CE: PM
I
NSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 21 (2004) 1–13 PII: S0264-9381(04)76772-6
Jerk, snap and the cosmological equation of state
Matt Visser
School of Mathematical and Computing Sciences, Victoria University of Wellington,
PO Box 600, Wellington, New Zealand
E-mail: matt.visser@mcs.vuw.ac.nz
Received 27 February 2004
Published DD MMM 2004
Online at stacks.iop.org/CQG/21/1 (
DOI: 10.1088/0264-9381/21/0/000)
Abstract
Taylor expanding the cosmological equation of state around the current epoch
p = p
0
+ κ
0
(ρ − ρ
0
) +
1
2
d
2
p
dρ
2
0
(ρ − ρ
0
)
2
+ O[(ρ − ρ
0
)
3
],
is the simplest model one can consider that does not make any apriori
restrictions on the nature of the cosmological fluid. Most popular cosmological
models attempt to be ‘predictive’, in the sense that once some aprioriequation
of state is chosen the Friedmann equations are used to determine the evolution of
the FRW scale factor a(t). In contrast, a ‘retrodictive’ approach might usefully
take observational data concerning the scale factor, and use the Friedmann
equations to infer an observed cosmological equation of state. In particular, the
value and derivatives of the scale factor determined at the current epoch place
constraints on the value and derivatives of the cosmological equation of state at
the current epoch. Determining the first three Taylor coefficients of the equation
of state at the current epoch requires a measurement of the deceleration, jerk
and snap—the second, third and fourth derivatives of the scale factor with
respect to time. Higher-order Taylor coefficients in the equation of state are
related to higher-order time derivatives of the scale factor. Since the jerk and
snap are rather difficult to measure, being related to the third and fourth terms
in the Taylor series expansion of the Hubble law, it becomes clear why direct
observational constraints on the cosmological equation of state are so relatively
weak; and are likely to remain weak for the foreseeable future.
PACS number:
See endnote 1
1. Introduction
This paper develops a ‘phenomenological’ approach to the equation of state (EOS) of the
cosmological fluid, and investigates what would have to be done in order to observationally
0264-9381/04/000001+13$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1
2 M Visser
determine the EOS. Even at the linearized level, where
p = p
0
+ κ
0
(ρ − ρ
0
) + O[(ρ − ρ
0
)
2
], (1)
the first nontrivial coefficient in the EOS will be seen to be related to the cosmological jerk—
the third derivative of the scale factor with respect to time, and thence to the third-order
term in the Taylor series expansion of the Hubble law. This is the fundamental reason why
observational determinations of the EOS are relatively poor, and why it is possible to choose
so many wildly differing apriorimodels for the EOS that nevertheless give good agreement
with the coarse features of the present epoch.
More generally, if we describe the cosmological equation of state in terms of a Taylor
series expansion around the current epoch
p = p
0
+
N−1
n=1
1
n!
d
n
p
dρ
n
0
(ρ − ρ
0
)
n
+ O[(ρ − ρ
0
)
N
], (2)
then the nth-order Taylor coefficient
d
n
p
dρ
n
0
, (3)
will be shown to depend on the (n + 2)th time derivative of the scale factor
d
n+2
a(t)
dt
n+2
0
, (4)
and thence to the (n +2)thterm,theO(z
n+2
) term, in the Taylor expansion of the Hubble law.
In most attempts at cosmological model building one takes a FRW cosmology
ds
2
=−c
2
dt
2
+ a(t)
2
dr
2
1 − kr
2
+ r
2
(dθ
2
+sin
2
θ dφ
2
)
, (5)
plus the conservation of stress–energy
˙ρa
3
+3[ρ + p]a
2
˙
a = 0, (6)
and chooses some aprioriequation of state
ρ = ρ(p), or p = p(ρ), (7)
to derive ρ(a), and equivalently p(a). The Einstein equations then reduce to the single
Friedmann equation, which can be written in the form
˙
a =
8πG
N
ρ(a)a
2
3
− k, (8)
and used to determine a(t). (See, for example, any standard text such as [1–3].)
In contrast, let us assume we have a FRW universe with good observational data on
a(t)—in Weinberg’s terminology we have a good ‘cosmography’ [1]. In this situation we can
use the Einstein equations in reverse to calculate the energy density ρ(t) and pressure p(t) via
8πG
N
ρ(t) = 3c
2
˙
a
2
a
2
+
kc
2
a
2
, (9)
8πG
N
p(t) =−c
2
˙
a
2
a
2
+
kc
2
a
2
+2
¨
a
a
. (10)
Under mild conditions on the existence and nonzero value of appropriate derivatives, we can
appeal to the inverse function theorem to assert the existence of a t(ρ) or t(p) and hence, in
principle, deduce an observational equation of state
ρ(p) = ρ(t = t (p)), p(ρ) = p(t = t (ρ)). (11)
Jerk, snap and the cosmological equation of state 3
In view of the many controversies currently surrounding the cosmological equation of state,
and the large number of speculative models presently being considered, such an observationally
driven reconstruction is of interest in its own right.
Now in observational cosmology we do not have direct access to a(t) over the entire
history of the universe—we do, however, have access (however imprecise) to the current
value of the scale factor and its derivatives, as encoded in the Hubble parameter, deceleration
parameter, etc. This more limited information can still be used to extract useful information
about the cosmological equation of state, in particular it yields information about the present
value of the w-parameter and the slope parameter κ
0
defined as
w
0
=
p
ρ
0
,κ
0
=
dp
dρ
0
. (12)
The value of the w-parameter in particular has recently become the centre of considerable
interest, driven by speculation that w
0
< −1 is compatible with present observations. Such a
value of w
0
would correspond to present-day classical and cosmologically significant violations
of the null energy condition. The associated ‘phantom matter’ (almost identical to the notion of
‘exotic matter’ in the sense of Morris and Thorne [4]) leads to a cosmological energy density
that is future increasing rather than future decreasing. (See, for example, [5, 6]). If w(t)
subsequently remains strictly less than −1, this will lead to a ‘big rip’ [7]—the catastrophic
infinite expansion of the universe in finite elapsed time.
Unfortunately, it is very difficult to measure w
0
and κ
0
with any accuracy—I will make
this point explicit by relating the measurement of w
0
to the deceleration parameter, and the
measurement of κ
0
to the ‘jerk’ of the cosmological scale factor—the third derivative with
respect to time.
For related comments see [8–13]. The ‘cubic’ term of Chiba and Nakamura [8] is identical
to the jerk, as is the ‘statefinder’ variable called r by Sahni et al [9–11]. The other ‘statefinder’
variable (called s, not to be confused with the snap) is a particular linear combination of the
jerk and deceleration parameters. Padmanabhan and Choudhury [12] have also emphasized
the need for constructing models for the cosmological fluid that are unprejudiced by apriori
theoretical assumptions. A good recent survey of the status of the cosmological fluid is [13].
2. Hubble, deceleration, jerk and snap parameters
It is standard terminology in mechanics that the first four time derivatives of position are
referred to as velocity, acceleration, jerk and snap
1
. In a cosmological setting this makes it
appropriate to define Hubble, deceleration, jerk and snap parameters as
H(t) = +
1
a
da
dt
, (13)
q(t) =−
1
a
d
2
a
dt
2
1
a
da
dt
−2
, (14)
j(t) = +
1
a
d
3
a
dt
3
1
a
da
dt
−3
, (15)
s(t) = +
1
a
d
4
a
dt
4
1
a
da
dt
−4
. (16)
1
Jerk (the third time derivative) is also sometimes referred to as jolt. Less common alternative terminologies are
pulse, impulse, bounce, surge, shock and super-acceleration. Snap (the fourth time derivative) is also sometimes
called jounce. The fifth and sixth time derivatives are sometimes somewhat facetiously referred to as crackle and pop.
4 M Visser
The deceleration, jerk and snap parameters are dimensionless, and we can write
a(t) = a
0
1+H
0
(t −t
0
) −
1
2
q
0
H
2
0
(t −t
0
)
2
+
1
3!
j
0
H
3
0
(t −t
0
)
3
+
1
4!
s
0
H
4
0
(t −t
0
)
4
+ O([t − t
0
]
5
)
. (17)
In particular, at arbitrary time t
w(t) =
p
ρ
=−
H
2
(1 − 2q) + kc
2
/a
2
3(H
2
+ kc
2
/a
2
)
=−
(1 − 2q) + kc
2
/(H
2
a
2
)
3[1 + kc
2
/(H
2
a
2
)]
. (18)
While observation is currently not good enough to distinguish between the three cases
k =−1/0/ + 1 with any degree of certainty, there is nevertheless widespread agreement
that at the present epoch H
0
a
0
/c 1 (equivalent to |
0
− 1|1).
Warning . From a theoretical perspective, H
0
a
0
/c 1 is a generic prediction of inflationary
cosmology—this is not the same as saying that cosmological inflation predicts k = 0. What
generic cosmological inflation predicts is the weaker statement that for all practical purposes
the present day universe is indistinguishable from a k = 0 spatially flat universe. If our
universe happens to be a topologically trivial k = 0 FRW cosmology, then we will never
be able to prove it. Simply as a matter of formal logic, all we will ever be able to do is to
place increasingly stringent lower bounds on H
0
a
0
, but this will never rigorously permit us to
conclude that k = 0. The fundamental reason for this often overlooked but trivial observation
is that a topologically trivial k = 0 FRW universe can be mimicked to arbitrary accuracy
by a k =±1 FRW universe provided the scale factor is big enough
2
. In contrast, if the
true state of affairs is k =±1, then with good enough data on H
0
a
0
we will in principle be
able to determine upper bounds which (at some appropriate level of statistical uncertainty)
demonstrate that k = 0. Also note that even in inflationary cosmologies it is not true that
H(t)a(t)/c 1 at all times, and in particular this inequality may be violated (and often is
violated) in the pre-inflationary epoch.
Now the w-parameter in cosmology is related to the Morris–Thorne exoticity parameter
[4] which was introduced by them to characterize the presence of ‘exotic matter’, matter
violating the null energy condition (NEC):
ξ =
ρ + p
|ρ|
= sign(ρ)[1 + w] =
2
3
sign(ρ)
1+q + kc
2
/(H
2
a
2
)
1+kc
2
/(H
2
a
2
)
. (19)
Thus if w<−1 and ρ>0wehaveξ<0 and the NEC is violated. In contrast, if w<−1but
ρ<0wehaveξ>0, the NEC is satisfied but the weak energy condition (WEC) is violated.
That is, ‘phantom matter’ (matter with w<1) is not quite the same as ‘exotic matter’ (for
which ξ<0), but the two are intimately related.
Accepting the approximation that H
0
a
0
/c 1wehave
ρ
0
≈
3
8πG
N
H
2
0
> 0,w
0
≈−
(1 − 2q
0
)
3
, and ξ
0
≈
2
3
(1+q
0
), (20)
so that in this situation the w
0
-parameter and exoticity parameter ξ
0
are intimately related
to the deceleration parameter q
0
. In particular if w
0
< −1, so that the universe is at the
current epoch dominated by ‘phantom matter’, we also (because in this approximation ρ
0
is
guaranteed to be positive) have ξ
0
< 0 so that at the current epoch this phantom matter is also
2
If the universe has nontrivial spatial topology there is a possibility of using the compactification scale, which might
be (but does not have to be) much smaller than the scale factor, to indirectly distinguish between k =−1/0/ +1.
Jerk, snap and the cosmological equation of state 5
‘exotic matter’. Exotic matter is powerful stuff: apart from possibly destroying the universe
in a future ‘big rip’ singularity [7], if the exotic matter clumps to any extent there is real risk
of even more seriously bizarre behaviour—everything from violations of the positive mass
condition (that is, objects with negative asymptotic mass), through traversable wormholes, to
time warps [4, 14–17].
3. Taylor series equation of state
Linearize the cosmological EOS around the present epoch as
p = p
0
+ κ
0
(ρ − ρ
0
) + O[(ρ − ρ
0
)
2
]. (21)
To calculate κ
0
we use
κ
0
=
dp/dt
|
0
dρ/dt
|
0
, (22)
where numerator and denominator can be obtained by differentiating the Friedmann equations
for ρ(t) and p(t). It is easy to see that at all times, simply from the definition of deceleration
and jerk parameters, we have
8πG
N
dρ
dt
=−6c
2
H
(1+q)H
2
+
kc
2
a
2
, (23)
8πG
N
dp
dt
= 2c
2
H
(1 − j)H
2
+
kc
2
a
2
, (24)
leading to
κ
0
=−
1
3
1 − j
0
+ kc
2
H
2
0
a
2
0
1+q
0
+ kc
2
H
2
0
a
2
0
, (25)
which approximates (using H
0
a
0
/c 1) to
κ
0
=−
1
3
1 − j
0
1+q
0
. (26)
The key observation here is that to obtain the linearized equation of state you need significantly
more information than the deceleration parameter q
0
; you also need to measure the jerk
parameter j
0
. If the only observations you have are measurements of the deceleration parameter
then you can of course determine w
0
= p
0
/ρ
0
, but this is not an equation of state for the
cosmological fluid. Determining w
0
merely provides information about the present-day value
of p/ρ but makes no prediction as to what this ratio will do in the future—not even in the near
future. (This point is also forcefully made in [12].) For this reason there have been several
attempts to observationally determine w(z),thevalueofw as a function of redshift. See
for example [12] and [9–11]. Since z is a function of lookback time D/c, this is ultimately
equivalent to determining w(t) = p(t)/ρ(t), and implicitly equivalent to reconstructing a
phenomenological equation of state p(ρ). I prefer to phrase the discussion directly in terms of
the EOS as that will make it clear what parameters have to be physically measured. In terms
of the history of the scale factor a(t), it is only when one goes to third order by including the
jerk parameter j
0
that one obtains even a linearized equation of state.
Going one step higher in the expansion, by using the chain rule and the implicit function
theorem it is easy to see that
d
2
p
dρ
2
=
¨
p − κ ¨ρ
( ˙ρ)
2
. (27)