Bounding the Hubble flow in terms of the parameter
Abstract
The last decade has seen increasing efforts to circumscribe and bound the cosmological Hubble flow in terms of modelindependent constraints on the cosmological fluid — such as, for instance, the classical energy conditions of general relativity. Quite a bit can certainly be said in this regard, but much more refined bounds can be obtained by placing more precise constraints (either theoretical or observational) on the cosmological fluid. In particular, the use of the parameter () has become increasingly common as a surrogate for trying to say something about the cosmological equation of state. Herein we explore the extent to which a constraint on the parameter leads to useful and nontrivial constraints on the Hubble flow, in terms of constraints on density , Hubble parameter , density parameter , cosmological distances , and lookback time . In contrast to other partial results in the literature, we carry out the computations for arbitrary values of the space curvature , equivalently for arbitrary .
Keywords:
arXiv 0806.nnnn [grqc]; 13 June 2008;
LaTeXed \DayOfWeek, August 19, 2019; \daytime.
Contents
1 Introduction
The classical energy conditions of general relativity [1, 2], despite their wellknown limitations [3, 4, 5], are nevertheless very useful surrogates for controlling the extent to which one is willing to countenance extreme and unusual physics. As applied to cosmology, in addition to the very general cosmological singularity theorem presented in [1], the classical energy conditions have (at the cost of additional hypotheses) been used to place more precise limits on the expansion of an idealized FLRW universe [6, 7, 8, 9, 10]. More recently, these ideas have been extended in various ways in [11, 12, 13, 14]. In all of these analyses there is a tradeoff between the precision and generality of the constraints one obtains — the art lies in choosing a form of the input assumptions that is as general as possible, but not too general, for the precision of the output constraints one wishes to derive.
In the current article we shall derive some very general bounds in terms of assumptions about the parameter, where as usual . Specifically, we shall ask the question: If we know for theoretical reasons, or can observationally determine, that lies in some restricted range
(1) 
between redshift zero and redshift , what constraint does that place on the cosmological expansion? We shall see that considerable useful information can be extracted regarding the density , Hubble parameter , density parameter , various cosmological distances , and lookback time . Specifically, for some generic cosmological parameter , we shall be looking for bounds of the form
(2) 
Conversely, observational constraints on these cosmological parameters can be used to infer features of the cosmological fluid in a largely modelindependent manner. In contrast to other partial results scattered throughout the literature, we carry out the computations for arbitrary values of the space curvature , equivalently for arbitrary .
2 Strategy
Our strategy will be to adopt a standard FLRW cosmology
(3) 
then, (setting , but explicitly retaining the speed of light ), we have the two Friedmann equations:
(4) 
Together, these two equations imply the standard conservation law:
(5) 
We also have the fundamental definitions
(6) 
and
(7) 
For intermediate steps of the calculation we shall work with the very simple linear equation of state
(8) 
where is taken to be a constant. Picking some generic cosmological parameter , we shall first calculate , and then (by assuming that from redshift zero out to redshift , and depending on the direction of the relevant inequality) use this to derive bounds of the form
(9) 
or
(10) 
We shall also make the extremely mild assumption that the density is positive
(11) 
This is certainly a completely redundant assumption for and FLRW universes. Only for universes does this provide the extremely mild additional constraint , that is, .
3 Density
We now apply this strategy to the density. From
(12) 
we have
(13) 
So integrating, for constant we obtain the wellknown result
(14) 
But now ask what happens if we only know that ? (Where in the real observable universe certainly need not be a constant.) Following the above analysis, we find that we must replace equalities by inequalities and so deduce
(15) 
Note that for we are looking into the past; in contrast for , we are looking into the future [15], and the inequality reverses to
(16) 
Of course, these simple constraints on the density are by far the most elementary of the inequalities we shall deduce — some of the other inequalities derived below will prove to be much more subtle.
If we now in addition relax our initial constraint on , by assuming we only know that the present epoch density lies in some bounded interval
(17) 
then these two bounds generalize to
(18) 
(19) 
4 Density parameter
We have the following identity
(20) 
This leads to the useful result
(21) 
Therefore, a bound on automatically implies a bound on . From the result for presented above, we deduce that bounds on can be given in terms of
(22) 
which we can equivalently recast as
(23) 
We can now use this quantity, which was derived for strictly constant , to bound the density parameter for a more realistic matter model satisfying the milder condition . We obtain:

If (but remember that by assumption ) then
(24) (25) 
If then .

If ,
(26) (27) but note that the bound can break down when the denominator of equals zero — this occurs at
(28) The failure of the bound might occur either in the past or the future depending on the value of .

If then the bound is useful only for , implying a limitation in the past.

If then the bound is valid for all .

If then the bound is useful only for , implying a limitation in the future.
Note that nothing unusual need happen to the universe itself at , it is only the bound that loses its predictive usefulness. Combining these observations we see that for it is better (in the sense of reducing the amount of special case exceptions to the general rule) to recast the bounds in the form:
(29) for , and
(30) for .

5 Hubble parameter
Let us now use the density equation (the first Friedmann equation) and the definition of the density parameter to write
(31) 
That is, as an identity:
(32) 
But we have already derived a formula for , whence
(33) 
which we can recast as
(34) 
For realistic matter, satisfying some constraint , we then deduce
(35) 
(36) 
Note that the Hubble bound ceases to provide useful information once the argument of the square root occurring in becomes negative.

For there is no limitation in the physical region .

For this limitation manifests itself at , the same place that the bound on ran into difficulties. (Some numerical estimates of where the bounds fail, based on current consensus observational data, are discussed in [14].)
Finally, suppose that we do not have precise information regarding and , and only have the more limited information
(37) 
then these two Hubble bounds further generalize to
(38) 
(39) 
subject to the caveat that for we should not push the bound past .
6 Cosmological distances
Let us for the time being focus on Peebles’ definition of “angular diameter distance”. This is what Weinberg calls the “proper motion distance” [17, 18], for more definitions and a discussion regarding the physical interpretation of the cosmological distance scales see [19], see also [15, 16]. We make this choice to minimize the number of factors of in subsequent formulae. Then the standard definition is
(40) 
But since
(41) 
this can be rewritten more suggestively as
(42) 
When interpreting this last formula for we make use of the fact that . Substituting and performing the integral, after considerable effort both Mathematica and Maple yield
(43) 
whence
(44)  
This simplifies slightly
(45)  
We now note
(46) 
(cross multiply top and bottom), which finally permits us to write the most tractable form of our exact result for Peebles’ angular diameter distance (in a constant FLRW universe):
(47)  
In this final expression we are always raising quantities to the same power, and the difference between the two terms is just in the placement of and signs. (Note that this expression is guaranteed to be real whatever the value of ; for the two terms are complex conjugates of each other and after taking the prefactor into account, the overall combination is guaranteed to be real.)
Note that once we have an explicit formula for the (Peebles) angular diameter distance , any of the other standard cosmological distances can easily be obtained by multiplying by suitable powers of [17, 18, 19], see also [15, 16]. In particular the luminosity distance is
(48)  
Returning to Peebles’ angular diameter distance, the Taylor series expansion in can be computed as
(49)  
Perhaps of more interest is the Taylor series expansion in (since observationally we have good reasons for expecting ). The leading term is easy to calculate
(50) 
Extracting the next term is not too difficult, but is somewhat tedious
(51)  
In any realistic situation (provided you accept the standard consensus cosmology) the uncertainties in will completely dwarf any possible effect due to uncertainties in , so carrying the expansion to higher order is not warranted.
As usual, [or ] can be used to bound [or ]. Specifically, let lie in the range then independent of :

,

.
Here is given by the rather formidable equation (47).
7 Lookback time
Finally, consider the “lookback time” defined by:
(52) 
That is:
(53) 
Using the known form of we define
(54) 
and shall use this quantity to place bounds on the actual lookback time .
It is easy to obtain the leading term for :
(55) 
The next subleading term again is trickier. We eventually obtain
(56)  
Again, in any realistic situation (provided you accept the standard consensus cosmology) the uncertainties in will completely dwarf any possible effect due to uncertainties in . Exact integration and subsequent evaluation of the result for can only be performed in terms of hypergeometric functions. Let us first be a little more careful about the use of the dummy variable in the integration and write
(57) 
and then, (following the procedure of [7, 14]), apply the binomial theorem
(58) 
Now this particular binomial series will converge provided
(59) 
That is, provided
(60) 
More explicitly, the integral will make sense provided
(61) 
which is equivalent to
(62) 

In all cases, to ensure convergence at redshift zero, we must certainly have
(63) 
If and , (), or if and , (): Then , and no additional limitation is imposed.

If and , (), or if and , (): In this situation , therefore we now obtain an additional limitation on that is necessary to ensure convergence:

If , then we need
(64) 
If then we need
(65)


In view of equation (22) these last conditions can also be interpreted as constraints on the parameter at the redshift one wishes to probe:
(66)
Subject to this convergence condition we can integrate term by term, and obtain the convergent series
(67) 
As a practical matter, for many purposes this series representation may be enough, but we can tidy things up somewhat by first defining
(68) 
in which case
(69) 
Finally we can recognize that is itself a particular example of a hypergeometric function,
(70) 
Therefore
(71)  
As usual, can be used to bound . Specifically, let lie in the bounded range , then independent of :

,

.
Here is given by the rather formidable equation (71), based on the use of hypergeometric functions.
8 Special cases and consistency checks
Useful special cases, and consistency checks we can perform on the formalism, include:
 Dust:

For pure dust, , we have simple exact results
(72) (73)