Dynamical Analysis of the Redshift Drift in FLRW Universes

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Citation: Lobo, F.S.N.; Mimoso, J.P.;

Santiago, J.; Visser, M. Dynamical

Analysis of the Redshift Drift in FLRW

Universes. Universe 2024,10, 162.

https://doi.org/

10.3390/universe10040162

Academic Editor: Pier Stefano

Corasaniti

Received: 23 February 2024

Revised: 21 March 2024

Accepted: 27 March 2024

Published: 29 March 2024

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

universe

Article

Dynamical Analysis of the Redshift Drift in FLRW Universes

Francisco S. N. Lobo 1,2 , José Pedro Mimoso 1,2 , Jessica Santiago 3,* , and Matt Visser 4

Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências, Universidade de Lisboa, Campo Grande,

Edifício C8, 1749-016 Lisboa, Portugal; fslobo@fc.ul.pt (F.S.N.L.); jpmimoso@fc.ul.pt (J.P.M.)

2Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C8,

1749-016 Lisboa, Portugal

3Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan

4School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600,

Wellington 6140, New Zealand; matt.visser@sms.vuw.ac.nz

*Correspondence: jessicasantiago@ntu.edu.tw

Abstract: Redshift drift is the phenomenon whereby the observed redshift between an emitter and

observer comoving with the Hubble ﬂow in an expanding FLRW universe will slowly evolve—on a

timescale comparable to the Hubble time. In a previous article, three of the current authors performed

a cosmographic analysis of the redshift drift in an FLRW universe, temporarily putting aside the issue

of dynamics (the Friedmann equations). In the current article, we add dynamics while still remaining

within the framework of an exact FLRW universe. We developed a suitable generic matter model

and applied it to both standard FLRW and various dark energy models. Furthermore, we present an

analysis of the utility of alternative cosmographic variables to describe the redshift drift data.

Keywords: redshift drift; dark energy models; cosmography; cosmodynamics; astrophysics;

cosmology

1. Introduction

The concept of “redshift drift” (RD) dates back (at least) some 60 years, to 1962,

arising in coupled papers by Sandage [

] and McVittie [

]. Relatively little direct follow-up

work took place in the 20th century, with Loeb’s 1998 article [

] as a stand-out exception.

However, with technological advances and new observational surveys on the horizon,

the possibilities of measuring RDs have become much more concrete [

–

]. The basic

idea is this: If in any FLRW universe emitter and observer are comoving with the Hubble

ﬂow, then the null curve connecting them slowly evolves on a timescale set by the Hubble

parameter; this implies that the redshift is slowly evolving. In any FLRW universe, the key

result is [1–4]:

z= (1+z)H0−H(z). (1)

Measuring this effect will certainly be a challenging enterprise, with typical estimates

suggesting the need for a decade-long observational window. Starting from an estimated

detection time of a couple of decades—using the ﬁrst observational feasibility study of the

Extremely Large Telescope (ELT) [

]—recent experimental proposals suggest a detection

time as low as 6 years [

] (though the constraints provided on cosmological parameters

could potentially be greatly diminished by the time-reduction [

]). Furthermore, other

future prospects from RD measurements are proposed to test the cosmological principle

(i.e., isotropy

and homogeneity) by taking into account large-scale structures and distin-

guishing between non-FLRW cosmological models [

–

]. More boldly, some authors

have recently speculated on what might be do-able with millennia-long observational

windows [30].

Universe 2024,10, 162. https://doi.org/10.3390/universe10100162 https://www.mdpi.com/journal/universe

arXiv:2210.13946v3 [gr-qc] 2 Apr 2024

Universe 2024,10, 162 2 of 24

In [

], three of the current authors performed a general cosmographic analysis (for

additional background, see [

–

]), both in terms of the regular

-redshift and in terms of

the y-redshift, deﬁned by

1−y=a

a0=1

1+z, (2)

so that for

z∈[

∞)

, one has

y∈[

0, 1

)

. There we have proved the closely related exact

result:

y= (1−y){H0−(1−y)H(y)}. (3)

We shall now analyze and extend these and related results by using the dynamical Fried-

mann equations of general relativistic cosmology.

Without choosing the exact composition of the universe, one can rewrite the RD in

terms of a suitably deﬁned density parameter,

Ω(z)

(one that includes the effect of spatial

curvature), as

z=(1+z)−qΩ(z)H0;(Ω0≡1). (4)

Similarly, for the yredshift, we have

y= (1−y)1−(1−y)qΩ(y)H0;(Ω0≡1). (5)

From this, we can proceed by building a physically plausible matter model for

Ω(z)

(or equivalently

Ω(y)

) and investigate its properties. We will allow for an arbitrary admix-

ture of non-interacting components, all individually satisfying linear equations of state

pi=wiρi.

Such a model is general enough to include

-CDM and many variants thereof

but is simple enough to allow explicit calculations of

Ω(z)

or, equivalently,

Ω(y)

and its

various approximations.

We also present an RD analysis applied to different dark energy (DE) models, such

CDM, the linear model, BAZS equation of state, CPL, logarithmic evolution, and a

couple of interactive models. This is followed by a discussion on whether RD data have

the power to distinguish distinct equations of state for dark energy or not.

The article is outlined as follows: In Section 2, we develop the notation, set the stage,

and subsequently develop a dynamical analysis in terms of the usual

-redshift. We also

present the relations between the RD signal peak,

zpeak

, and

zequality

, and the turning point

of the acceleration rate of the universe,

0, for the

CDM case. In Section 3, we start by

introducing the dark energy models discussed in this work, followed by the predicted RD

signal of each model for different values of the relevant free parameters. We then proceed

in Section 3to discuss the power of redshift data in distinguishing different DE models

from each other. In Section 4.1, we move towards a cosmographic analysis, presenting

the general results in terms of the

-redshift deﬁned by

y=z/(1+z)

. Other auxiliary

variables for describing the redshift are then presented in a table in Section 4.2. Finally, we

conclude in Section 5.

2. Dynamics of the RD in Terms of z

As is well known, the dynamical behavior of the Friedmann (FLRW) cosmological

models is determined by two equations:

(i)

The second-order Raychaudhuri equation (or second Friedmann equation):

a=−κ2(ρ+3p)

6, (6)

where

κ2=

πGN

and

1. Here,

and

are the energy density and the pressure of

the matter content of the universe, respectively, described by an isotropic perfect ﬂuid.

Note that we have adopted the simpliﬁcation of absorbing the cosmological constant,

if present, into the stress-energy tensor.

Universe 2024,10, 162 3 of 24

(ii)

The ﬁrst Friedmann equation:

˙

a2

=−k

a2+κ2ρ

3. (7)

The latter equation acts as a ﬁrst integral of Equation (6) and constrains the solutions

in connection with the possible spatial curvature cases set by k=0, ±1.

Furthermore, it is useful to absorb the spatial curvature term into the density by deﬁning

ρk=−3k

8πa2;ρeffective =ρ+ρk. (8)

Similarly, it is useful to deﬁne

pk=k

8πa2;peffective =p+pk. (9)

In this way, we have

H2=˙

a2

=κ2ρeffective

3and ¨

a=−κ2(ρeffective +3peffective)

6, (10)

where the Raychaudhuri equation can be rewritten in terms of

and

ωeffective =peffective/

ρeffective as

H+H2=−κ21+3ωeffective

6ρeffective , (11)

giving us

H=−κ21+ωeffective

2ρeffective (12)

Now, making use of Equation (10), one can rewrite H(z)as

H=H0rρeffective

ρeffective,0 , (13)

where

ρeffective

and

ρeffective,0

represent the energy density at redshift

and at the present

time, respectively. Introducing the appropriate notion of critical density (sometimes called

Hubble density) and Omega (or density) parameter

ρcrit =3H2

κ2=ρeffective,0;Ω=ρeffective

ρcrit ; (14)

respectively, we see that these deﬁnitions automatically imply

Ω0≡

1, and hence,

H(z) =

H0pΩ(z)

. Thence, for the RD, even before choosing a speciﬁc cosmological model, we

have the quite general exact (FLRW) result:

z=(1+z)−qΩ(z)H0;(Ω0≡1). (15)

The RD will exhibit a zero whenever

Ω(z∗) = (1+z∗)2. (16)

One obvious (trivial) root occurs at

z∗=

0. We shall soon see that, typically, there will be

at least one other nontrivial root. Again, we emphasize that, up to this point, all quoted

results are exact, at least within the context of FLRW spacetime.

Universe 2024,10, 162 4 of 24

2.1. Choosing a Speciﬁc Cosmological Model

If we can approximate the cosmological ﬂuid by a collection of

non-interacting ﬂuid

components with individual, strictly linear equations of state (EOS) pi=wiρi, then

Ω(z) =

∑

i=1

Ω0i(1+z)3(1+wi);N

∑

i=1

Ω0i=1; (17)

and so

H(z) = H0v

∑

i=1

Ω0i(1+z)3(1+wi);N

∑

i=1

Ω0i=1. (18)

Note that we are explicitly allowing possible spatial curvature, so both

and

Ωk

are allowed

to be nonzero, with the corresponding value of

being

wk=−

3. Once you allow

nonzero

Ωk

, then the sum of all the

Ω0i

is, by deﬁnition, set to unity. We can characterize

the components of this cosmological model as follows:

wi





>−1/3 matter-like (dust, radiation, etc.);

=−1/3 spatial curvature (the marginal case);

<−1/3 dark-energy-like (quintessence, cosmological constant, etc.).

(19)

Typically, one has

wi∈[−

]

or even

wi∈[−

]

, but such a restriction is not

absolutely necessary. Note that the “matter-like” components (

wi>−

3) dominate at

early times (small

), while the “dark-energy-like” components (

wi<−

3) dominate

at later times (large

). We shall now investigate the implications of this model for

Ω(z)

in some detail. We shall ﬁnd it advantageous to work with weighted moments of the

-parameters (for an akin deﬁnition of an averaged adiabatic index

γ=

, see [

]).

Speciﬁcally, at the current epoch, we deﬁne

⟨wn⟩0=∑N

i=1Ωi0wn

∑N

i=1Ωi0

∑

i=1

Ωi0wn

i, (20)

while at redshift z, we deﬁne

⟨wn⟩z=∑N

i=1Ωi(z)wn

∑N

i=1Ωi(z)=∑N

i=1Ωi0(1+z)3(1+wi)wn

∑N

i=1Ωi0(1+z)3(1+wi). (21)

Note that Taylor series expansions of

Ω(z)

will mathematically converge only for

|z|<

and will be astrophysically most useful only for 0

≤z≪

1. Furthermore, within the context

of our {Ω0i,wi}matter model, for ωeffective =peffective/ρeffective, we have

ωeffective =∑N

i=1Ωi(z)wi

∑N

i=1Ωi(z)=⟨w⟩z. (22)

The dominance of some individual component,

j∗

, with regard to the others can be charac-

terized by

Ωj∗(z)>

2 (this follows trivially from

Ωj∗(z)>∑i̸=j∗Ωi(z) = (

−Ωj∗(z))

Furthermore, the emergence of late-time accelerated expansion happens, of course, for

⟨w⟩z<−

3 (for some

z<zcrit

and, in particular, for

z→

0). We shall have more to say

on these issues later on.

Given the convergence issue with the

-redshift expansions for

1—namely all the

regions of interest for RD experiments—one can resort, for example, to the y-redshift:

Ω(y) =

∑

i=1

Ω0i(1−y)−3(1+wi);N

∑

i=1

Ω0i=1; (23)

Universe 2024,10, 162 5 of 24

whence

H(y) = H0v

∑

i=1

Ω0i(1−y)−3(1+wi);N

∑

i=1

Ω0i=1. (24)

Again, the Taylor series expansions of

Ω(y)

will mathematically converge only for

|y|<

and will be astrophysically most useful only for 0

≤y≪

1. Fortunately, 0

≤y<

1 covers

the entire physically relevant region 0

≤z<∞

. Again, it is important to highlight this is

the reason why so much effort is put into working with the y-redshift (or its variants).

2.1.1. Generic Model

By inserting

(18)

into

(15)

(this is still exact in FLRW with this particular model for the

cosmological EOS), we obtain the generic result:

z=H0





(1+z)−v

∑

i=1

Ω0i(1+z)3(1+wi)





∑

i=1

Ω0i=1. (25)

Observationally, one now merely needs to ﬁt the

{Ω0i

wi}

to the empirical data. In contrast,

a theorist need only ﬁt the {Ω0i,wi}to their preferred toy model.

2.1.2. Λ-CDM

For a general four-component

-CDM model, when explicitly allowing the inclusion

of both spatial curvature Ωkand radiation Ωrcomponents, one has

H=H0qΩΛ+Ωk(1+z)2+Ωm(1+z)3+Ωr(1+z)4, (26)

with

ΩΛ+Ωk+Ωm+Ωr=1. (27)

This particular model is commonly believed to be an accurate representation of the evolu-

tion of our own universe from the current epoch to at least as far back as the surface of last

scattering (the CMB) at z≈1100. Thence, by eliminating Ωk, one has

H=H0qΩΛ+ (1−ΩΛ−Ωm−Ωr)(1+z)2+Ωm(1+z)3+Ωr(1+z)4, (28)

which leads to the RD equation in terms of the cosmological parameters:

z=H0(1+z)−qΩΛ+ (1−ΩΛ−Ωm−Ωr)(1+z)2+Ωm(1+z)3+Ωr(1+z)4. (29)

Using the latest PDG 2022 data, the density parameters at the current epoch are estimated

to be [46,47]

ΩΛ=0.685(7),Ωm=0.315(7),Ωr=5.38(15)×10−5,Ωk=0.0007(19).

By using those data to plot the RD vs.

, we can see (Figure 1) that the RD has a maximum

z≈

0.875, where

z≈

0.213

, which then, subsequently, has a zero at

z≈

1.918, beyond

which the RD becomes negative—a well-known result due to the fact that the universe is

matter-dominated for

z≳

2. While the existence of this peak in the RD is tolerably well

known [23,25], we will have considerably more to say on this point later.

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Citation:Lobo,F.S.N.;Mimoso,J.P.;Santiago,J.;Visser,M.DynamicalAnalysisoftheRedshiftDriftinFLRWUniverses.Universe2024,10,162.https://doi.org/10.3390/universe10040162AcademicEditor:PierStefanoCorasanitiReceived:23February2024Revised:21March2024Accepted:27March2024Published:29March2024Copyright:©2024b...

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