In general relativity, a gravitational horizon (more commonly known as the “apparent horizon”) is an imaginary surface beyond which all null geodesics recede from the observer. The Universe has an apparent (gravitational) horizon, but unlike its counterpart in the Schwarzschild and Kerr metrics, it is not static. It may eventually turn into an event horizon—an asymptotically defined membrane that forever separates causally connected events from those that are not—depending on the equation of state of the cosmic fluid. In this paper, we examine how and why an apparent (gravitational) horizon is manifested in the Friedmann–Robertson–Walker metric, and why it is becoming so pivotal to our correct interpretation of the cosmological data. We discuss its observational signature and demonstrate how it alone defines the proper size of our visible Universe. In so doing, we affirm its physical reality and its impact on cosmological models.

1.
W.
Rindler
, “
Visual horizons in world models
,”
MNRAS
116
,
662
678
(
1956
).
2.
R. J.
Nemiroff
and
B.
Patla
, “
Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations
,”
Am. J. Phys.
76
,
265
276
(
2008
).
3.
V.
Faraoni
,
Cosmological and Black Hole Apparent Horizons
(
Springer
,
New York
,
2015
).
4.
R.
Gautreau
, “
Cosmological Schwarzschild radii and Newtonian gravitational theory
,”
Am. J. Phys.
64
,
1457
1467
(
1996
).
5.
G.
Birkhoff
,
Relativity and Modern Physics
(
Harvard U.P.
,
Cambridge
,
1923
).
6.
G. F. R.
Ellis
and
T.
Rothman
, “
Lost horizons
,”
Am. J. Phys.
61
,
883
893
(
1993
).
7.
I.
Ben-Dov
, “
Outer trapped surfaces in Vaidya spacetimes
,”
Phys. Rev. D
75
,
064007
(
2007
).
8.
V.
Faraoni
, “
Cosmological apparent and trapping horizons
,”
Phys. Rev. D
84
,
024003
(
2011
).
9.
I.
Bengtsson
and
J. M. M.
Senovilla
, “
Region with trapped surfaces in spherical symmetry, its core, and their boundaries
,”
Phys. Rev. D
83
,
044012
(
2011
).
10.
F.
Melia
, “
The cosmic horizon
,”
MNRAS
382
,
1917
1921
(
2007
).
11.
W.
de Sitter
, “
On the relativity of inertia. Remarks concerning Einstein's latest hypothesis
,”
Proc. Akad. Wetensch Amsterdam
19
,
1217
1225
(
1917
) ADS bibliographic code 1917KNAB…19.1217D.
12.
A.
Friedmann
, “
Über die Krümmung des Raumes
,”
Z. Phys.
10
,
377
386
(
1922
).
13.
T. M.
Davis
and
T. H.
Lineweaver
, “
Expanding confusion: Common misconceptions of cosmological horizons and the superluminal expansion of the universe
,”
PASA
21
,
97
109
(
2004
).
14.
P.
van Oirschot
,
J.
Kwan
, and
G. F.
Lewis
, “
Through the looking glass: Why the ‘cosmic horizon’ is not a horizon
,”
MNRAS
404
,
1633
1638
(
2010
).
15.
G. F.
Lewis
, “
Matter matters: Unphysical properties of the Rh = ct universe
,”
MNRAS
432
,
2324
2330
(
2013
).
16.
D. Y.
Kim
,
A. N.
Lasenby
, and
M. P.
Hobson
, “
Spherically-symmetric solutions in general relativity using a tetrad approach
,”
GRG
50
,
i.d. 29, 37
(
2018
).
17.
O.
Bikwa
,
F.
Melia
, and
A. S. H.
Shevchuk
, “
Photon geodesics in Friedmann-Robertson-Walker cosmologies
,”
MNRAS
421
,
3356
3361
(
2012
).
18.
F.
Melia
, “
The gravitational horizon for a Universe with phantom energy
,”
JCAP
09
,
029
038
(
2012
).
19.
F.
Melia
, “
Proper size of the visible Universe in FRW metrics with a constant spacetime curvature
,”
CQG
30
,
155007
(
2013
).
20.
W. M.
Stuckey
, “
Can galaxies exist within our particle horizon with Hubble recessional velocities greater than c?
,”
Am. J. Phys.
60
,
142
146
(
1992
).
21.
T. M.
Davis
and
C. H.
Lineweaver
, “
Superluminal recession velocities
,”
AIP Conf. Proc.
555
,
348
351
(
2001
).
22.
A.
Kaya
, “
Hubble's law and faster than light expansion speeds
,”
Am. J. Phys.
79
,
1151
1154
(
2011
).
23.
F.
Melia
,
The Edge of Infinity–Supermassive Black Holes in the Universe
(
Cambridge U.P.
,
Cambridge
,
2003
).
24.
F.
Melia
and
M.
Abdelqader
, “
The cosmological spacetime
,”
Int. J. Mod. Phys.
18
,
1889
1901
(
2009
).
25.
F.
Melia
and
A. S. H.
Shevchuk
, “
The Rh = ct universe
,”
MNRAS
419
,
2579
2586
(
2012
).
26.
F.
Melia
, “
Physical basis for the symmetries in the Friedmann-Robertson-Walker metric
,”
Front. Phys.
11
,
119801
(
2016
).
27.
F.
Melia
, “
The zero active mass condition in Friedmann-Robertson-Walker cosmologies
,”
Front. Phys.
12
,
129802
(
2017
).
28.
E. R.
Harrison
, “
Mining energy in an expanding universe
,”
Astrophys. J.
446
,
63
66
(
1995
).
29.
M. J.
Chodorowski
, “
A direct consequence of the Expansion of Space?
,”
MNRAS
378
,
239
244
(
2007
).
30.
M. J.
Chodorowski
, “
The kinematic component of the cosmological redshift
,”
MNRAS
413
,
585
594
(
2011
).
31.
Yu.
Baryshev
,
Practical Cosmology, Proceedings of the International Conference on Problems of Practical Cosmology
, edited by
Yurij V.
Baryshev
,
Igor N.
Taganov
, and
Pekka
Teerikorpi
(
TIN
,
St.-Petersburg
,
2008
), pp.
60
67
.
32.
E. F.
Bunn
and
D. W.
Hogg
, “
The kinematic origin of the cosmological redshift
,”
Am. J. Phys.
77
,
688
694
(
2009
).
33.
R. J.
Cook
and
M. S.
Burns
, “
Interpretation of the cosmological metric
,”
Am. J. Phys.
77
,
59
66
(
2009
).
34.
Ø.
Grøn
and
Ø.
Elgaroy
, “
Is space expanding in the Friedmann universe models?
,”
Am. J. Phys.
75
,
151
157
(
2007
).
35.
F.
Melia
, “
Cosmological redshift in FRW metrics with constant spacetime curvature
,”
MNRAS
422
,
1418
1424
(
2012
).
36.
Planck Collaboration
 et al., “
Planck 2015 results. XIII. Cosmological parameters
,”
A&A
594
,
A13
A76
(
2016
).
37.
S.
Weinberg
,
Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity
(
Wiley
,
New York
,
1972
).
38.
J. T.
Jebsen
, “
On the general spherically symmetric solutions of Einstein's gravitational equations in Vacuo
,”
Ark Mat Ast Fys (Stockholm)
15
,
18
24
(
1921
) [Reprinted in Gen. Relativ. Grav. 37, 2253–2259 (2005)].
39.
C. W.
Misner
and
D. H.
Sharp
, “
Relativistic equations for adiabatic, spherically symmetric gravitational collapse
,”
Phys. Rev.
136
,
571
576
(
1964
).
40.
W. C.
Hernandez
, Jr.
and
C. W.
Misner
, “
Observer time as a coordinate in relativistic spherical hydrodynamics
,”
Astrophys. J.
143
,
452
465
(
1966
).
41.

As demonstrated by these authors, the mass M(R) emerging from the grr coefficient of the FRW metric is defined in terms of the comoving density ρ and the proper radius R = a(t)r, where a(t) is the expansion factor and r is the (unchanging) comoving radius (see Eq. (3)). Therefore, as one can see in Eq. (15), Eqs. (1) and (2) lead to an identification of Rh as the Hubble radius only if Rh is itself a proper distance, in contrast to the claim in Ref. 16 that Rh is not a proper radius.

42.
A.
Prain
,
V.
Vitagliano
,
V.
Faraoni
, and
L. M.
Lapierre-Léonard
, “
Hawking-Hayward quasi-local energy under conformal transformations
,”
CQG
33
,
145008
(
2016
).
43.
S.
Chakraborty
and
N.
Dadhich
, “
Brown-York quasilocal energy in Lanczos-Lovelock gravity and black hole horizons
,”
J. High Energy Phys.
2015
,
1
19
(
2015
).
44.
H.
Zeng
,
F.
Melia
, and
L.
Zhang
, “
Cosmological tests with the FSRQ gamma-ray luminosity function
,”
MNRAS
462
,
3094
3103
(
2016
).
45.
C. L.
Bennett
 et al., “
First-year Wilkinson microwave anisotropy probe (WMAP) observations: Preliminary maps and basic results
,”
ApJS
148
,
1
27
(
2003
).
46.
D. N.
Spergel
 et al., “
First-year Wilkinson microwave anisotropy probe (WMAP) observations: Determination of cosmological parameters
,”
ApJS
148
,
175
194
(
2003
).
47.
Planck Collaboration
and
P. A. R.
Ade
 et al., “
Planck 2013 results. XXIII. Isotropy and statistics of the CMB
,”
A&A
571
,
A23
A71
(
2014
).
48.
H.
Weyl
, “
Zur allgemeinen relativitätstheorie
,”
Z. Phys.
24
,
230
232
(
1923
).
49.
A. B.
Nielsen
and
M.
Visser
, “
Production and decay of evolving horizons
,”
CQG
23
,
4637
4658
(
2006
).
50.
G.
Abreu
and
M.
Visser
, “
Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes
,”
Phys. Rev. D
82
,
044027
(
2010
).
51.
J. R.
Oppenheimer
and
G. M.
Volkoff
, “
On massive neutron cores
,”
Phys. Rev.
55
,
374
381
(
1939
).
52.
P. S.
Florides
, “
The Robertson-Walker metrics expressible in static form
,”
Gen. Rel. Grav.
12
,
563
574
(
1980
).
53.

Of course, H is necessarily “constant” at every position only on a given time slice t, but may change with time, depending on the expansion dynamics.

54.
G. F.
Lewis
and
P.
van Oirschot
, “
How does the Hubble sphere limit our view of the Universe?
,”
MNRAS
423
,
L26
L29
(
2012
).
55.
R.
Caldwell
, “
A phantom menace? Cosmological consequences of a dark energy component with super-negative equation of state
,”
PLB
545
,
23
29
(
2002
).
56.
R.
Caldwell
,
M.
Kamionkowski
, and
N. N.
Weinberg
, “
Phantom energy: Dark energy with w < −1 causes a cosmic doomsday
,”
Phys. Rev. Lett.
91
,
071301
(
2003
).
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