#
Mctp-02-02

Cardassian Expansion: a Model in which the Universe is Flat,
Matter Dominated, and Accelerating

###### Abstract

A modification to the Friedmann Robertson Walker equation is proposed in which the universe is flat, matter dominated, and accelerating. An additional term, which contains only matter or radiation (no vacuum contribution), becomes the dominant driver of expansion at a late epoch of the universe. During the epoch when the new term dominates, the universe accelerates; we call this period of acceleration the Cardassian era. The universe can be flat and yet consist of only matter and radiation, and still be compatible with observations. The energy density required to close the universe is much smaller than in a standard cosmology, so that matter can be sufficient to provide a flat geometry. The new term required may arise, e.g., as a consequence of our observable universe living as a 3-dimensional brane in a higher dimensional universe. The Cardassian model survives several observational tests, including the cosmic background radiation, the age of the universe, the cluster baryon fraction, and structure formation.

###### pacs:

Recent observations of Type IA Supernovae [1, 2] as well as concordance with other observations (including the microwave background and galaxy power spectra) indicate that the universe is accelerating. Many authors have explored a cosmological constant, a decaying vacuum energy [3, 4], and quintessence [5, 6, 7] as possible explanations for such an acceleration.

Here we propose an alternative which invokes no vacuum energy whatsoever. In our model the universe is flat and yet consists only of matter and radiation. Pure matter (or radiation) alone can drive an accelerated expansion if the Friedmann Robertson Walker (FRW) equation is modified by the addition of a new term on the right hand side as follows:

(1) |

where is the Hubble constant (as a function of time), is the scale factor of the universe, the energy density contains only ordinary matter and radiation, and we will take

(2) |

In the usual FRW equation . To be consistent with the usual FRW result, we take . We note here that the geometry is flat, as required by measurements of the cosmic background radiation [8], so that there are no curvature terms in the equation. There is no vacuum term in the equation. This paper does not address the cosmological constant () problem; we simply set .

In this paper, we first study the phenomenology of the ansatz in
Eq.(1), and then turn to a discussion of the origin of
this equation^{1}^{1}1As discussed below, we were
motivated to study an equation of this form
by work of Chung and Freese [10] who showed that terms
of the form can generically appear in the FRW equation
as a consequence of embedding our observable universe as a brane
in extra dimensions.. Directions for a future search for a fundamental
theory will be discussed.

The Role of the Cardassian Term^{2}^{2}2The name Cardassian
refers to a humanoid race in Star Trek whose goal is to take over the universe,
i.e., accelerated expansion. This race looks foreign to us
and yet is made entirely of matter.:
The new term in the equation (the second term on the right
hand side) is initially negligible. It only
comes to dominate recently, at the redshift indicated
by the supernovae observations. Once the second term dominates,
it causes the universe to accelerate. We can consider the contribution
of ordinary matter, with

(3) |

to this new term. Once the new term dominates the right hand side of the equation, we have accelerated expansion. When the new term is so large that the ordinary first term can be neglected, we find

(4) |

so that the expansion is superluminal (accelerated) for . As examples, for we have ; for we have ; and for we have . The case of produces a term in the FRW equation ; such a term looks similar to a curvature term but is generated here by matter in a universe with a flat geometry. Note that for the acceleration is constant, for the acceleration is diminishing in time, while for the acceleration is increasing (the cosmic jerk).

The second term starts to dominate at a redshift when , i.e., when

(8) |

We have two parameters in the model: and , or, equivalently, and . Note that here is chosen to make the second term kick in at the right time to explain the observations. As yet we have no explanation of the coincidence problem; i.e., we have no explanation for the timing of . Such an explanation would arise if we had a reason for the required mass scale of . The parameter has units of mass. Later, we will discuss the origin of the Cardassian term in terms of extra dimensions, and discuss the origin of the mass scale of . As discussed below, to match the CMB and supernovae data we take , but this value can easily be refined to better fit upcoming observations.

What is the Current Energy Density of the Universe?

Observations of the cosmic background radiation show that the geometry of the universe is flat with . In the Cardassian model we need to revisit the question of what value of energy density today, , corresponds to a flat geometry. We will show that the energy density required to close the universe is much smaller than in a standard cosmology, so that matter can be sufficient to provide a flat geometry.

The energy density that satisfies Eq.(6) is, by definition, the critical density. From Eqs.(1) and (5), we can write

(9) |

Evaluating this equation today with , we have

(10) |

Defining we find that the critical density has been modified from its usual value, i.e., the number has changed. We find

(11) |

Thus

(12) |

where

(13) |

and

(14) |

and is the Hubble constant today in units of 100 km/s/Mpc. In Figure 1, we have plotted the new critical density as a function of the two parameters and . For example, if we take , we find

(15) |

We see that the value of the critical density can be much lower
than previously estimated.
Since , we have today’s energy density as
as given above ^{3}^{3}3An alternate possible definition
would be to keep the standard value of and discuss the
contribution to it from the two terms on the right hand side
of the modified FRW equation. Then there would be a contribution
to from the term and another contribution from
the term, with the two terms adding to 1. This is the
approach taken when one discusses a cosmological constant in lieu
of our second term. However, the situation here is different in
that we have only matter in the equation. The disadvantage of this
second choice of definitions would be that a value of the energy density today
equal to according to this second definition would not correspond
to a flat geometry.,
i.e.,

(16) |

For larger values of , the modification to the value of can be even larger. Note the amusing result that for and , we have so that baryons would close the universe (not a universe we advocate).

Cluster Baryon Fraction

For the past ten years, a multitude of observations has pointed towards a value of the matter density . The cluster baryon fraction [11, 12] as well as the observed galaxy power spectrum are best fit if the matter density is 0.3 of the old critical density. Recent results from the CMB [8, 9] also obtain this value. In the standard cosmology this result implied that matter could not provide the entire closure density. Here, on the other hand, the value of the critical density can be much lower than previously estimated. Hence the cluster motivated value for is now compatible with a closure density of matter, , all in the form of matter. For example, if with , or if with , a critical density of matter corresponds to , as required by the cluster baryon fraction and other data. In Figure 1, one can see which combination of values of and produce the required factor of 0.3. If we assume that the value is correct, for a given value of (that is constant in time) we can compute the value of for our model from Eq.(13). Table I lists these values of and . Henceforth, we shall focus on these combinations of parameters.

Age of the Universe

In the Cardassian model, the universe is older due to the presence of the second term. In Table I, we present the age of the universe for various values of (under the assumption that ).

0.60 | 1.00 | 0.73 |

0.50 | 0.76 | 0.78 |

0.40 | 0.60 | 0.83 |

0.30 | 0.50 | 0.87 |

0.20 | 0.42 | 0.92 |

0.10 | 0.37 | 0.95 |

0.00 | 0.33 | 0.99 |

If one takes Gyr as the lower bound on globular cluster ages, then one requires for . If one requires globular cluster ages greater than 11 Gyr [23], then for . All values in Table I satisfy these bounds.

Structure Formation

Since the new (Cardassian) term becomes important only at , early structure formation is not affected. Below we discuss the impact on late structure formation during the era where the Cardassian term is important. This term accelerates the expansion of the universe, and freezes out perturbation growth once it dominates (much like when a curvature term dominates); this freezeout happens late enough that it is relatively unimportant. To obtain an idea of the type of effects that we may find, instead of analyzing the exact perturbation equations with metric perturbations included, we will merely modify the time dependence of the scale factor in the usual Jeans analysis equation. For now we take the standard equation for perturbation growth; as a caveat, we warn that recent structure formation may be further modified due to a change in Poisson’s equations as described below. For we now we take

(17) |

where is the fluid overdensity. Now one must substitute Eq.(1) for . In the standard FRW cosmology with matter domination, , and there is one growing solution to with . This standard result still applies throughout most of the (matter dominated) history of the universe in our new model, so that structure forms in the usual way.

Modifications set in once the new Cardassian term becomes important. When , Eq.(17) can be written

(18) |

where with denoting the time today and superscript prime refers to . This equation can generally be solved in terms of Bessel functions for constant (such as is the case once the Cardassian term completely overrides the old term). A simple example is the case of and ; in the limit , the last term in Eq.(18) can be dropped and the equation is solved as . Perturbations cease growing and become frozen in. This result agrees with the expectation that in a universe that is expanding more rapidly, the overdensity will grow more slowly with the scale factor. As mentioned at the outset, as long as the Cardassian term becomes important only very late in the history of the universe, much of the structure we see will have already formed and be unaffected. Further comments on late structure formation (e.g. cluster abundances) follow below.

Doppler Peak in Cosmic Background Radiation

Here we argue that the location of the first Doppler peak is only mildly affected by the new Cardassian cosmology. We need to calculate the angle subtended by the sound horizon at recombination. In the standard FRW cosmology with flat geometry, this value corresponds to a spherical harmonic with . A peak at this angular scale has indeed been confirmed [8]. In the Cardassian cosmology we still have a flat geometry. Hence, we can still write

(19) |

where is the sound horizon at the time of recombination and is the distance a light ray travels from recombination to today. To calculate these lengths, we use the fact that for a light ray to write

(20) |

Following the notation of Peebles [13], we define the redshift dependence of as

(21) |

so that Eq.(20) can be written

(22) |

Similarly, the sound horizon at recombination is

(23) |

In standard matter dominated FRW cosmology with , in Eq.(22) and .

For the cosmology of Eq.(1), we have

(24) |

with given in Eq.(13). As discussed previously, as our standard case we will take . With this assumption, and by using expression Eq.(24) in Eq.(22), we find that changes by a factor of (1.47, 1.88, 2.04, and 2.23) for =(0.6, 0.3, 0.2, and 0.1) respectively compared to the the usual (nonCardassian) FRW universe with . In addition and and we use and . We find that changes by a factor of (1.44, 1.62, 1.67, 1.29) for =(0.6, 0.3, 0.2, and 0.1) respectively compared to the usual FRW universe with . The angle subtended by the sound horizon on the surface of last scattering decreases and the location () of the first Doppler peak increases by roughly a factor of where

(25) |

compared to the usual FRW universe with . This shift still lies within the experimental uncertainty on measurements of the location of the Doppler peak.

We note the following: in the same way that a nonzero may make the current CBR observations compatible with a small but nonzero curvature, indeed a nonzero Cardassian term could also allow for a nonzero curvature in the data. A more accurate study of the effects of Cardassian expansion on the cosmic background radiation (including the first and higher peaks) is the subject of a future study.

The Cutoff Energy Density

An alternate way to write Eq.(1) is

(26) |

where . This notation offers a new interpretation; it indicates that the second term only becomes important once the energy density of the universe drops below , which has a value a few times the critical density. Hence, regions of the universe where the density exceeds this cutoff density will not experience effects associated with the Cardassian term. In particular, we can be reassured that the new term won’t affect gravity on the Earth or the Solar System. The density of water on the Earth is 1 gm/cm, which is 28 orders of magnitude higher than the critical density.

Comparing to Quintessence

We note that, with regard to observational tests, one can make a correspondence between the Cardassian and Quintessence models; we stress, however, that the two models are entirely different. Quintessence requires a dark energy component with a specific equation of state (), whereas the only ingredients in the Cardassian model are ordinary matter () and radiation (). However, as far as any observation that involves only , or equivalently , the two models predict the same effects on the observation. Regarding such observations, we can make the following identifications between the Cardassian and quintessence models: , , and , where is the quintessence equation of state parameter, is the ratio of matter density to the (old) critical density in the standard FRW cosmology appropriate to quintessence, is the ratio of quintessence energy density to the (old) critical density, and F is given by Eq.(13). In this way, the Cardassian model can make contact with quintessence with regard to observational tests.

All observational constraints on quintessence that depend only on the scale factor, (or, equivalently, ) can also be used to constrain the Cardassian model. However, because the Cardassian model requires modified Einstein equations (see below), the gravitational Poisson’s equations and consequently late-time structure formation may be changed; e.g., the redshift dependence of cluster abundance should be different in the two models. These effects (and others, such as the fact that quintessence clumps) may serve to distinguish the Cardassian and quintessence models. The correspondence with quintessence, as well as discussion of distinguishing tests will be the subject of a future paper.

Best Fit of Parameters to Current Data

We can find the best fit of the Cardassian parameters and to current CMB and Supernova data. The current best fit is obtained for (as we have discussed above) and (equivalently, ) [20, 21]. In Table I one can see the values of compatible with this bound, as well as the resultant age of the universe. As an example, for (equivalently, ), we find that . Then the position of the first Doppler peak is shifted by a factor of 1.12. The age of the universe is 13 Gyr. The cutoff energy density is , so that the new term is important only for . Hence, as mentioned above, the Cardassian term won’t affect the physics of the Earth or solar system in any way.

We note the enormous uncertainty in the current data; future experiments (such as SNAP [22]) will constrain these parameters further.

Extra Dimensions

A Cardassian term may arise as a consequence of embedding our observable universe as a 3+1 dimensional brane in extra dimensions. Chung and Freese [10] showed that, in a 5-dimensional universe with metric

(27) |

where is the coordinate of the fifth dimension, one may obtain a modified FRW equation on our observable brane with for any (see also [19]). This result was obtained with 5-dimensional Einstein equations plus the Israel boundary conditions relating the energy-momentum on our brane to the derivatives of the metric in the bulk.

We do not yet have a fundamental higher dimensional theory, i.e., a higher dimensional , which we believe describes our universe. Once we have this, we can write down the modified four-dimensional Einstein’s equations and compute the modified Poisson’s equations, as would be required, e.g., to fully understand latetime structure formation.

There is no unique 5-dimensional energy momentum tensor that gives rise to Eq.(1) on our brane. Hence, in this paper we construct an example which is easy to find but is clearly not our universe, simply as a proof that such an example can be written down. Following [10] (see Eqs. (24) and (25) there with ), we have constructed an example of a bulk for arbitrary in , matter on the brane as in Eq.(3), and with in Eq.(27). We display only here (the other components will be published in a future paper):

(28) |

where

(29) |

and the constant is related to the 5-dimensional Newton’s constant and 5-D reduced Planck mass by the relation . This is merely one (inelegant) example of many bulk that produce Cardassian expansion.

We may now investigate the meaning of the values of required by Eq.(8), where is the parameter in front of the new Cardassian term in Eq.(1). As mentioned previously, the mass scale of has units of . We find that the corresponding mass scale is very small for , is singular at , and then goes over to a very large value for . Specifically, for and , we obtain GeV which corresponds to a mass scale of GeV. In the context of extra dimensions, this large mass scale turns out to cancel against other large numbers in such a way that it corresponds to reasonable values of the energy momentum tensor in the bulk. We find that is roughly the age of the universe and we have for all . Then we have

(30) |

Although this value is not motivated, it is not unreasonable. In other words, reasonable bulk values can generate the required parameters in Eq.(1). Numerical values for other components of are the same order of magnitude, with the exception that . For the case of , we obtain a mass scale of , which cancels other small numbers in such a way as to again require roughly Eq.(30) to be satisfied. The form of given in Eq.(28) is by no means unique and has been presented merely as an existence proof; we hope a more elegant may be found, perhaps with a motivation for the required value of .

Discussion

We have presented Eq.(1) as a modification to the FRW equations in order to suggest an explanation of the recent acceleration of the universe. In the Cardassian model, the universe can be flat and yet matter dominated. We have found that the new Cardassian term can dominate the expansion of the universe after and can drive an acceleration. We have found that matter alone can be responsible for this behavior (but see the comments below). The current value of the energy density of the universe is then smaller than in the standard model and yet is at the critical value for a flat geometry. Structure formation is unaffected before . The age of the universe is somewhat longer. The first Doppler peak of the cosmic background radiation is shifted only slightly and remains consistent with experimental results. Such a modified FRW equation may result from the existence of extra dimensions. Further work is required to find a simple fundamental theory responsible for Eq.(1).

Questions of interpretation remain. We have said that matter alone is responsible for the accelerated behavior. However, if the Cardassian behavior results from integrating out extra dimensions, then one may ask what behavior of the radii of the extra dimensions is required. The Israel conditions connect the energy density on the brane to fields in the bulk. The required behavior of bulk fields is not transparent when one writes the modified FRW equation. We have found a large or small mass scale to be required, which must result from the extra dimensions. In principle one would like to have a complete 5-dimensional theory so as to perform post-Newtonian tests on the model and also to check other consequences. For example, with a 5-dimensional model, one would like to compare with limits from fifth force experiments and to check that none of the higher dimensional fields are overcontributing to the energy density of the universe at any point (the moduli problem).

One might attempt to use a Cardassian term (the second term in Eq.(1)) to drive an early inflationary era in the universe as well. For one could have a superluminal expansion during the radiation dominated era. However, once accelerated expansion begins, Eq.(1) without a potential provides no way for inflation to stop. Hence we have focused on using this new term to generate acceleration today rather than to cause an inflationary mechanism early on. However, it may be possible to combine such a term with a different way to end inflation.

###### Acknowledgements.

K.F. thanks Ted Baltz, Daniel Chung, Richard Easther, Gus Evrard, Paolo Gondolo, Wayne Hu, Lam Hui, Will Kinney, Risa Wechsler, and especially Jim Liu for many useful conversations and helpful suggestions. We acknowledge support from the Department of Energy via the University of Michigan.## References

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