Chapter 15
Solutions for the Present Universe
Abstract The universe is presently dominated by vacuum or dark energy, described
by the cosmological constant lambda, and cold dark matter; it is thus referred to as
the LCDM universe. In the spatially flat case the Friedmann master equation may be
solved for the two ingredients separately and also for both together; the combined
solution is remarkably simple and useful in understanding properties of the universe.
15.1 The Positive Cosmological Constant
Preparatory to solving the master dynamical equation (14.19) we will show that the
cosmological constant must be positive and establish an important relation among
the cosmic fluid constituent densities.
As we have indicated in previous chapters, observations of distant supernovae
show that the universe is accelerating, with a negative deceleration parameter of
about q 0 = −0.55. This clearly implies that a
is positive according to the definition
(13.28). It is easy to show that the universe can only be accelerating if the cosmological constant is positive. To see this we differentiate the master equation (14.19)
and find that the second derivative of the scale factor at the present time is
a
= a 0
V 0 −
m0
2
+ r 0
H
2
0 , , V 0 ≡
c
2
3H
2
0
.
(15.1)
Since this second derivative is positive V 0 must also be positive and so must the
cosmological constant .
Equation (15.1) may be put into a simpler and useful form in terms of the
deceleration parameter defined in (13.28). We find
q 0 =
m0
2
+ r 0
− V 0 .
(15.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_15
233
Solutions for the Present Universe
Abstract The universe is presently dominated by vacuum or dark energy, described
by the cosmological constant lambda, and cold dark matter; it is thus referred to as
the LCDM universe. In the spatially flat case the Friedmann master equation may be
solved for the two ingredients separately and also for both together; the combined
solution is remarkably simple and useful in understanding properties of the universe.
15.1 The Positive Cosmological Constant
Preparatory to solving the master dynamical equation (14.19) we will show that the
cosmological constant must be positive and establish an important relation among
the cosmic fluid constituent densities.
As we have indicated in previous chapters, observations of distant supernovae
show that the universe is accelerating, with a negative deceleration parameter of
about q 0 = −0.55. This clearly implies that a
is positive according to the definition
(13.28). It is easy to show that the universe can only be accelerating if the cosmological constant is positive. To see this we differentiate the master equation (14.19)
and find that the second derivative of the scale factor at the present time is
a
= a 0
V 0 −
m0
2
+ r 0
H
2
0 , , V 0 ≡
c
2
3H
2
0
.
(15.1)
Since this second derivative is positive V 0 must also be positive and so must the
cosmological constant .
Equation (15.1) may be put into a simpler and useful form in terms of the
deceleration parameter defined in (13.28). We find
q 0 =
m0
2
+ r 0
− V 0 .
(15.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. J. Adler, General Relativity and Cosmology, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-61574-1_15
233