † INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica di Milano, Via A. Corti 12, I – 20133 Milano, Italy INFN, Sezione di Pavia, Via A. Bassi 6, I – 27100 Pavia, Italy INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I – 23807 Merate, Italy (Dated: February 3, 2021)So far, the standard attitude to solve the Friedmann equations in the simultaneous presence ofradiation R , matter M and cosmological constant Λ is to find solutions R R ( t ), R M ( t ) and R Λ ( t ) separately for each individual component alone , and next to join them together, thereby obtaining a piecewise solution R pw ( t ). We instead find the exact and analytic solution R ( t ) of the same equationsin flat space. Moreover, we quantify the error made when R pw ( t ) is used in place of R ( t ). Introduction – A wide spread belief is that cosmologyhas entered its precision phase. Of course, we still do notknow the nature of non-baryonic dark matter and darkenergy – represented here by the cosmological constantΛ – and the cosmological parameters H , Ω R, , Ω M, and Ω Λ are not determined very precisely [1]. But wheneverything is put together a very remarkably consistentscenario emerges.Nevertheless, the Friedmann equations are still solvedin a too naive fashion. Basically, their solutions R R ( t ), R M ( t ) and R Λ ( t ) are found separately for each individ-ual component Ω R ( t ), Ω M ( t ) and Ω Λ , respectively, alone .And next R R ( t ), R M ( t ) and R Λ ( t ) are joined together,thereby providing a piecewise solution R pw ( t ). Such aprocedure looks correct whenever one component dom-inates over the others, so that effectively only a singlecomponent is relevant. Yet, we know that there is a time t M Λ when Ω M ( t M Λ ) = Ω Λ and a previous time t RM whenΩ R ( t RM ) = Ω M ( t RM ). Around t M Λ and t RM two com-ponents contribute almost equally to the energy budgetof the Universe: in such a situation the piecewise ap-proximation manifestly breaks down. In addition, beingthe Friedmann equations non-linear in R ( t ), the contri-bution of, say, Ω M ( t ) plus Ω Λ gives rise to a scale factor R M Λ ( t ) different from the sum of the scale factors R M ( t )and R Λ ( t ) as computed by taking Ω M ( t ) alone and Ω Λ alone , respectively, into account. In some textbook thebehavior of the Universe in the presence of two compo-nents is discussed rather cursorily (see e.g. [2]), but – tothe best of our knowledge – so far the scale factor R ( t )has never been computed exactly and analytically in thesimultaneous presence of Ω R ( t ), Ω M ( t ) and Ω Λ withinthe Friedmann equations.The aim of the present Letter is to fill this gap, and todiscuss some of its implications. Notations and conventions – It is more useful to workwith the normalized scale factor a ( t ) ≡ R ( t ) /R (with R ≡ R ( t )), so that the Friedmann equations are (cid:18) ˙ a ( t ) a ( t ) (cid:19) = 8 πG c (cid:15) ( t ) − kc R a ( t ) , (1) ¨ a ( t ) a ( t ) = − πG c (cid:16) (cid:15) ( t ) + 3 P ( t ) (cid:17) , (2)where G is the Newton gravitational constant, c is thespeed of light in vacuo , k is the curvature constant, while (cid:15) ( t ) represents the total energy density of the cosmic fluid (cid:15) ( t ) = 3 c H πG (cid:18) Ω R, a ( t ) + Ω M, a ( t ) + Ω Λ (cid:19) , (3)and P ( t ) denotes the pressure of the cosmic fluid P ( t ) = c H πG (cid:18) Ω R, a ( t ) − Λ (cid:19) . (4)We henceforth restrict our attention to a metricallyflat there-dimensional space ( k = 0), for which there isnowadays a general consensus. Accordingly, by insertingEq. (3) into Eq. (1) with k = 0, and Eqs. (3) and (4) intoEq. (2), we end up with (cid:18) ˙ a ( t ) a ( t ) (cid:19) = H (cid:18) Ω R, a ( t ) + Ω M, a ( t ) + Ω Λ (cid:19) , (5)¨ a ( t ) a ( t ) = − H (cid:18) Ω R, a ( t ) + 12 Ω M, a ( t ) − Ω Λ (cid:19) . (6)Moreover, by exploiting Eq. (3), we find that the scalefactor when radiation and matter contribute equallyreads a RM ≡ a ( t RM ) = Ω R, Ω M, , (7)while the scale factor when the matter and dark energycontributions are equal is given by a M Λ ≡ a ( t M Λ ) = (cid:18) Ω M, Ω Λ (cid:19) / . (8) Piecewise solution – As a preliminary step, we brieflyrecall the piecewise solution a pw ( t ) ≡ R pw ( t ) /R usuallyfound in the literature – describing separately the threeepochs of radiation, matter and cosmological constant a r X i v : . [ a s t r o - ph . C O ] F e b domination – since it provides a benchmark for compar-ison with our exact solution. By solving Eq. (5) in thesingle-component case, we find a pw ( t ) = K R t / t ≤ ˜ t RM ,K M t / ˜ t RM < t ≤ ˜ t M Λ ,K Λ exp (cid:16) Ω / H t (cid:17) t > ˜ t M Λ , (9)where K R , K M and K Λ are normalization constants, and˜ t RM and ˜ t M Λ have the same conceptual meaning of t RM and t M Λ , respectively [3]. We obtain K Λ by imposingthe condition a = 1 today, so that we get K Λ = exp (cid:16) − Ω / H t (cid:17) , (10)while K M and K R are fixed by requiring continuity of a ( t ) at the junction points ˜ t M Λ and ˜ t RM . Because thelatter two quantities are still unknown, we turn the ar-gument around by demanding continuity of the inversefunction t pw ( a ) (which is always single-valued since a ( t )is monotonically increasing) t pw ( a ) = K − R a a ≤ a RM ,K − / M a / a RM < a ≤ a M Λ ,t + Ω − / H − ln a a > a M Λ . (11)Specifically, from the second and third of Eq. (11) wefind K − / M a / M Λ = t + Ω − / H − ln a M Λ , which implies K M = (cid:18) Ω M, Ω Λ (cid:19) / (cid:20) H t + 13 Ω − / ln (cid:18) Ω M, Ω Λ (cid:19)(cid:21) − / H / , (12)where we have employed Eq. (8) with t M Λ → ˜ t M Λ . Sim-ilarly, from the first and second of Eq. (11) we furtherobtain K − R a RM = K − / M a / RM , which entails K R = (cid:18) Ω R, Ω Λ (cid:19) / (cid:20) H t + 13 Ω − / ln (cid:18) Ω M, Ω Λ (cid:19)(cid:21) − / H / , (13)where now we have used Eq. (7) with t RM → ˜ t RM . What remains to be done is the determination of ˜ t RM and ˜ t M Λ . Being by definition ˜ t RM ≡ t pw ( a RM ), thanksto the first of Eq. (11) combined with Eqs. (7) and (13)we get˜ t RM = Ω / R, Ω − M, Ω / (cid:20) t + 13 Ω − / H − ln (cid:18) Ω M, Ω Λ (cid:19)(cid:21) . (14)Moreover – since by definition ˜ t M Λ ≡ t pw ( a M Λ ) – owingto the second of Eq. (11) in conjunction with Eqs. (8)and (12) we find˜ t M Λ = t + 13 Ω − / H − ln (cid:18) Ω M, Ω Λ (cid:19) . (15)With the benchmark values of the numerical parametersreported in [5] we have ˜ t RM = 6 . × yr and ˜ t M Λ =8 .
75 Gyr.
Our solution – We start by rewriting Eq. (5) as H − d a ( t )d t = (cid:18) Ω R, a ( t ) + Ω M, a ( t ) + Ω Λ a ( t ) (cid:19) / , (16)and by performing the separation of variables, it takesthe integral form H t = (cid:90) a ( t )0 d a (cid:48) (cid:18) Ω R, a (cid:48) + Ω M, a (cid:48) + Ω Λ a (cid:48) (cid:19) − / , (17)which allows us to express t as a function of a . Ourultimate goal is to have instead a as a function of t .A direct attempt at inverting Eq. (17) is extremelycumbersome. We can save work by splitting up such anequation – which contains three Ω parameters – into apair of equations, each including only two of them. Thistask can be accomplished as follows. Neglecting primor-dial inflation, standard cosmology tells us that the Uni-verse is radiation dominated for 0 ≤ t ≤ t RM , matterdominated for t RM < t ≤ t MΛ and dark energy domi-nated for t MΛ < t ≤ t . So, we can define a time t s suchthat t RM (cid:28) t s (cid:28) t MΛ (more about this choice, later).Being t s well inside the regime of matter domination, it issure that for t ≤ t s only radiation or matter dominates,whereas for t > t s only matter or dark energy dominates.As a consequence, Eq. (17) becomes equivalent to the twoequations H t = (cid:90) a ( t )0 d a (cid:48) (cid:18) Ω R, a (cid:48) + Ω M, a (cid:48) (cid:19) − / , a ≤ a s ,H t s + (cid:90) a ( t ) a s d a (cid:48) (cid:18) Ω M, a (cid:48) + Ω Λ a (cid:48) (cid:19) − / , a > a s , (18)where a s ≡ a ( t s ). The solution of Eq. (18) is H t = M, (cid:104) (Ω M, a − R, )(Ω M, a + Ω R, ) / + 2Ω / R, (cid:105) , a ≤ a s ,H t s + 23Ω / ln (cid:34) Ω Λ a / + Ω / (cid:0) Ω M, + Ω Λ a (cid:1) / Ω Λ a / s + Ω / (Ω M, + Ω Λ a s ) / (cid:35) , a > a s . (19)In order to proceed towards our goal – and so gettingthe explicit function a ( t ) – we turn both expressions inEq. (19) into two third-order algebraic equations. Then,the fundamental theorem of algebra ensures that theyhave three roots, which can be either all real or one realand two complex conjugate, since the equations in ques-tion have real coefficients. The calculation of a ( t ) in thesecond of Eq. (19) is rather straightforward, and by in-spection one can see that we are in the case of one realand two complex conjugate roots. Being the latter ones physically unacceptable, the behavior of a ( t ) for t > t s isreported in the third line of Eq. (20) below.The calculation of a ( t ) in the first of Eq. (19) is in-stead not straightforward at all, and deserves a detaileddiscussion which is reported in the Supplementary Ma-terial (SM). Accordingly, we find the behavior of a ( t ) for t ≤ t s as reported in the first and second line of Eq. (20)below.Thus, our full exact solution a ( t ) of the first Friedmannequation (5) is a ( t ) = Ω R, Ω M, (cid:40) − (cid:34)
13 arcsin (cid:32) − M, Ω / R, H t + 98 Ω M, Ω R, H t (cid:33)(cid:35)(cid:41) , ≤ t ≤ t ∗ , Ω R, Ω M, (cid:40) (cid:34)
13 arccos (cid:32) − M, Ω / R, H t + 98 Ω M, Ω R, H t (cid:33)(cid:35)(cid:41) , t ∗ < t ≤ t s , (cid:40) a / s cosh (cid:20)
32 Ω / H ( t − t s ) (cid:21) + (cid:18) a s + Ω M, Ω Λ (cid:19) / sinh (cid:20)
32 Ω / H ( t − t s ) (cid:21)(cid:41) / , t > t s , (20)where we have set t ∗ ≡ / R, / (3Ω M, H ) (see also theSM), and a s is calculated by employing the second ofEq. (20) with t = t s .It can be checked that Eq. (20) is solution of Eq. (5) bytaking Ω Λ = 0 for t ≤ t s (where only radiation and mat-ter are important) and Ω R ( t ) = 0 for t > t s (where onlymatter and dark energy are relevant). And by employingthese prescriptions, it is possible to see that Eq. (20) issolution also of the second Friedmann equation (6) eventhough this is unnecessary, since such an equation can bereplaced by the conservation equation [4]˙ (cid:15) ( t ) + 3 (cid:0) (cid:15) ( t ) + P ( t ) (cid:1) ˙ a ( t ) a ( t ) = 0 , (21)which is identically satisfied thanks to Eqs. (3) and (4).Thus, Eq. (20) describes the whole evolution of a ( t ) inthe present situation. Discussion – We are now ready to investigate the prop-erties of our exact solution a ( t ). In order to find out thebehavior of a as a function of H t , we take the bench- mark values of the relevant parameters reported in [5].Accordingly, in the upper panel of Figure 2 we plot a versus H t , along with the asymptotic behavior as givenby a pw . Moreover, thanks to Eq. (7) and the first ofEq. (19), t RM is provided by t RM = 4Ω / R M H (cid:16) − − / (cid:17) , (22)while on account of Eq. (8) and of the second of Eq. (19) t M Λ can be written as t M Λ = t s + 23Ω / H ln (cid:34) Ω / M, (cid:0) / (cid:1) Ω / a / s + (Ω M, + Ω Λ a s ) / (cid:35) . (23)Superficially, Eq. (23) might look suspicious owing to thedependence of t M Λ on t s and a s – given the freedom toarbitrarily select t s within the rather wide range t RM (cid:28) t s (cid:28) t MΛ – which correspondingly fixes a s . The best wayto clarify this point amounts to recall that Eq. (23) isthe solution of the Friedmann equations – in the form of Figure 1.
Upper panel : using our benchmark values, we plotthe scale factor a and the asymptotic behaviors given by a pw versus H t . Lower panel : exact solution a and piecewise one a pw as plotted versus H t . For the reader’s convenience, wereport this Figure in a larger format in the SM. the second of Eq. (19) – specialized to the case a = a M Λ according to Eq. (8). As we shall shortly see, a very goodchoice is t s = ( t RM t ) / . Correspondingly, from Eqs.(22) and (23) we find t RM = 4 . × yr, t s = 25 .
23 Myr, a s = 0 .
013 and t M Λ = 9 .
82 Gyr. So, t s indeed turns outto be just halfway between t RM and t M Λ , and in addition t ∗ = 1 . × yr, thereby implying t ∗ (cid:28) t s as previouslyassumed. Conclusions – We have derived the exact analytic solu-tion of the Friedmann equations in the presence of radia-tion, matter and cosmological constant as time increases,assuming that three-dimensional space is metrically flat.We close this Letter by comparing our solution with thepiecewise one. This is best done by plotting both of them in the lower panel of Figure 2 versus H t . From thatwe see that a pw ( t ) roughly approximates a at any time.Specifically, as t increases in the radiation dominatedepoch a pw ( t ) underestimates a ( t ), while in the matterdominated era a pw ( t ) crosses a ( t ), thus first underesti-mating and then overestimating the exact solution. Onlyinside the dark energy dominated epoch appears a pw ( t )as a good approximation of a . In order to quantify theaccuracy of the piecewise solution we estimate the er-ror with respect to the exact solution (20) in Table I,where t M is a generic time when the Universe is mat-ter dominated. Finally, note that t RM / ˜ t RM = 0 .
75 and t M Λ / ˜ t M Λ = 1 . t ( a pw ( t ) − a ( t )) /a ( t ) [%]0 . t RM − . t RM − . t RM − . t M − . ÷ . t M Λ . t M Λ . t a pw ( t ) with respect to a ( t ) at differ-ent cosmic times. The error at t is zero by construction. Thus, the largest discrepancy between a pw ( t ) and a ( t )occurs in the matter dominated regime even around t RM ,and next progressively decreases at smaller and smallertimes. ACKNOWLEDGMENTS
G. G. acknowledges contribution from the grant ASI-INAF 2015-023-R.1, while the work of M. R. is supportedby an INFN TAsP grant. ∗ [email protected] † [email protected][1] As usual, we denote by H the value of the Hubble con-stant, by Ω R, , Ω M, and Ω Λ the present values of thedimensionless cosmic density parameters pertaining to ra-diation, matter and cosmological constant, respectively.[2] B. Ryden, Introduction to Cosmology (Cambridge Univer-sity Press, Cambridge, 2017).[3] We use t RM and t M Λ in connection with our exact solu-tion. Below, in this Section we simply write a ( t ) rather a pw ( t ) for notational simplicity.[4] This can be seen multiplying Eq. (1) by a ( t ), taking itstime derivative and getting rid of ¨ a ( t ) using Eq. (2).[5] As benchmark values we take Ω R, = 8 . × − , Ω M, =0 .
30, Ω Λ = 0 . H = 70 km s − Mpc − , resulting in t =13 .
47 Gyr.
SUPPLEMENTARY MATERIAL
Solution of the first of Eq. (19)
For the reader’s convenience, we report below the first of Eq. (19) H t = 23Ω M, (cid:104) (Ω M, a − R, )(Ω M, a + Ω R, ) / + 2Ω / R, (cid:105) , a ≤ a s , (24)where a s ≡ a ( t s ) is defined in the main text. Our aim is to study its solutions and their behavior in order to singleout the physical one. To this end, we define the polynomial P ( a, t ) ≡ M, a − M, Ω R, a − M, H t + 24Ω M, Ω / R, H t , (25)where a should be regarded as a variable while t as a parameter. In view of our subsequent needs, we also quote itsderivative d P ( a, t )d a = 12Ω M, a − M, Ω R, a . (26)Hence, we have the following: Lemma .1. P ( a, t ) is a decreasing function of a for < a < R, / Ω M, and an increasing function of a for a > R, / Ω M, . In addition, P ( a, t ) is an increasing function of a also for a < . Manifestly, by setting P ( a, t ) = 0 we recover Eq. (24), a fact which allows us to find the desired solution a ( t ).Correspondingly, we are dealing with a third order algebraic equation. The fundamental theorem of algebra ensuresthat for each value of t the equation P ( a, t ) = 0 possesses three roots, which can be all real or one real and twocomplex conjugated since P ( a, t ) has real coefficients. Owing to the physical meaning of t , we assume t ≥
0. From analgebraic point of view P ( a, t ) = 0 possesses the three solutions for each value of ta ( t ) = Ω R, Ω M, (cid:20) − (cid:18)
13 arcsin X ( t ) (cid:19)(cid:21) , (27) a ( t ) = Ω R, Ω M, (cid:20) (cid:18)
13 arccos X ( t ) (cid:19)(cid:21) , (28) a ( t ) = Ω R, Ω M, (cid:26) (cid:20) (cid:0) π + arccos X ( t ) (cid:1)(cid:21)(cid:27) , (29)where we have set for notational simplicity X ( t ) ≡ − M, Ω / R, H t + 98 Ω M, Ω R, H t . (30)In order to understand which of them is physically meaningful, we perform an analytical study of the function P ( a, t ). We stress that in order for arcsin X ( t ) and arccos X ( t ) to be real-valued we must have − ≤ X ( t ) ≤
1, butfor X ( t ) > X ( t ) is complex while arccos X ( t ) is imaginary. Explicitlyarcsin X ( t ) = − i ln (cid:16) iX ( t ) + (cid:112) − X ( t ) (cid:17) , (31)arccos X ( t ) = − i ln (cid:16) X ( t ) + (cid:112) X ( t ) − (cid:17) . (32)We are now ready to accomplish our task, keeping in mind that Ω M, , Ω R, and H are real positive quantities.We start by observing that from Eq. (25) we havelim a → + ∞ P ( a, t ) = + ∞ , (33)and in addition P (0 , t ) < t satisfies the condition t > t (34)with t defined by t ≡ / R, M, H . (35)Assuming that Eq. (34) is met, owing to Eq. (33) and P (0 , t ) < P ( a, t ) obeys the Bolzano theorem,which entails that P ( a, t ) = 0 possesses at least one root in the range 0 ≤ a < ∞ . Moreover, by combining P (0 , t ) < P ( a, t ) = 0 possesses only one real root in the range 0 ≤ a < ∞ . Further, Eq. (34)implies that X ( t ) >
1. Therefore, from Eq. (31) we find that arcsin X ( t ) is complex, and so its sine is complex either.So, Eq. (27) is one of the two complex conjugated roots. Turning our attention to Eq. (28), from X ( t ) > X ( t ) is imaginary, and so its cosine is real, since cos i x = cosh x for any real number x .Thus, we see that Eq. (28) is a real root. Finally, we address Eq. (29). By the same token, we see that the argumentof the cosine is complex – owing to the presence of 2 π – so that this is the remaining complex conjugated root. Inconclusion, for t > t only Eq. (28) is physically acceptable.Let us next consider the opposite case, namely 0 ≤ t < t . (36)Accordingly, we now have P (0 , t ) >
0, and since lim a →−∞ P ( a, t ) = −∞ (37)the Bolzano theorem ensures that P ( a, t ) possesses at least one root in the range −∞ < a ≤
0. Which one? Inorder to settle this issue, it is of paramount importance to realize that condition (36) implies − ≤ X ( t ) ≤ ≤ arccos X ( t ) ≤ π . Therefore, we have 2 π/ ≤ (2 π + arccos X ( t )) / ≤ π , which means that − . ≤ cos [(2 π + arccos X ( t )) / ≤ −
1. Correspondingly, we find that it is Eq. (29) which has a real negative root: thisis manifestly unphysical and will be discarded. What about the other two solutions? Because we presently have P (0 , t ) > a priori physicallyacceptable. These solutions are represented by Eqs. (27) and (28). And by taking the limit for t → a → t →
0. As a result, when either radiation ormatter dominates, we discover that close enough to t = 0 the solution of the Friedmann equation is given by Eq. (27),while for t satisfying condition (34) the solution becomes Eq. (28). When do these two solutions join? Because theymust soothly join, there will be a time t ∗ such thatsin (cid:18) −
13 arcsin X ( t ∗ ) (cid:19) = cos (cid:18)
13 arccos X ( t ∗ ) (cid:19) (38)where we have employed the fact that the sine is an odd function. By using the trigonometric relation cos ( π/ − α ) = sin α we can equal the arguments of the cosines, and we are left with the condition 3 π/ X ( t ∗ )) =arccos ( X ( t ∗ )), which can be simplified as arcsin ( X ( t ∗ )) = − π/ y = π/ − arcsin y . Thus, we obtain that condition (38) is verified when X ( t ∗ ) = − t ∗ = 4Ω / R, M, H . (39)What remains to be done is to consider the case t = t . Accordingly, it is easy to show that P ( a, t ) = 0 possessestwo coincident roots a ( t ) = 0. They can be found by inserting Eq. (35) into Eq. (30), thereby getting X ( t ) = 1,which – once further inserted into Eqs. (27) and (29) – yields a ( t ) = 0 in either case: these solutions are not physicallyacceptable because a ( t ) must be a monotonically increasing function of t starting from t = 0 and t (cid:54) = 0. The case ofEq. (28) is different, since it gives a ( t ) = 3Ω M, / Ω R, , which is the physically acceptable solution.As a result, the final unique physical solution of Eq. (24) – namely of Eq. (19) of the main text – can be expressedas a ( t ) = Ω R, Ω M, (cid:40) − (cid:34)
13 arcsin (cid:32) − M, Ω / R, H t + 98 Ω M, Ω R, H t (cid:33)(cid:35)(cid:41) , t ≤ t ∗ , Ω R, Ω M, (cid:40) (cid:34)
13 arccos (cid:32) − M, Ω / R, H t + 98 Ω M, Ω R, H t (cid:33)(cid:35)(cid:41) , t ∗ < t ≤ t s , (40)where t s is defined in the main text. Thus, Eq. (40) represents the solution of the Friedmann equation when eitherradiation or matter is important. Alternative form of the Friedmann equations
Here, we derive an alternative mathematical expression of Eq. (28). By employing Eq. (32), we find that for t < t ≤ t s , a ( t ) can be rewritten as a ( t ) = Ω R, Ω M, (cid:26) (cid:20)
13 ln (cid:16) X ( t ) + (cid:112) X ( t ) − (cid:17)(cid:21)(cid:27) , (41)in order not to encounter the imaginary unit during the calculation. The quantity X ( t ) is given by Eq. (30). Noteindeed that Eq. (28) produces at the end real positive value for a even if the imaginary part appears during thecalculation, as previously discussed. Enlarged Figures of the main text
Figure 2. Using our benchmark values, we plot the scale factor a and the asymptotic behaviors versus H t . Figure 3. Exact scale factor a and piecewise function approximation a pw as plotted versus H tt