- Email: [email protected]
A stringy origin of the recent acceleration
Physics Letters B 634 (2006) 437–441 www.elsevier.com/locate/physletb
A stringy origin of the recent acceleration Tirthabir Biswas ∗ , Anupam Mazumdar CHEP, McGill University, Montréal, QC, H3A 2T8, Canada Received 22 July 2005; received in revised form 1 February 2006; accepted 6 February 2006 Available online 13 February 2006 Editor: M. Cvetiˇc
Abstract Inspired by the current observations that the ratio of the abundances of dark energy and matter density is close to one, we provide a string inspired phenomenological model where we try to explain this order one ratio, smallness of the cosmological constant, and also the recent cosmic acceleration. We observe that any effective theory motivated by higher-dimensional physics provides radion/dilaton couplings to the standard model and the dark matter component with different strengths. Provided radion/dilaton is a dynamical field we then show that the dark energy component tracks matter density and can start to dominate very recently. © 2006 Elsevier B.V. All rights reserved.
The cosmological constant problem is one of the most difficult problems of theoretical physics. What requires an explanation is that, the observed energy content of the Universe, ∼ 4 × 10−47 (GeV)4 , is many orders of magnitude smaller than that of the theoretical prediction for the cosmological constant energy density alone. The mismatch is of order 10−120 × Mp4 , where Mp ∼ 2.4 × 1018 GeV. One would naively expect that even if the bare cosmological constant can be made to vanish, quantum corrections would eventually lead to quadratic divergences ∼ Mp4 (supersymmetry only ameliorates the mismatch). Thus the puzzle is “why the bare cosmological constant is so small and moreover, stable under quantum corrections” [1]. There is also a kind of a coincidence problem, sometimes dubbed as a why now problem. Recent observations from supernovae [2] and the cosmic microwave background (CMB) anisotropy measurements [3] suggest that the majority of the energy density ∼ 70% is in the form of dark energy, whose constituent is largely unknown, but usually believed to be the cosmological constant with an equation of state ω = −1, which is also responsible for the current acceleration. In this respect not only the cosmological constant is small, but it also happens to be dominating the energy density of the Universe. A priori,
* Corresponding author.
E-mail address: [email protected] (T. Biswas). 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.02.015
the physics behind the value of the cosmological constant and its “evolution” need not be related to the redshifting of matter/radiation density. Is it then just by coincidence that today ρΛ is close to the value ρm or are there deeper connections between the two? A related question would be why the cosmological constant is dominating right now and for example not during the big bang nucleosynthesis (BBN) at a temperature T ∼ 1 MeV? Both the smallness and the why now questions can be answered in part if we believe that the physics of the dark energy is somehow related to the rest of the energy density of our Universe. Attempts have been made to construct such tracking mechanisms in single and multi-fields, such as dynamical quintessence [4], k-essence [5] type models, non-Abelian vacuum structure with non-vanishing winding modes [6], modified Friedmann equation at late times [7], or large distances [8], or due to inflationary backreaction [9]. In this Letter we propose a simple model which can arise naturally from the compactifications of a higher-dimensional theory. For simplicity we assume that all the moduli/dilaton fields have been stabilized at higher scales except a single linear combination, Q, which has a runaway potential, see [10]. In an effective four-dimensional theory the Q field will couple to the Standard Model (SM) degrees of freedom as well as the dark matter. We will show that by virtue of this coupling to the matter fields “the effective potential” for Q can have a
438
T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441
local minimum which can track the matter density of the Universe. Further observe that the Q field in general is expected to have different couplings to different species of fermions (f), scalars (s), and gauge bosons (r) by virtue of its running below the string compactification scale (see for example [11]), and depending on the strength of this coupling our Universe will either accelerate or not. In particular it will be crucial in our model to have only a small component of dark matter to couple strongly to Q, while the rest of the matter couple very weakly. This is one of the crucial differences as compared to some of the earlier models [12,13] that were considered where the Q field couples uniformly to the entire dark matter sector. In fact it is this difference, along with the adiabatic tracking mechanism that help us in explaining the current acceleration without making the couplings time-dependent [13]. In our analysis we also assume that at least some relevant (beyond SM) degrees of freedom have already been thermally/non-thermally excited in the early Universe, e.g. see [14]. Let us imagine that our world was originally higher-dimensional, such that the three spatial dimensions, along with the origin of the SM are all due to some interesting compactification of the extra spatial dimensions. After dimensional reduction, in the Einstein frame we typically obtain a scalar-tensor action [15] of the form S = Sgrav+Q + Sfermions + Sgauge bosons + Sscalars , where √ Sgrav+Q = Mp2 d 4 x −g R/2 − (∂Q)2 /2 − V (Q) , Sfermions =
√ d 4 x −g ψ¯ f ∇ / + imf Bf (Q) ψf ,
Sgauge bosons = (1/4πα0 )
d 4x
√
curvature of internal manifold, etc. We therefore assume a simple form for the potential, V (Q) = V0 e−2βQ .
(2)
Henceforth we identify the dark energy density (d) as ˙ 2 /2 + V (Q). ρ˜d = Q
(3)
Note however, that Q effectively obtains an additional contribution to its potential via its coupling to matter, e2μi Q ρi . Veff (Q) = V0 e−2βQ + (4) i=s,f
The “bare density”, ρi ≡ e−2μi Q ρ˜i , is defined such that it does not depend on Q; here ρ˜i is the observed energy density of the ith component. For simplicity we may assume that radiation from gauge bosons do not contribute to an effective potential for Q. This is strictly true if on average electric field and the magnetic field vanishes. There could be other forms of radiative matter which can induce an effective coupling as discussed for example in [19], but we do not consider them here. Given all the components, we can write down the Friedmann equation for a Robertson–Walker metric, ρ˜I , H 2 = 1/3Mp2 ρ˜t = 1/3Mp2 I =s,f,r,d
2μi Q 2 −2βQ ˙ e ρi + Q /2 + V0 e . = (1/3)
(5)
i=r,s,f
The evolution equations for Q, ρi read as, ¨ + 3H Q ˙ = 2 βV0 e−2βQ − Q μi e2μi Q ρi ,
−gBr (Q)F 2 ,
(6)
i=s,f
Sscalars =
ρ˙i + 3H (pi + ρi ) = 0
√ d x −g −(∂φs )2 /2 − m2s Bs2 (Q)φs2 . 4
Although in general Bi (Q) could be an arbitrary functions, in supergravity/String theory reductions one most commonly obtains exponential couplings [16] and hence we restrict our attention to the case when Bi (Q) = e2μi Q ,
(1)
where i = {f, r, s}, runs over fermionic, radiation and scalar species, respectively. The radiation and baryonic matter content of the Universe is determined by the SM degrees of freedom, while the cold dark matter component arises from either scalars, fermions or gauge bosons of SM or beyond the SM gauge group. Note that due to the couplings various masses ∼ mi e2μi Q and fine structure constants ∼ α0 e−2μr Q now depend on Q. We will see later that observational data on variation of masses and fine structure constants, as well as fifth force experiments constrain the μ’s for SM particles to be very small < 10−3 –10−4 [15,17], and henceforth we will assume this. As far as the potential is concerned one often encounters exponential run away types in M-theory/supergravity compactifactions for various modulii/dilaton [18], coming from fluxes,
(7)
with an equation of state, pi = ωi ρi . Note that although ρ˜i is the energy density we measure, it is the bare energy density, ρi , which obeys the usual equation of state. This can be seen, for example, in the case of a non-relativistic dust. The density is given by e2μi Q mi N/V ≡ e2μi Q ρi , where N is the total number of particles, a constant, and V is the volume of the Universe, which redshifts as a −3 leading to ρi ∼ a −3 . From Eq. (7) we obtain the standard result ρi = ρ0i (a/a0 )−3(1+ωi ) . We will first work within the approximation that there are only two dominating components in the Universe, the dark energy determined by Q and the matter component. If μ’s and β have the same sign (we will furnish examples later where this can happen) then the effective potential for the Q field, see Eq. (4), provides a dynamical stabilizing mechanism because the two overall exponents differ in signs. In this case the Q field will track the minimum of the potential, see Eq. (4), as it slowly evolves due to the redshifting of the matter contributions. For our purpose we may assume that the field evolves adiabatically, which holds when, β(μi + β) > 3(1 + ωi )/4, a generalization of the result derived in [20] for μ = 0. Otherwise, Q rolls too slowly along V (Q) and cannot keep up with the shifting of the minimum. The evolution of Q can be obtained
T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441 (Q) = 0 implies, in terms of ρ. Veff
e2Q = (βV0 /μi ρi )1/(μi +β) .
(8)
After some algebra we can evaluate the fraction of the matter density over the critical energy density of the Universe Ωi = ρ˜i /ρ˜t = ρ˜i /(ρ˜d + ρ˜i ) = β/(μi + β).
(9)
We can also estimate the Hubble expansion rate, H 2 = V0 (1 + β/μi )(βV0 /μi )−β/(μi +β) ρ β/(μi +β) ,
which scales like H ∼ a −(3/2)(1+ωi )β/(μi +β) . The above equation can be solved to give a(t) = a0 (t/t0 )(2/3β)[(μi +β)/(1+ωi )] .
(11)
In a matter (m) dominated epoch with ωm = 0, we obtain an interesting relationship in order to have an accelerated expansion, μm /β > 1/2.
(12)
Consider now the present abundance for the cold dark matter (c), Ωc0 ∼ 0.27; from Eq. (9) we obtain, μc /β ∼ 2.7, indicating an accelerated expansion. In other words μc /β ∼ 2.7 can explain why ρ˜d0 ∼ 2.7ρ˜c ∼ 10−120 Mp4 , as well as the current acceleration. During the radiation era one can ignore the matter energy density and the evolution of the Q field is such that ρ˜d tracks ρ˜r [20], ρ˜Q =
4 ρ˜r , 4β 2 − 3
the acceleration epoch, we must study the three component evolution involving ρ˜m , ρ˜c and ρ˜d . The evolution equations are now given by H 2 = 1/3Mp2 (ρ˜c + ρ˜m + ρ˜d ), (14) ¨ + 3H Q˙ = −2β ρ˜d + 2μm ρ˜m + 2μc ρ˜c . Q
(13)
and thus the scale factor goes as a(t) ∼ t 1/2 just as in the case of standard radiation. From the above analysis it may appear that the moment we enter the matter dominated era the Universe starts to accelerate, but this is not correct. The baryonic matter (b) and cold dark matter (CDM) redshift differently since μc μb indicating that baryons were dominating over the CDM not so long ago and since β μb during this phase we do not have any acceleration. One can compute approximately when the CDM began to dominate over the baryons, which will also roughly correspond to the beginning of the accelerated expansion, 1 + zacc = (Ωc0 /Ωb0 )[(μc +β)/3μc ] . Taking μc /β ∼ 2.7, as obtained above, we find zacc ≈ 1.8. Note that this result is determined solely by Ωd0 /Ωc0 and Ωb0 /Ωc0 ! However in this setup we end up with more baryons than CDM at early epochs, say during the CMB formation. This issue can be addressed by realizing that one can have two competing dark matter candidates; one with ρ˜w , weakly coupled to Q, say μw = μb , and the other ρ˜c strongly coupled to dark energy with μc as before. In this scenario the “why now” problem is answered by simply postulating that today is the epoch when the strongly coupled dark matter is catching up with the weakly coupled matter, ρ˜m (= ρ˜w + ρ˜b ). Note that since μw = μb , ρ˜w redshifts according to the baryons, and hence maintains a constant ratio with ρ˜b throughout the evolution until very recently when the ratio increases slightly due to the emergence of ρ˜c . To estimate how much this ratio changes and its relationship with
(15)
In the adiabatic approximation the latter implies −2β ρ˜d + 2μm ρ˜m + 2μc ρ˜c = 0.
(10)
439
(16)
Notice, since μc μm the third term starts to dominate over the second long before ρ˜c ∼ ρ˜m , so that the dark energy starts to track ρ˜c long before today. Then from Eq. (14), we find that the Universe enters the acceleration phase approximately when ρ˜c + ρ˜d = (1 + μc /β)ρ˜c becomes equal to ρ˜m , so that [(μc +β)/3μc ] . 1 + zacc = (1 + μc /β)Ωc0 /Ωm0 (17) From (17) one can compute Ωc0 /Ωm0 for a given zacc and μc /β. On the other hand, since during CMB ρ˜c was negligible the baryon to dark matter ratio that WMAP data measures is just ρ˜b /ρ˜m = Ωb0 /Ωm0 ≈ 0.17 [3], as this remains constant through out the evolution. Thus for a given zacc and μc /β one can find the current baryon to dark matter ratio Ωb0 /(Ωm0 + Ωc0 ). For zacc = 0.5, μc /β ∼ 3 gives Ωb0 h2 = 0.014 while μc /β ∼ 9 gives Ωb0 h2 = 0.017. Observe that these estimates are smaller than the CMB estimate of Ωb0 h2 = 0.0224 which assumes that the baryon to dark matter ratio has not changed since CMB. Thus our model might be able to explain why the baryonic abundances coming from BBN considerations, 0.009 < Ωb0 h2 < 0.0223 [21], which are based on current observational data (and hence measures the true baryon to dark matter ratio, including ρ˜c ) tend to give slightly lower values than CMB. BBN constraints also clearly imply that zacc has to be quite recent. For example, for zacc = 1, μc /β ∼ 3 yields Ωb0 h2 = 0.01 which is just within the BBN range. In other words, to ensure that the baryon to dark matter ratio has not changed significantly since CMB, the acceleration epoch has to be quite recent. As a final check let us estimate the equation of state for the dark energy, ωQ , which is defined via V (Q) ∼ a −3(1+ωQ ) . From Eq. (8) we find, ωQ = −(μc /β − ωc )/(1 + μc /β).
(18)
In particular for ωc = 0, we find, for instance, μc /β ∼ 3.5 gives ωQ ≈ −0.78 while μc /β ∼ 9 yields ωQ ≈ −0.9, in good agreement with the current observations [2]. However, one really has to do a numerical fit to the supernova data to ensure the viability of our model and also check that the parameter space in μ − β is compatible with CMB data. Such a work is currently underway [22]. Let us now focus on some of the other important observational constraints. Although the homogeneous Q field is stabilized, the Q quanta are virtually massless, i.e. mQ ∼ 10−33 eV, therefore mediating a fifth force which may violate the equivalence principle. There are also constraints from observations of time variation of fine structure constant and the Newton’s constants (or equivalently the time
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T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441
variation in masses). All of them typically give bounds of the order μ < 10−4 [15,17], for ordinary matter and radiation. Note that the cold dark matter coupling to Q is unconstrained from the above mentioned experiments. In our model we require μc > β/2 > μr , μb and β 1, therefore even if μr,b ∼ 10−4 , the above relationship can be satisfied. The running of the coupling constants due to stringy loop corrections may naturally lead to such small values for μ’s [11]. On the other hand, recently it has been also realized that even if μb ’s are significantly larger (∼ 0.1), it might avoid any detection in the fifth force experiments due to “chameleon” type mechanisms, see [23]. Also, what is crucial for our mechanism is that the coupling exponent for the strongly coupled dark matter, μ, have a sign opposite to that of β. For the purpose of illustration let us consider a (D + 4)dimensional supergravity/string theory model where branes are wrapping all the internal dimensions. They appear as point like objects to us and hence redshift as a −3 , just like non-relativistic dust, and indeed they have been proposed as dark matter candidates, see Ref. [24]. From the Brandenberger/Vafa mechanism [25] it is intuitively clear that since the branes try to compress the extra dimensions that they are wrapping, the coupling of the brane-gas to the internal volume will have a positive (confining) exponent. In this simple configuration after making suitable conformal redefinitions one can compute the effective four-dimensional brane energy density to be (in [26] this is derived for strings)
D , T00 = ρ˜ = ρ0 a −3 e2μv with 2μ = (19) 2(D + 2) and indeed the coupling exponent comes out to be positive. Although in this simple example the exponent is smaller than what cosmological phenomenology seems to be requiring, nevertheless, one can imagine more interesting brane configurations (they can even wrap one of the ordinary space dimensions and hence appear as cosmic strings) and compactification schemes where one might realize the desired values. Of course, the success of our model relies on a very small energy density of the strongly coupled matter at early Universe, and unless one has a concrete way of realizing this, we cannot claim to have solved the cosmic coincidence problem. Nevertheless, this may be a step in the right direction. Several other interesting questions remain, such as the fluctuations of Q during its slow roll evolution, the origin of the two types of CDM and their role in galaxy formation. It is also important to go beyond the two component approximation and perform a numerical simulation to determine the dynamics exactly. These questions we leave for future investigation. Nevertheless we may conclude that our result hints at the possibility of connecting string/supergravity theory to explaining the current discrepancy between CMB and BBN observations of baryons, recent acceleration due to the dark energy contribution and the observed small cosmological constant.
Acknowledgements This work is supported in part by the NSERC. We would like to thank Luca Amendola, Guy Moore, Tuomas Multamäki and Andrew Liddle for discussions. In particular, Luca Amendola pointed out to us a very similar work of his (last reference in [13]) that had appeared a week before our Letter was put in the arxiv. References [1] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. [2] S. Perlmutter, et al., Supernova Cosmology Project Collaboration, Astrophys. J. 517 (1999) 565, astro-ph/9812133; A.G. Riess, et al., Supernova Search Team Collaboration, Astron. J. 116 (1998) 1009, astro-ph/9805201; A.G. Riess, et al., Supernova Search Team Collaboration, Astrophys. J. 607 (2004) 665, astro-ph/0402512. [3] D.N. Spergel, et al., WMAP Collaboration, Astrophys. J. Suppl. 148 (2003) 175, astro-ph/0302209. [4] C. Wetterich, Nucl. Phys. B 302 (1988) 668; R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582, astro-ph/9708069. [5] C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B 458 (1999) 209, hep-th/9904075. [6] P. Jaikumar, A. Mazumdar, Phys. Rev. Lett. 90 (2003) 191301, hepph/0301086. [7] K. Freese, M. Lewis, Phys. Lett. B 540 (2002) 1, astro-ph/0201229. [8] G.R. Dvali, G. Gabadadze, Phys. Rev. D 63 (2001) 065007, hepth/0008054. [9] R.H. Brandenberger, hep-th/0210165; R. Brandenberger, A. Mazumdar, JCAP 0408 (2004) 015, hep-th/0402205; R. Brandenberger, A. Mazumdar, JCAP 0408 (2004) 015, hep-th/0402205. [10] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Univ. Press, Cambridge, 1987. [11] T. Damour, A.M. Polyakov, Nucl. Phys. B 423 (1994) 532, hepth/9401069. [12] C. Wetterich, Astron. Astrophys. 301 (1995) 321, hep-th/9408025. [13] L. Amendola, Phys. Rev. D 62 (2000) 043511, astro-ph/9908023; L. Amendola, C. Quercellini, Phys. Rev. D 68 (2003) 023514, astroph/0303228; L. Amendola, D. Tocchini-Valentini, Phys. Rev. D 64 (2001) 043509, astro-ph/0011243; L. Amendola, M. Gasperini, F. Piazza, astro-ph/0407573. [14] K. Enqvist, A. Mazumdar, Phys. Rep. 380 (2003) 99, hep-ph/0209244. [15] C.M. Will, Living Rev. Relativ. 4 (2001) 4, gr-qc/0103036. [16] T. Appelquist, A. Chodos, P.G.O. Freund, Introduction to Modern Kaluza– Klein Theories, Addison–Wesley, Reading, MA, 1987; M.J. Duff, B.E.W. Nilsson, C.N. Pope, Phys. Rep. 130 (1986) 1; A. Mazumdar, R.N. Mohapatra, A. Perez-Lorenzana, JCAP 0406 (2004) 004, hep-ph/0310258. [17] C.J.A. Martins, A. Melchiorri, G. Rocha, R. Trotta, P.P. Avelino, P. Viana, Phys. Lett. B 585 (2004) 29, astro-ph/0302295. [18] M.S. Bremer, M.J. Duff, H. Lu, C.N. Pope, K.S. Stelle, Nucl. Phys. B 543 (1999) 321, hep-th/9807051; T. Biswas, P. Jaikumar, JHEP 0408 (2004) 053, hep-th/0407063; T. Biswas, P. Jaikumar, Int. J. Mod. Phys. A 19 (2004) 5443; T. Biswas, P. Jaikumar, Phys. Rev. D 70 (2004) 044011, hep-th/0310172; A.R. Frey, A. Mazumdar, Phys. Rev. D 67 (2003) 046006, hepth/0210254. [19] G. Huey, P.J. Steinhardt, B.A. Ovrut, D. Waldram, Phys. Lett. B 476 (2000) 379, hep-th/0001112. [20] E.J. Copeland, A.R. Liddle, D. Wands, Phys. Rev. D 57 (1998) 4686, grqc/9711068.
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[21] S. Sarkar, astro-ph/0205116. [22] T. Biswas, R. Brandenberger, A. Mazumdar, T. Multamaki, hepth/0507199. [23] D.F. Mota, J.D. Barrow, Phys. Lett. B 581 (2004) 141, astro-ph/0306047; D.F. Mota, J.D. Barrow, Mon. Not. R. Astron. Soc. 349 (2004) 281, astroph/0309273;
441
J. Khoury, A. Weltman, Phys. Rev. D 69 (2004) 044026, astroph/0309411. [24] G. Shiu, L.T. Wang, Phys. Rev. D 69 (2004) 126007, hep-ph/0311228. [25] R.H. Brandenberger, C. Vafa, Nucl. Phys. B 316 (1989) 391. [26] T. Battefeld, S. Watson, JCAP 0406 (2004) 001.
A stringy origin of the recent acceleration Tirthabir Biswas ∗ , Anupam Mazumdar CHEP, McGill University, Montréal, QC, H3A 2T8, Canada Received 22 July 2005; received in revised form 1 February 2006; accepted 6 February 2006 Available online 13 February 2006 Editor: M. Cvetiˇc
Abstract Inspired by the current observations that the ratio of the abundances of dark energy and matter density is close to one, we provide a string inspired phenomenological model where we try to explain this order one ratio, smallness of the cosmological constant, and also the recent cosmic acceleration. We observe that any effective theory motivated by higher-dimensional physics provides radion/dilaton couplings to the standard model and the dark matter component with different strengths. Provided radion/dilaton is a dynamical field we then show that the dark energy component tracks matter density and can start to dominate very recently. © 2006 Elsevier B.V. All rights reserved.
The cosmological constant problem is one of the most difficult problems of theoretical physics. What requires an explanation is that, the observed energy content of the Universe, ∼ 4 × 10−47 (GeV)4 , is many orders of magnitude smaller than that of the theoretical prediction for the cosmological constant energy density alone. The mismatch is of order 10−120 × Mp4 , where Mp ∼ 2.4 × 1018 GeV. One would naively expect that even if the bare cosmological constant can be made to vanish, quantum corrections would eventually lead to quadratic divergences ∼ Mp4 (supersymmetry only ameliorates the mismatch). Thus the puzzle is “why the bare cosmological constant is so small and moreover, stable under quantum corrections” [1]. There is also a kind of a coincidence problem, sometimes dubbed as a why now problem. Recent observations from supernovae [2] and the cosmic microwave background (CMB) anisotropy measurements [3] suggest that the majority of the energy density ∼ 70% is in the form of dark energy, whose constituent is largely unknown, but usually believed to be the cosmological constant with an equation of state ω = −1, which is also responsible for the current acceleration. In this respect not only the cosmological constant is small, but it also happens to be dominating the energy density of the Universe. A priori,
* Corresponding author.
E-mail address: [email protected] (T. Biswas). 0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.02.015
the physics behind the value of the cosmological constant and its “evolution” need not be related to the redshifting of matter/radiation density. Is it then just by coincidence that today ρΛ is close to the value ρm or are there deeper connections between the two? A related question would be why the cosmological constant is dominating right now and for example not during the big bang nucleosynthesis (BBN) at a temperature T ∼ 1 MeV? Both the smallness and the why now questions can be answered in part if we believe that the physics of the dark energy is somehow related to the rest of the energy density of our Universe. Attempts have been made to construct such tracking mechanisms in single and multi-fields, such as dynamical quintessence [4], k-essence [5] type models, non-Abelian vacuum structure with non-vanishing winding modes [6], modified Friedmann equation at late times [7], or large distances [8], or due to inflationary backreaction [9]. In this Letter we propose a simple model which can arise naturally from the compactifications of a higher-dimensional theory. For simplicity we assume that all the moduli/dilaton fields have been stabilized at higher scales except a single linear combination, Q, which has a runaway potential, see [10]. In an effective four-dimensional theory the Q field will couple to the Standard Model (SM) degrees of freedom as well as the dark matter. We will show that by virtue of this coupling to the matter fields “the effective potential” for Q can have a
438
T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441
local minimum which can track the matter density of the Universe. Further observe that the Q field in general is expected to have different couplings to different species of fermions (f), scalars (s), and gauge bosons (r) by virtue of its running below the string compactification scale (see for example [11]), and depending on the strength of this coupling our Universe will either accelerate or not. In particular it will be crucial in our model to have only a small component of dark matter to couple strongly to Q, while the rest of the matter couple very weakly. This is one of the crucial differences as compared to some of the earlier models [12,13] that were considered where the Q field couples uniformly to the entire dark matter sector. In fact it is this difference, along with the adiabatic tracking mechanism that help us in explaining the current acceleration without making the couplings time-dependent [13]. In our analysis we also assume that at least some relevant (beyond SM) degrees of freedom have already been thermally/non-thermally excited in the early Universe, e.g. see [14]. Let us imagine that our world was originally higher-dimensional, such that the three spatial dimensions, along with the origin of the SM are all due to some interesting compactification of the extra spatial dimensions. After dimensional reduction, in the Einstein frame we typically obtain a scalar-tensor action [15] of the form S = Sgrav+Q + Sfermions + Sgauge bosons + Sscalars , where √ Sgrav+Q = Mp2 d 4 x −g R/2 − (∂Q)2 /2 − V (Q) , Sfermions =
√ d 4 x −g ψ¯ f ∇ / + imf Bf (Q) ψf ,
Sgauge bosons = (1/4πα0 )
d 4x
√
curvature of internal manifold, etc. We therefore assume a simple form for the potential, V (Q) = V0 e−2βQ .
(2)
Henceforth we identify the dark energy density (d) as ˙ 2 /2 + V (Q). ρ˜d = Q
(3)
Note however, that Q effectively obtains an additional contribution to its potential via its coupling to matter, e2μi Q ρi . Veff (Q) = V0 e−2βQ + (4) i=s,f
The “bare density”, ρi ≡ e−2μi Q ρ˜i , is defined such that it does not depend on Q; here ρ˜i is the observed energy density of the ith component. For simplicity we may assume that radiation from gauge bosons do not contribute to an effective potential for Q. This is strictly true if on average electric field and the magnetic field vanishes. There could be other forms of radiative matter which can induce an effective coupling as discussed for example in [19], but we do not consider them here. Given all the components, we can write down the Friedmann equation for a Robertson–Walker metric, ρ˜I , H 2 = 1/3Mp2 ρ˜t = 1/3Mp2 I =s,f,r,d
2μi Q 2 −2βQ ˙ e ρi + Q /2 + V0 e . = (1/3)
(5)
i=r,s,f
The evolution equations for Q, ρi read as, ¨ + 3H Q ˙ = 2 βV0 e−2βQ − Q μi e2μi Q ρi ,
−gBr (Q)F 2 ,
(6)
i=s,f
Sscalars =
ρ˙i + 3H (pi + ρi ) = 0
√ d x −g −(∂φs )2 /2 − m2s Bs2 (Q)φs2 . 4
Although in general Bi (Q) could be an arbitrary functions, in supergravity/String theory reductions one most commonly obtains exponential couplings [16] and hence we restrict our attention to the case when Bi (Q) = e2μi Q ,
(1)
where i = {f, r, s}, runs over fermionic, radiation and scalar species, respectively. The radiation and baryonic matter content of the Universe is determined by the SM degrees of freedom, while the cold dark matter component arises from either scalars, fermions or gauge bosons of SM or beyond the SM gauge group. Note that due to the couplings various masses ∼ mi e2μi Q and fine structure constants ∼ α0 e−2μr Q now depend on Q. We will see later that observational data on variation of masses and fine structure constants, as well as fifth force experiments constrain the μ’s for SM particles to be very small < 10−3 –10−4 [15,17], and henceforth we will assume this. As far as the potential is concerned one often encounters exponential run away types in M-theory/supergravity compactifactions for various modulii/dilaton [18], coming from fluxes,
(7)
with an equation of state, pi = ωi ρi . Note that although ρ˜i is the energy density we measure, it is the bare energy density, ρi , which obeys the usual equation of state. This can be seen, for example, in the case of a non-relativistic dust. The density is given by e2μi Q mi N/V ≡ e2μi Q ρi , where N is the total number of particles, a constant, and V is the volume of the Universe, which redshifts as a −3 leading to ρi ∼ a −3 . From Eq. (7) we obtain the standard result ρi = ρ0i (a/a0 )−3(1+ωi ) . We will first work within the approximation that there are only two dominating components in the Universe, the dark energy determined by Q and the matter component. If μ’s and β have the same sign (we will furnish examples later where this can happen) then the effective potential for the Q field, see Eq. (4), provides a dynamical stabilizing mechanism because the two overall exponents differ in signs. In this case the Q field will track the minimum of the potential, see Eq. (4), as it slowly evolves due to the redshifting of the matter contributions. For our purpose we may assume that the field evolves adiabatically, which holds when, β(μi + β) > 3(1 + ωi )/4, a generalization of the result derived in [20] for μ = 0. Otherwise, Q rolls too slowly along V (Q) and cannot keep up with the shifting of the minimum. The evolution of Q can be obtained
T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441 (Q) = 0 implies, in terms of ρ. Veff
e2Q = (βV0 /μi ρi )1/(μi +β) .
(8)
After some algebra we can evaluate the fraction of the matter density over the critical energy density of the Universe Ωi = ρ˜i /ρ˜t = ρ˜i /(ρ˜d + ρ˜i ) = β/(μi + β).
(9)
We can also estimate the Hubble expansion rate, H 2 = V0 (1 + β/μi )(βV0 /μi )−β/(μi +β) ρ β/(μi +β) ,
which scales like H ∼ a −(3/2)(1+ωi )β/(μi +β) . The above equation can be solved to give a(t) = a0 (t/t0 )(2/3β)[(μi +β)/(1+ωi )] .
(11)
In a matter (m) dominated epoch with ωm = 0, we obtain an interesting relationship in order to have an accelerated expansion, μm /β > 1/2.
(12)
Consider now the present abundance for the cold dark matter (c), Ωc0 ∼ 0.27; from Eq. (9) we obtain, μc /β ∼ 2.7, indicating an accelerated expansion. In other words μc /β ∼ 2.7 can explain why ρ˜d0 ∼ 2.7ρ˜c ∼ 10−120 Mp4 , as well as the current acceleration. During the radiation era one can ignore the matter energy density and the evolution of the Q field is such that ρ˜d tracks ρ˜r [20], ρ˜Q =
4 ρ˜r , 4β 2 − 3
the acceleration epoch, we must study the three component evolution involving ρ˜m , ρ˜c and ρ˜d . The evolution equations are now given by H 2 = 1/3Mp2 (ρ˜c + ρ˜m + ρ˜d ), (14) ¨ + 3H Q˙ = −2β ρ˜d + 2μm ρ˜m + 2μc ρ˜c . Q
(13)
and thus the scale factor goes as a(t) ∼ t 1/2 just as in the case of standard radiation. From the above analysis it may appear that the moment we enter the matter dominated era the Universe starts to accelerate, but this is not correct. The baryonic matter (b) and cold dark matter (CDM) redshift differently since μc μb indicating that baryons were dominating over the CDM not so long ago and since β μb during this phase we do not have any acceleration. One can compute approximately when the CDM began to dominate over the baryons, which will also roughly correspond to the beginning of the accelerated expansion, 1 + zacc = (Ωc0 /Ωb0 )[(μc +β)/3μc ] . Taking μc /β ∼ 2.7, as obtained above, we find zacc ≈ 1.8. Note that this result is determined solely by Ωd0 /Ωc0 and Ωb0 /Ωc0 ! However in this setup we end up with more baryons than CDM at early epochs, say during the CMB formation. This issue can be addressed by realizing that one can have two competing dark matter candidates; one with ρ˜w , weakly coupled to Q, say μw = μb , and the other ρ˜c strongly coupled to dark energy with μc as before. In this scenario the “why now” problem is answered by simply postulating that today is the epoch when the strongly coupled dark matter is catching up with the weakly coupled matter, ρ˜m (= ρ˜w + ρ˜b ). Note that since μw = μb , ρ˜w redshifts according to the baryons, and hence maintains a constant ratio with ρ˜b throughout the evolution until very recently when the ratio increases slightly due to the emergence of ρ˜c . To estimate how much this ratio changes and its relationship with
(15)
In the adiabatic approximation the latter implies −2β ρ˜d + 2μm ρ˜m + 2μc ρ˜c = 0.
(10)
439
(16)
Notice, since μc μm the third term starts to dominate over the second long before ρ˜c ∼ ρ˜m , so that the dark energy starts to track ρ˜c long before today. Then from Eq. (14), we find that the Universe enters the acceleration phase approximately when ρ˜c + ρ˜d = (1 + μc /β)ρ˜c becomes equal to ρ˜m , so that [(μc +β)/3μc ] . 1 + zacc = (1 + μc /β)Ωc0 /Ωm0 (17) From (17) one can compute Ωc0 /Ωm0 for a given zacc and μc /β. On the other hand, since during CMB ρ˜c was negligible the baryon to dark matter ratio that WMAP data measures is just ρ˜b /ρ˜m = Ωb0 /Ωm0 ≈ 0.17 [3], as this remains constant through out the evolution. Thus for a given zacc and μc /β one can find the current baryon to dark matter ratio Ωb0 /(Ωm0 + Ωc0 ). For zacc = 0.5, μc /β ∼ 3 gives Ωb0 h2 = 0.014 while μc /β ∼ 9 gives Ωb0 h2 = 0.017. Observe that these estimates are smaller than the CMB estimate of Ωb0 h2 = 0.0224 which assumes that the baryon to dark matter ratio has not changed since CMB. Thus our model might be able to explain why the baryonic abundances coming from BBN considerations, 0.009 < Ωb0 h2 < 0.0223 [21], which are based on current observational data (and hence measures the true baryon to dark matter ratio, including ρ˜c ) tend to give slightly lower values than CMB. BBN constraints also clearly imply that zacc has to be quite recent. For example, for zacc = 1, μc /β ∼ 3 yields Ωb0 h2 = 0.01 which is just within the BBN range. In other words, to ensure that the baryon to dark matter ratio has not changed significantly since CMB, the acceleration epoch has to be quite recent. As a final check let us estimate the equation of state for the dark energy, ωQ , which is defined via V (Q) ∼ a −3(1+ωQ ) . From Eq. (8) we find, ωQ = −(μc /β − ωc )/(1 + μc /β).
(18)
In particular for ωc = 0, we find, for instance, μc /β ∼ 3.5 gives ωQ ≈ −0.78 while μc /β ∼ 9 yields ωQ ≈ −0.9, in good agreement with the current observations [2]. However, one really has to do a numerical fit to the supernova data to ensure the viability of our model and also check that the parameter space in μ − β is compatible with CMB data. Such a work is currently underway [22]. Let us now focus on some of the other important observational constraints. Although the homogeneous Q field is stabilized, the Q quanta are virtually massless, i.e. mQ ∼ 10−33 eV, therefore mediating a fifth force which may violate the equivalence principle. There are also constraints from observations of time variation of fine structure constant and the Newton’s constants (or equivalently the time
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variation in masses). All of them typically give bounds of the order μ < 10−4 [15,17], for ordinary matter and radiation. Note that the cold dark matter coupling to Q is unconstrained from the above mentioned experiments. In our model we require μc > β/2 > μr , μb and β 1, therefore even if μr,b ∼ 10−4 , the above relationship can be satisfied. The running of the coupling constants due to stringy loop corrections may naturally lead to such small values for μ’s [11]. On the other hand, recently it has been also realized that even if μb ’s are significantly larger (∼ 0.1), it might avoid any detection in the fifth force experiments due to “chameleon” type mechanisms, see [23]. Also, what is crucial for our mechanism is that the coupling exponent for the strongly coupled dark matter, μ, have a sign opposite to that of β. For the purpose of illustration let us consider a (D + 4)dimensional supergravity/string theory model where branes are wrapping all the internal dimensions. They appear as point like objects to us and hence redshift as a −3 , just like non-relativistic dust, and indeed they have been proposed as dark matter candidates, see Ref. [24]. From the Brandenberger/Vafa mechanism [25] it is intuitively clear that since the branes try to compress the extra dimensions that they are wrapping, the coupling of the brane-gas to the internal volume will have a positive (confining) exponent. In this simple configuration after making suitable conformal redefinitions one can compute the effective four-dimensional brane energy density to be (in [26] this is derived for strings)
D , T00 = ρ˜ = ρ0 a −3 e2μv with 2μ = (19) 2(D + 2) and indeed the coupling exponent comes out to be positive. Although in this simple example the exponent is smaller than what cosmological phenomenology seems to be requiring, nevertheless, one can imagine more interesting brane configurations (they can even wrap one of the ordinary space dimensions and hence appear as cosmic strings) and compactification schemes where one might realize the desired values. Of course, the success of our model relies on a very small energy density of the strongly coupled matter at early Universe, and unless one has a concrete way of realizing this, we cannot claim to have solved the cosmic coincidence problem. Nevertheless, this may be a step in the right direction. Several other interesting questions remain, such as the fluctuations of Q during its slow roll evolution, the origin of the two types of CDM and their role in galaxy formation. It is also important to go beyond the two component approximation and perform a numerical simulation to determine the dynamics exactly. These questions we leave for future investigation. Nevertheless we may conclude that our result hints at the possibility of connecting string/supergravity theory to explaining the current discrepancy between CMB and BBN observations of baryons, recent acceleration due to the dark energy contribution and the observed small cosmological constant.
Acknowledgements This work is supported in part by the NSERC. We would like to thank Luca Amendola, Guy Moore, Tuomas Multamäki and Andrew Liddle for discussions. In particular, Luca Amendola pointed out to us a very similar work of his (last reference in [13]) that had appeared a week before our Letter was put in the arxiv. References [1] S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. [2] S. Perlmutter, et al., Supernova Cosmology Project Collaboration, Astrophys. J. 517 (1999) 565, astro-ph/9812133; A.G. Riess, et al., Supernova Search Team Collaboration, Astron. J. 116 (1998) 1009, astro-ph/9805201; A.G. Riess, et al., Supernova Search Team Collaboration, Astrophys. J. 607 (2004) 665, astro-ph/0402512. [3] D.N. Spergel, et al., WMAP Collaboration, Astrophys. J. Suppl. 148 (2003) 175, astro-ph/0302209. [4] C. Wetterich, Nucl. Phys. B 302 (1988) 668; R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582, astro-ph/9708069. [5] C. Armendariz-Picon, T. Damour, V. Mukhanov, Phys. Lett. B 458 (1999) 209, hep-th/9904075. [6] P. Jaikumar, A. Mazumdar, Phys. Rev. Lett. 90 (2003) 191301, hepph/0301086. [7] K. Freese, M. Lewis, Phys. Lett. B 540 (2002) 1, astro-ph/0201229. [8] G.R. Dvali, G. Gabadadze, Phys. Rev. D 63 (2001) 065007, hepth/0008054. [9] R.H. Brandenberger, hep-th/0210165; R. Brandenberger, A. Mazumdar, JCAP 0408 (2004) 015, hep-th/0402205; R. Brandenberger, A. Mazumdar, JCAP 0408 (2004) 015, hep-th/0402205. [10] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, vol. 2: Loop Amplitudes, Anomalies and Phenomenology, Cambridge Univ. Press, Cambridge, 1987. [11] T. Damour, A.M. Polyakov, Nucl. Phys. B 423 (1994) 532, hepth/9401069. [12] C. Wetterich, Astron. Astrophys. 301 (1995) 321, hep-th/9408025. [13] L. Amendola, Phys. Rev. D 62 (2000) 043511, astro-ph/9908023; L. Amendola, C. Quercellini, Phys. Rev. D 68 (2003) 023514, astroph/0303228; L. Amendola, D. Tocchini-Valentini, Phys. Rev. D 64 (2001) 043509, astro-ph/0011243; L. Amendola, M. Gasperini, F. Piazza, astro-ph/0407573. [14] K. Enqvist, A. Mazumdar, Phys. Rep. 380 (2003) 99, hep-ph/0209244. [15] C.M. Will, Living Rev. Relativ. 4 (2001) 4, gr-qc/0103036. [16] T. Appelquist, A. Chodos, P.G.O. Freund, Introduction to Modern Kaluza– Klein Theories, Addison–Wesley, Reading, MA, 1987; M.J. Duff, B.E.W. Nilsson, C.N. Pope, Phys. Rep. 130 (1986) 1; A. Mazumdar, R.N. Mohapatra, A. Perez-Lorenzana, JCAP 0406 (2004) 004, hep-ph/0310258. [17] C.J.A. Martins, A. Melchiorri, G. Rocha, R. Trotta, P.P. Avelino, P. Viana, Phys. Lett. B 585 (2004) 29, astro-ph/0302295. [18] M.S. Bremer, M.J. Duff, H. Lu, C.N. Pope, K.S. Stelle, Nucl. Phys. B 543 (1999) 321, hep-th/9807051; T. Biswas, P. Jaikumar, JHEP 0408 (2004) 053, hep-th/0407063; T. Biswas, P. Jaikumar, Int. J. Mod. Phys. A 19 (2004) 5443; T. Biswas, P. Jaikumar, Phys. Rev. D 70 (2004) 044011, hep-th/0310172; A.R. Frey, A. Mazumdar, Phys. Rev. D 67 (2003) 046006, hepth/0210254. [19] G. Huey, P.J. Steinhardt, B.A. Ovrut, D. Waldram, Phys. Lett. B 476 (2000) 379, hep-th/0001112. [20] E.J. Copeland, A.R. Liddle, D. Wands, Phys. Rev. D 57 (1998) 4686, grqc/9711068.
T. Biswas, A. Mazumdar / Physics Letters B 634 (2006) 437–441
[21] S. Sarkar, astro-ph/0205116. [22] T. Biswas, R. Brandenberger, A. Mazumdar, T. Multamaki, hepth/0507199. [23] D.F. Mota, J.D. Barrow, Phys. Lett. B 581 (2004) 141, astro-ph/0306047; D.F. Mota, J.D. Barrow, Mon. Not. R. Astron. Soc. 349 (2004) 281, astroph/0309273;
441
J. Khoury, A. Weltman, Phys. Rev. D 69 (2004) 044026, astroph/0309411. [24] G. Shiu, L.T. Wang, Phys. Rev. D 69 (2004) 126007, hep-ph/0311228. [25] R.H. Brandenberger, C. Vafa, Nucl. Phys. B 316 (1989) 391. [26] T. Battefeld, S. Watson, JCAP 0406 (2004) 001.