The Shared Causal Pasts and Futures of Cosmological Events
TThe Shared Causal Pasts and Futures of Cosmological Events
Andrew S. Friedman , David I. Kaiser , and Jason Gallicchio ∗ Center for Theoretical Physics and Department of Physics,Massachusetts Institute of Technology,Cambridge, Massachusetts 02139 USA Kavli Institute for Cosmological Physics,University of Chicago,Chicago, Illinois 60637 USA (Dated: July 16, 2018)We derive criteria for whether two cosmological events can have a shared causalpast or a shared causal future, assuming a Friedmann-Lemaitre-Robertson-Walkeruniverse with best-fit ΛCDM cosmological parameters from the
Planck satellite. Wefurther derive criteria for whether either cosmic event could have been in past causalcontact with our own worldline since the time of the hot “big bang,” which we taketo be the end of early-universe inflation. We find that pairs of objects such as quasarson opposite sides of the sky with redshifts z ≥ .
65 have no shared causal past witheach other or with our past worldline. More complicated constraints apply if theobjects are at different redshifts from each other or appear at some relative angleless than 180 ◦ , as seen from Earth. We present examples of observed quasar pairsthat satisfy all, some, or none of the criteria for past causal independence. Givendark energy and the recent accelerated expansion, our observable universe has a finiteconformal lifetime, and hence a cosmic event horizon at current redshift z = 1 . PACS numbers: 04.20.Gz; 98.80.-k; Preprint MIT-CTP 4440 ∗ Email addresses: [email protected]; [email protected]; [email protected] a r X i v : . [ a s t r o - ph . C O ] A ug I. INTRODUCTION
Universes (such as our own) that expand or contract over time can have nontrivial causalstructure. Even in the absence of physical singularities, cosmic expansion can create horizonsthat separate observers from various objects or events [1–4]. Our observable Universe hashad a nontrivial expansion history: it likely underwent cosmic inflation during its earliestmoments [5–7]; and observations strongly indicate that our Universe was decelerating afterinflation and is presently undergoing a phase of accelerated expansion again, driven by darkenergy consistent with a cosmological constant [8–13]. The late-time acceleration creates acosmic event horizon that bounds the furthest distances observers will be able to see, evenin infinite cosmic proper time [14–16].One of the best-known examples of how nontrivial expansion history can affect causalstructure concerns the cosmic microwave background radiation (CMB). At the time theCMB was emitted at redshift z ≈ z (cid:38) .
1) with which we may explore causal structurebeyond the example of the CMB (e.g. [20–29]). We may ask, for example, whether twoquasars that we observe today have been in causal contact with each other in the past. Howfar away do such objects need to be to have been out of causal contact between the hot bigbang and the time they emitted the light we receive today? Previous work investigatingthe uniformity of physical laws on cosmological scales has long emphasized the importanceof observing causally disjoint quasars (e.g. [30, 31]), culminating in recent searches forspatiotemporal variation of fundamental dimensionless constants such as the hydrogen fine-structure constant and the proton-to-electron mass ratio using quasar absorption lines (e.g.see [32, 33] and references therein, although see [34]). We add to such longstanding causalstructure applications by outlining a novel formalism unifying past and future causal rela-tions for cosmic event pairs, generalized for arbitrary space-time curvature, and applyingit to the current best measurements of the cosmological parameters for our own Universefrom the
Planck satellite [17]. An additional application, which will be explored in futurework, centers on fundamental aspects of quantum mechanics where it is important to clarifywhether physical systems are prepared independently on causal grounds alone. Experimentsdesigned along these lines might be able to test both fundamental physics and perhaps evenspecific models of inflation. This strongly motivates developing a secure handle on the the-oretical conditions for past causal independence of cosmic event pairs, which is the primaryfocus of this work.In this paper we derive criteria for events to have a shared causal past — that is, whetherthe past-directed lightcones from distant emission events overlap with each other or withour own worldline since the time of the big bang (at the end of inflation). If event pairs haveno shared causal past with each other, no additional events could have jointly influencedboth of them with any signals prior to the time they emitted the light that we observetoday. Similarly, if an event’s past lightcone does not intersect our worldline, no eventsalong Earth’s comoving worldline could have influenced that event with any signals beforethe time of emission. We find, for example, that objects like quasars on opposite sides ofthe sky with redshifts z ≥ .
65 had been out of causal contact with each other and with ourworldline between the big bang and the time they emitted the light we receive today. Thiscritical value, which we call the causal-independence redshift, z ind = 3 .
65, is not particularlylarge by present astronomical standards; tens of thousands of objects have been observedwith redshifts z > z ind (e.g. quasars from the Sloan Digital Sky Survey and other surveys[22, 23]). More complicated past causal independence constraints apply if the objects are atdifferent redshifts from each other or appear at some relative angle (as seen from Earth) lessthan 180 ◦ . The criteria depend on cosmological parameters such as the Hubble constant andthe relative contributions to our Universe from matter, radiation, and dark energy. Usingthe current best-fit parameters for a spatially flat cosmology with dark energy and colddark matter (ΛCDM), we derive conditions for past causal independence for pairs of cosmicobjects at arbitrary redshift and angle. We also generalize these relationships for spacetimeswith nonzero spatial curvature.In addition to considering objects’ shared causal pasts, we also investigate whether theywill be able to exchange signals in the future, despite the late-time cosmic acceleration andthe associated cosmic event horizon. By studying the overlap of objects’ future lightconeswith each other’s worldlines, we determine under what conditions signals from various ob-jects (including Earth) could ever reach other distant objects. Our discussion of both theshared causal futures and causal pasts of cosmic event pairs is presented within a unifiedformalism.Throughout the paper we assume that our observable Universe may be represented bya simply-connected, non-compact Friedmann-Lemaitre-Robertson-Walker (FLRW) metric,which is consistent with recent measurements of large-scale homogeneity and isotropy [35–37]. In Section II we establish units and notation for distances, times, and redshifts. InSection III we derive the conditions required for past causal independence in the case of aspatially flat FLRW metric, and in Section IV we derive comparable relations for FLRWmetrics of nonzero spatial curvature. Section V considers future lightcone intersections,and concluding remarks follow in Section VI. Appendix A revisits early-universe inflationand cosmic horizons within the formalism established in Sections II - III, and AppendixB examines the evolution of the “Hubble sphere,” beyond which objects recede from ourworldline faster than light. II. DISTANCES, TIMES, AND REDSHIFTS
For arbitrary spatial curvature, we may write the FLRW line-element in the form ds = − c dt + R a ( t ) (cid:20) d ˜ r (1 − k ˜ r ) + ˜ r (cid:0) dθ + sin θ dϕ (cid:1)(cid:21) , (1)where a ( t ) is the scale factor, c is the speed of light, R is a constant with units of length,and the dimensionless constant k = 0 , ± R , we take a ( t ) and ˜ r to be dimensionless for any spatial curvature k .) Theangular coordinates range over 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ π , and in the case k = 1, the radialcoordinate satisfies ˜ r ≤
1. We normalize a ( t ) = 1, where t is the present time.For arbitrary curvature k , the (dimensionless) comoving radial distance χ between anobject at coordinate ˜ r and the origin is given by χ = (cid:90) ˜ r d ˜ r (cid:48) √ − k ˜ r (cid:48) = arcsin ˜ r for k = 1 , ˜ r for k = 0 , arcsinh ˜ r for k = − . (2)We may likewise define a (dimensionless) conformal time, τ , via the relation dτ ≡ cR dta ( t ) . (3)Then we may rewrite the line-element of Eq. (1) as ds = R a ( τ ) (cid:2) − dτ + dχ + S k ( χ ) (cid:0) dθ + sin θdϕ (cid:1)(cid:3) , (4)where S k ( χ ) = sin χ for k = 1 ,χ for k = 0 , sinh χ for k = − . (5)It is also convenient to define C k ( χ ) ≡ (cid:113) − kS k ( χ ) = cos χ for k = 1 , k = 0 , cosh χ for k = − . (6)Given Eq. (4), light rays traveling along radial null geodesics ( dθ = dϕ = 0) obey dχ = dτ. (7)For any spatial curvature k , we set the dimensionful constant R , with units of length,to be R = cH , (8)where H is the present value of the Hubble constant with best-fit value H = 67 . − Mpc − =(14 .
53 Gyr) − [17]. In the case k = 1, the coordinates (˜ r, θ, ϕ ) only cover half the spatialmanifold. In that case, ˜ r = sin(0) = 0 at the north pole and ˜ r = sin( π/
2) = 1 at theequator, so for a single-valued radial coordinate ˜ r , we may only cover the upper (or lower)half of the manifold. We may avoid this problem by working with the coordinate χ in the k = 1 case and allowing χ to range between 0 ≤ χ ≤ π rather than 0 ≤ χ ≤ π/ z , of an object whose light was emitted at some time t e andwhich we observe today at t is given by1 + z = a ( t ) a ( t e ) = 1 a e , (9)upon using our normalization convention a ( t ) = 1 and defining a e ≡ a ( t e ). Following[38, 40], we parameterize the Friedmann equation governing the evolution of a ( t ) in termsof the function E ( a ) ≡ H ( a ) H = (cid:112) Ω Λ + Ω k a − + Ω M a − + Ω R a − , (10)where H ( a ) is the Hubble parameter for a given scale factor a = a ( t ). The Ω i are the ratiosof the energy densities contributed by dark energy (Ω Λ ), cold matter (Ω M ), and radiation(Ω R ) to the critical density ρ c = 3 H / (8 πG ), where G is Newton’s gravitational constant.We also define a fractional density associated with spatial curvature (Ω k ≡ − Ω T ) andthe total fractional density of matter, dark energy, and radiation (Ω T ≡ Ω M + Ω Λ + Ω R ).We assume that Ω Λ arises from a genuine cosmological constant with equation of state w = p/ρ = −
1, which is consistent with recent measurements [11, 12, 17, 35, 36, 41], andhence Ω Λ a − w ) = Ω Λ . Current observations yield best-fit cosmological parameters for ourUniverse consistent with (cid:126) Ω = ( h, Ω M , Ω Λ , Ω R , Ω k , Ω T ) = (0 . , . , . , . × − , , , (11)where we define the dimensionless Hubble constant as h ≡ H / (100 km s − Mpc − ). Valuesfor Eq. (11) are taken from Table 2, column 6 of [17] including the most recent CMBtemperature data from the Planck satellite and low multipole polarization data from the 9-year Wilkinson Microwave Anisotropy Probe (WMAP) release [42]. The fractional radiationdensity Ω R is derived from the relation Ω R = Ω M / (1 + z eq ) where Ω M = Ω b + Ω c is thefractional matter density given by the sum of the fractional baryon (Ω b ) and cold darkmatter (Ω c ) densities and z eq is the redshift of matter-radiation equality. The quantitiesΩ b h , Ω c h , and z eq are all listed in Table 2, column 6 of [17].Given Eqs. (3), (7), (9), and cosmological parameters from Eq. (11), we may evaluatecomoving distance along a (radial) null geodesic using either a ( t ) or z as our time-likevariable, χ = (cid:90) a e daa E ( a ) = (cid:90) z dz (cid:48) E ( z (cid:48) ) . (12)Although Eq. (12) does not permit analytic solutions for the general case in which thevarious Ω i are nonvanishing, the equation may be integrated numerically to relate comovingdistance to redshift.We may also consider how conformal time, τ , evolves. If τ = 0 is the beginning of timeand inflation did not occur, τ is equivalent to the comoving distance to the particle horizon, τ ( t ) = (cid:90) a e daa E ( a ) = (cid:90) ∞ z dz (cid:48) E ( z (cid:48) ) . (13)As above, τ is dimensionless and R τ /c = H − τ has dimensions of time. The present ageof the Universe, τ = τ ( t ), is given by τ ≡ (cid:90) daa E ( a ) = (cid:90) ∞ dzE ( z ) ≡ χ ∞ (14)which is equivalent to χ ∞ , the comoving distance to the particle horizon today (at thecomoving location corresponding to z = ∞ ).Even if inflation did occur, Eq. (13) is still a reliable way to calculate τ numerically fortimes after inflation, τ >
0. We consider inflation to begin at some early cosmic time t i andto persist until some time t end , where t end will typically be of the order t end ∼ O (10 − sec)[6, 7]. In this case, the limits of integration in Eq. (13) would be altered as τ ( t ) = (cid:90) a e a ( t end ) daa E ( a ) = (cid:90) z ( t end ) z dz (cid:48) E ( z (cid:48) ) , (15)where a ( t end ) is the scale factor at the end of inflation ( τ ( t end ) = 0) and z ( t end ) is theredshift for a hypothetical object we could observe today that emitted light at τ = 0.Although a ( t ) would have grown enormously during inflation, such that a ( t end ) (cid:29) a ( t i ), westill expect a ( t end ) (cid:28) a e for objects whose light was emitted well after the end of inflation.In particular, as discussed in Appendix A, for cosmological parameters as in Eq. (11) wehave a ( t end ) /a ( t ) ∼ O (10 − ), so that the nonzero lower bound to the scale-factor integralin Eq. (15) makes a negligible numerical contribution to the evolution of τ for τ > z ( t end ) ∼ O (10 )in the integral over redshift in Eq. (15). Thus we may still use Eq. (13) to evaluate τ numerically for times after the end of inflation.If inflation did occur, it would correspond to times τ <
0. For convenience we assume k = 0 for the explicit construction, though comparable results may be derived for k = ± τ ( t ) = 1 a ( t end ) (cid:18) H H I (cid:19) (cid:20) − a ( t end ) a ( t ) (cid:21) , (16)where H I is the value of the Hubble parameter during inflation, and we have used Eq. (8)for R . As usual, we find that τ < τ → − as t → t end . If weassume instant reheating to a radiation-dominated universe at t end , then we may matchsmoothly to a solution in which τ > a ( t ) = a ( t end ) (cid:18) tt end (cid:19) / (17)or τ ( t ) = 2 H t end a ( t end ) (cid:34)(cid:18) tt end (cid:19) / − (cid:35) (18)for t ≥ t end . Consistent with Eqs. (16) and (18), we therefore take the time of the big bangto be t end or τ ( t end ) = 0, after the end of early-universe inflation. III. SPATIALLY FLAT CASE
In this section we consider a spatially flat universe (like our own), and set k = Ω k = 0.We may then absorb the constant R into the definition of the comoving radial coordinateby introducing r ≡ R ˜ r = R χ . For the remainder of this section, we work in terms of acomoving radial coordinate r that carries dimensions of length, whereas the comoving radialcoordinate χ remains dimensionless, as does conformal time τ . In this section, boldfacesymbols represent spatial 3-vectors.With respect to the CMB dipole, we treat the Earth’s position in the CMB rest frameas the origin of the spatial coordinates. However, small corrections between the heliocentricand CMB frame or systematic redshift offsets from peculiar velocities do not affect ourresults, which are presented only to 2 decimal places in redshift. Typical random peculiarvelocities of σ pec v ≈
300 km s − lead to a systematic redshift error of only σ pec z ≈ .
001 [43].We now present the formalism for intersection of past lightcones for cosmic event pairsin a flat universe (see Fig. 1). An object A at comoving spatial location r A emits light atconformal time τ A which the observer on Earth receives at the present time, τ , while anobject B at comoving location r B emits light at conformal time τ B which the observer alsoreceives at τ . The light signals travel along null geodesics, ds = 0, and hence from Eq. (7)we immediately find τ − τ A = χ A = R − | r A | ,τ − τ B = χ B = R − | r B | . (19)The past-directed lightcones from the emission events A and B intersect at comoving location r AB at time τ AB , such that τ A − τ AB = R − | r A − r AB | ,τ B − τ AB = R − | r B − r AB | , (20)or, upon making use of Eq. (19), τ − τ AB = χ A + R − | r A − r AB | ,τ − τ AB = χ B + R − | r B − r AB | . (21)Without loss of generality, we consider event A to occur later than event B ( τ A > τ B and hence z A < z B ), in which case the past-directed lightcone centered on A must expandfurther before it intersects with the past-directed lightcone centered on B. By construction,we take event B to lie along the x axis and the vector r A to make an angle θ with respectto the x axis, so that an observer on Earth would see events A and B separated by an angle α = π − θ on the sky. See Fig. 2.Given the orientation of the vectors in Fig. 2b, we have | r A − r B | = | r A − r AB | + | r B − r AB | . (22)Using Eqs. (20) and (22), we then find τ AB = 12 ( τ A + τ B − χ L ) , (23)0 FIG. 1. Conformal diagram showing comoving distance, R χ in Glyr, versus conformal time, R τ /c in Gyr, for the case in which events A and B appear on opposite sides of the sky as seen fromEarth ( α = 180 ◦ ). The observer sits at Earth at χ = 0 at the present conformal time τ = τ .Light is emitted from A at ( χ A , τ A ) and from B at ( χ B , τ B ); both signals reach the Earth alongour past lightcone at (0, τ ). The past-directed lightcones from the emission events (red and bluefor A and B, respectively) intersect at ( χ AB , τ AB ) and overlap for 0 < τ < τ AB (purple region).For redshifts z A = 1 and z B = 3 and a flat ΛCDM cosmology with parameters given in Eq. (11),the events are located at comoving distances R χ A = 11 .
11 Glyr and R χ B = 21 .
25 Glyr, withemission at conformal times R τ A /c = 35 .
09 Gyr and R τ B /c = 24 .
95 Gyr. The past lightconesintersect at event AB at R χ AB = 10 .
14 Glyr at time R τ AB /c = 13 .
84 Gyr, while the presenttime is R τ /c = 46 .
20 Gyr. Also shown are the cosmic event horizon (line separating yellow andgray regions) and the future-directed lightcones from events A and B (thin dashed lines) and fromthe origin (0,0) (thick dashed lines). In a ΛCDM cosmology like ours, events in the yellow regionoutside our current past lightcone are space-like separated from us today but will be observablein the future, while events in the gray region outside the event horizon are space-like separatedfrom observers on Earth forever. Additional scales show redshift (top horizontal axis) and time asmeasured by the scale factor, a ( τ ), and by proper time, t , (right vertical axis) as measured by anobserver at rest at a fixed comoving location. where we have defined χ L as the (dimensionless) comoving spatial distance between eventsA and B: χ L ≡ R − | r A − r B | = R − (cid:112) ( r A − r B ) · ( r A − r B )= (cid:113) χ A + χ B − χ A χ B cos α . (24)In the special case α = π (see Fig. 1), for which χ L → χ A + χ B , Eq. (23) reduces to τ AB → τ A + τ B − τ (25)1 r A r B r A − r A B r B − r A B r AB FIG. 2. (
Left ) Plot of our past lightcone from τ (gray outer cone) and the past lightcones fromemission events A and B (red and blue cones, respectively). The green circles show the projectionof the past lightcones on the hypersurface τ = τ AB when the lightcones first intersect. For thecase shown here, α = 135 ◦ , z A = 1, and z B = 3. ( Right ) Plot of the spatial ( x, y ) plane for thehypersurface τ = τ AB , corresponding to the green circles in the left figure. Earth is at the origin.Event A occurs at comoving location r A (red vector) and event B occurs at comoving location r B (blue vector). The past-directed lightcones from A and B appear in the plane as circles centeredon A and B, respectively. The past lightcones intersect at event AB at comoving location r AB (green vector). The angle between events A and B as seen from Earth is α = π − θ . For animationsof the intersecting lightcones as one varies z i and α , see the online Supplementary Materials at http://prd.aps.org/supplemental/PRD/v88/i4/e044038 , which include 11 animationswith captions based on Figures 1 and 2, constructed for cosmological parameters from the Planck satellite given by Eq. 11. The animations vary or hold fixed the redshifts and angular separationsof cosmic event pairs to illustrate the conditions for events to have either a shared causal pastor no shared causal past since the big bang. These and other animations are also available at . upon using Eq. (19).We may also solve for the comoving spatial location, r AB , at which the past-directedlightcones intersect. Squaring both sides of the identity r A = r B + ( r A − r B ) yields r A = r B + r L − r B r L cos β, (26)where β is the angle between vectors r B and ( r B − r A ), as in Fig. 2b, and r L = | r A − r B | = R χ L . We likewise have r AB · r AB = [ r B − ( r B − r AB )] · [ r B − ( r B − r AB )] . (27)Upon using r AB = R χ AB and Eq. (20) to substitute | r B − r AB | = R ( τ B − τ AB ), Eq. (27)may be written χ AB = χ B − χ B ( τ B − τ AB ) cos β + ( τ B − τ AB ) . (28)From Eqs. (26) and (28), we then find χ AB = χ B + ( τ B − τ AB ) − χ B χ L ( τ B − τ AB ) ( χ B − χ A cos α ) . (29)2By fixing α and χ B and using Eqs. (19), (23), and (24), we may derive the condition onthe critical comoving distance ˆ χ A such that the past lightcones from A and B intersect attime τ AB , ˆ χ A = χ B − ( τ − τ AB ) (cid:104) χ B (1+cos α )2( τ − τ AB ) − (cid:105) . (30)Alternatively, we may fix χ A and χ B to derive the crititcal angle ˆ α such that the pastlightcones intersect at τ AB ,ˆ α = cos − (cid:18) χ A + χ B − ( τ A + τ B − τ AB ) χ A χ B (cid:19) . (31)When τ AB ≤
0, events A and B share no causal past after the end of inflation. Consideringevent pairs that just barely meet this condition ( τ AB = 0) leads to Figs. 3 and 4, wherewe use Eq. (30) with τ AB = 0 to plot the hyperbolic curves for different angles α in Fig.3a and Fig. 4. For Fig. 3b, we must invert Eq. (12) numerically to solve for the redshift z corresponding to a given comoving distance χ ( z ). Setting τ AB = 0, then for χ A ≥ ˆ χ A or α ≥ ˆ α , events A and B share no causal past since the big bang. In particular, if we fix α = π and consider the symmetric case in which χ A = χ B , then Eq. (30) for τ AB = 0 andcosmological parameters (cid:126) Ω as in Eq. (11) yields R χ ind = 23 .
10 Glyr, which, using Eq. (12),corresponds to the causal-independence redshift z ind = 3 . τ = 0. From Eq. (7), for τ ≥ χ, τ ) = (0 ,
0) is given by χ flc ( τ ) = τ. (32)See Fig. 1. If inflation did not occur and τ = 0 corresponds to t = 0, then χ flc ( τ ) = χ ph ( τ ),the comoving distance to the particle horizon for an observer at rest at χ = 0. Along theradial null geodesic extending backward from Earth at ( χ, τ ) = (0 , τ ) toward the event atA, the past-directed lightcone is given by χ plc ( τ ) = τ − τ. (33)The past-directed lightcone from (0, τ ) will intersect the future-directed lightcone from (0 , χ lc at conformal time τ lc χ plc ( τ lc ) = χ flc ( τ lc ) (34)3 FIG. 3. (
Left ) Comoving distance R χ A versus R χ B for pairs of objects separated by angle α , suchthat ( a ) their past-directed lightcones intersect at τ AB = 0 (colored curves for various angles), and( b ) neither object’s past-directed lightcone intersects our worldline after τ = 0 (white box in upperright corner). For a given α , comoving distances for event pairs that lie above the correspondingcolored curve (toward the upper right corner) satisfy τ AB < τ >
0; thus the Earth’scomoving location had been in causal contact with the event prior to emission. Objects in thelower left of the plot (dark gray region) have τ AB > α = 180 ◦ and χ A = χ B , objects with R χ > R χ ind = 23 .
10 Glyrshare no causal past with each other or with our worldine since τ = 0. ( Right ) The same plotin terms of redshift rather than comoving distance. For α = 180 ◦ and z A = z B , object pairswith z > z ind = 3 .
65 share no causal past with each other or with our worldline since τ = 0.Both plots are constructed for a flat ΛCDM cosmology with parameters (cid:126) Ω given in Eq. (11). Inboth figures, the dashed black box corresponds to the most distant object observed to date, at z max = 8 .
55 or R χ max = 30 .
31 Glyr, corresponding to the Gamma-Ray Burst in associated hostgalaxy UDFy-38135539 [44]. or τ lc = 12 τ . (35)As long as τ A < τ lc = τ /
2, then the past lightcone from event A will not intersect theobserver’s worldline since the big bang at τ = 0. By construction, since we have identified τ A ≥ τ B , the past lightcone of event B will likewise not intersect the observer’s worldlinesince τ = 0. For (cid:126) Ω as in Eq. (11), the requirement that τ A < τ / z A > z ind = 3 .
65. See Fig. 3.Requiring both τ AB ≤ and τ B ≤ τ A < τ / τ = 0. A quick examination of Fig. 1 illustrates that if4 FIG. 4. For various fixed values of z B , we plot the critical redshift ˆ z A vs. the angular separation α such that τ AB = 0. For each z B and α , ˆ z A is derived from ˆ χ A in Eq. (30) by inverting Eq.(12) numerically. For all values of z B , ˆ z A monotonically increases as α decreases: as the angularseparation between event pairs decreases, larger redshifts for object A (for a given z B ) are requiredfor the events to have no shared causal past. Event pairs with z A > ˆ z A that lie above the coloredcurve for a given α and z B have no shared causal past since the end of inflation. For any angle α ≤ ◦ , events A and B have no shared causal past with Earth’s worldline if z A > z ind = 3 . z B > z ind = 3 .
65. As in Fig. 3 the dashed horizontal linecorresponds to the most distant object observed to date, at z max = 8 . the emission events A and B have no shared causal past with each other or with us since τ = 0, then neither will any prior events along the worldlines of A and B. Many real objectsvisible in the sky today fulfill the conditions τ AB ≤ τ B ≤ τ A < τ /
2. Representativeastronomical objects (quasar pairs) that obey all, some, or none of these joint conditionsare displayed in Fig. 5 and listed in Table I.Of course, one may consider objects that have been out of causal contact with each otheronly during more recent times. For example, one may calculate the criteria for objects’past lightcones to have shared no overlap since the time of the formation of the thin diskof the Milky Way galaxy around 8 .
80 Gyr ago [45]; or since the formation of the Earth4 .
54 Gyr ago [46]; or since the first appearance on Earth of eukaryotic cells (precursors tomulticellular organisms) 1 .
65 Gyr ago [47]. Events more recent than around 1 .
35 Gyr agocorrespond to redshifts z ≤ .
1, and hence to distances where peculiar velocities are notnegligible compared to cosmic expansion [43]. For the α = 180 ◦ case, pushing the past-lightcone intersection time closer to the present day, τ AB → τ , yields curves in the z A - z B FIG. 5. Same as Fig. 3b, with three quasar pairs marked (see Table I). For pair 1 (red), the pastlightcones from each emission event share no overlap with each other or with our worldline since τ = 0. For pair 2 (green), the past lightcones from each emission event share no overlap with eachother, though the past lightcone from quasar A does overlap our worldline for τ >
0. For pair 3(blue), both emission events have past lightcones that intersect each other as well as our worldlineat times τ > Pair Separation Event Redshifts Object RA DEC R BAngle α i [deg] Labels z Ai , z Bi Names [deg] [deg] [mag] [mag]1 116 . A .
109 SDSS J031405.36-010403.8 48.5221 -1.0675 16 . . B .
606 SDSS J171919.54+602241.0 259.8313 60.3781 18 . .
92 130 . A .
167 KX 257 24.1229 15.0481 16 . . B .
086 SDSS J110521.50+174634.1 166.3396 17.7761 16 . .
13 154 . A .
950 Q 0023-4124 6.5496 -41.1381 14 . . B .
203 HS 1103+6416 166.5446 64.0025 14 . . TABLE I. Three quasar pairs from [23], as shown in Fig. 5. Redshift pairs ( z Ai , z Bi ) and angularseparations α i (in degrees) are chosen so that the pairs obey all (pair 1), some (pair 2), or none(pair 3) of the joint conditions of having no shared causal past with each other ( τ AB ≤
0) andeach having no shared causal past with our worldline ( τ A , τ B < τ / z A , z B > .
65. Basic properties of each quasar from[23] are also shown including: object names from the relevant quasar catalogs, celestial coordinates(
RA, DEC ) in degrees, and R and B band brightnesses (in magnitudes). plane that move down and to the left through the gray region of Fig. 3b. See Fig. 6 andTable II.6 FIG. 6. (
Left ) Redshifts z A vs. z B for the case α = 180 ◦ corresponding to various times at whichthe past-directed lightcones from emission events A and B last intersected. Lightcone intersectiontimes (in Gyr) are given in terms of conformal time since the big bang, H − τ AB , and lookbacktime t lAB , the cosmic time that has elapsed since the event in question. The black line toward theupper right corresponds to past-lightcone intersection at the big bang, τ AB = 0 as in Fig. 3. ( Right )Causal-independence redshift, z ind , vs. lookback time, t lAB , for the case z A = z B and α = 180 ◦ ,which asymptotes to z ind = 3 .
65 (dotted line) as the lightcone intersection approaches the time ofthe big bang, t lAB = 13 .
82 Gyr ago. All calculations assume parameters (cid:126)
Ω as in Eq. (11).
Event Redshift Lookback Time Proper Time Conformal Time causal-independence redshift z t l AB [Gyr] t AB [Gyr] H − τ AB [Gyr] ˜ z ind ( τ AB )Big Bang ∞ .
82 0 0 3 . .
23 8 .
80 5 .
01 33 .
32 0 . .
41 4 .
54 9 .
27 40 .
81 0 . .
124 1 .
65 12 .
16 44 .
45 0 . TABLE II. Table of sample lightcone intersection times equal to times of selected past cosmicevents from Fig. 6. Redshifts z in column 2 correspond to lookback, proper, and conformaltimes in columns 3-5. Pushing the past-lightcone intersection event forward, τ AB → τ , is highlynonlinear in redshift. Column 6 shows the causal-independence redshift ˜ z ind = ˜ z ind ( τ AB ) for eachconformal lightcone intersection time τ AB . For two sources on the sky with z A , z B > ˜ z ind ( τ AB ) and α = 180 ◦ , the past-directed lightcones from the emission events have not intersected each other orour worldline since τ AB . When the past lightcones intersect at the big bang, we have the familiar˜ z ind ( τ AB = 0) = z ind = 3 .
65. Computations are done for parameters (cid:126)
Ω from Eq. (11). FIG. 7. The curved-space analog of Fig. 2b, showing emission events A and B on the unit comovingspherical manifold ( k = 1). Earth is at the north pole (labeled point E). The center of the sphereis labeled O. The emission at event A occurs at angle χ A , which is the angle between the lines OEand OA; the emission at event B occurs at angle χ B . The past-directed lightcones from events Aand B intersect at point AB, which falls along the spatial geodesic connecting points A and B. Thecomoving arclength between events A and AB is u , and the comoving arclength between events Band AB is v . The angle between Earth (E) and the lightcone intersection event AB as seen fromevent B is β . As usual, α represents the angle between emission events A and B as seen fromEarth. IV. CURVED SPATIAL SECTIONS
We now consider how the results of Section III generalize to the cases of nonzero spatialcurvature. Given the FLRW line-element in Eq. (4), radial null geodesics satisfy Eq. (7)for arbitrary spatial curvature k . For concreteness, we consider first a space of positivecurvature, k = 1. As illustrated in Fig. 7, we place the Earth at point E at the north poleof the 3-sphere, with coordinates χ = θ = ϕ = 0. By construction, the coordinates χ and τ are dimensionless, while R a ( τ ) has dimensions of length. Thus we may take the comovingspatial manifold to be a unit sphere. In that case, the coordinate χ B (for example) givesthe angle between the radial line connecting the center of the sphere (point O) to the pointB on its surface, and the radial line connecting O to the point E at the north pole. Becausethe comoving spatial manifold has unit radius, χ B also gives the arclength along the surfacefrom the point B to the point E. At a given time τ , the physical distance between points Band E is then given by R a ( τ ) χ B . See Fig. 7.8As in the spatially flat case, we take the angle (as seen from Earth) between events Aand B to be α . The past-directed lightcones from events A and B intersect at a comovinglocation marked AB, which falls along the spatial geodesic connecting A and B. We labelthe comoving arclength between points A and B as χ L ; the comoving arclength from A toAB as u ; and the comoving arclength from point AB to B as v , such that χ L = u + v. (36)In our chosen coordinate system, neither A nor B is at the origin, and hence the pathconnecting points A and B does not appear to be a radial null geodesic. In particular, dθ/dλ (cid:54) = 0 along the path connecting points A and B, where λ is an affine parameter withwhich to parameterize the geodesic. But we may always rotate our coordinates such thatpoint A is the new origin (at χ (cid:48) = θ (cid:48) = ϕ (cid:48) = 0) and extend a radial null geodesic from thenew origin to point B (cid:48) . We may then exploit the spherical symmetry of the spatial manifoldto conclude that the arclength between points A (cid:48) and B (cid:48) will be the same as the arclengthbetween points A and B in our original coordinate system. Thus we find that the arclength u is the (comoving) radius of the past-directed lightcone between points A and AB, and fromEq. (7) we know that the radius of that lightcone at time τ AB must equal u = τ A − τ AB .Likewise, the arclength v = τ B − τ AB . Thus Eq. (36) is equivalent to τ AB = 12 ( τ A + τ B − χ L ) , (37)which is identical to Eq. (23) for the spatially flat case.We next wish to relate the arclength χ L to the inscribed angle α . Although Fig. 7 isconstructed explicitly for a positively curved space, we may use it to guide our applicationof the generalized law of cosines [38, 39] for either spherical ( k = 1) or hyperbolic ( k = − S k ( χ ) and C k ( χ ) defined in Eqs. (5) and (6), thearclength χ L between events A and B separated by an angle α may be written C k ( χ L ) = C k ( χ A ) C k ( χ B ) + kS k ( χ A ) S k ( χ B ) cos α. (38)The conformal time τ AB at which the past-directed lightcones intersect is thus given by Eq.(37), with χ L given by Eq. (38), which is equivalent to alternative expressions found in[48, 49] (but see [50]).We may likewise solve for the comoving spatial coordinate, χ AB , at which the past-directed lightcones intersect. Using Fig. 7, we again label the comoving arclength from9points AB to B as v = τ B − τ AB ; we label the inscribed angle between arclengths v and BEas β ; and we use the fact that the comoving arclength from point AB to E (the green arcin Fig. 7) is simply χ AB . Then for the triangle with vertices AB, E, and B, we have, in thegeneral curved case C k ( χ AB ) = C k ( v ) C k ( χ B ) + kS k ( v ) S k ( χ B ) cos β. (39)We may solve for the angle β by considering the larger triangle with vertices A, B, and E,for which we may write C k ( χ A ) = C k ( χ B ) C k ( χ L ) + kS k ( χ A ) S k ( χ L ) cos β, (40)where χ L is given by Eq. (38). Using Eq. (40) and the arclength v = τ B − τ AB , we mayrearrange Eq. (39) to yield C k ( χ AB ) = C k ( τ B − τ AB ) C k ( χ B ) + S k ( τ B − τ AB ) S k ( χ B ) S k ( χ A ) S k ( χ L ) [ C k ( χ A ) − C k ( χ B ) C k ( χ L )] , (41)with τ AB and C k ( χ L ) given by Eqs. (37) and (38), respectively.As in the flat case ( k = 0), for the spatially curved cases ( k = ±
1) if the past-directedlightcones from A and B intersect at time τ AB , given by Eq. (37), we can fix α and χ B toderive the condition on the critical comoving distance, ˆ χ A ,ˆ χ A = T − k (cid:18) C k ( χ B − τ + 2 τ AB ) − C k ( χ B ) k [ S k ( χ B ) cos α + S k ( χ B − τ + 2 τ AB )] (cid:19) , (42)where T k ( χ ) ≡ S k ( χ ) /C k ( χ ). Or we may fix χ A and χ B to determine the critical angle ˆ α such that the past lightcones of A and B intersect at time τ AB ,ˆ α = cos − (cid:18) C k ( τ A + τ B − τ AB ) − C k ( χ A ) C k ( χ B ) kS k ( χ A ) S k ( χ B ) (cid:19) . (43)Setting τ AB = 0, then for χ A ≥ ˆ χ A or α ≥ ˆ α the shared causal past of the events is pushedto τ ≤
0, into the inflationary epoch. We use Eq. (42) with τ AB = 0 to plot the hyperboliccurves for different angles α in the lefthand side of Fig. 8, and use Eq. (12) to relate χ to z for the plots in the righthand side of Fig. 8.Eqs. (42) and (43) are the curved-space generalizations of Eqs. (30) and (31). It iseasy to see that they reduce to the spatially flat case when k = 0. The limit k → χ i small compared to the radius of curvature. Since weare considering comoving distances on a unit comoving sphere (for k = 1) or on a unit0hyperbolic paraboloid (for k = − χ i (cid:28)
1. Then we may use theusual power-series expansions, S k ( χ ) = χ + O ( χ ) ,C k ( χ ) = 1 − k χ + O ( χ ) ,T k ( χ ) = χ + O ( χ ) (44)to write Eqs. (42) and (43) as ˆ χ A ( k ) = ˆ χ A (flat) + O ( χ i ) , ˆ α ( k ) = ˆ α (flat) + O ( χ i ) (45)in the limit χ i (cid:28)
1, where ˆ χ A (flat) and ˆ α (flat) are given by Eqs. (30) and (31), respectively.Comparing Figs. 3 and 8, one finds that FLRW universes with the same values of Ω M and Ω R as ours but with different values of Ω Λ yield different values of the critical angle ˆ α atwhich objects with redshifts z A and z B satisfy τ AB ≤
0. First note that Ω Λ ,f = 0 .
685 is thevalue of Ω Λ in Eq. (11) corresponding to our Universe. For a closed universe (Ω Λ > Ω Λ ,f )the range of critical angles ˆ α for which one may find objects with redshifts z A and z B thatsatisfy the condition τ AB ≤ Λ < Ω Λ ,f ) the range of critical angles ˆ α is narrower than in the spatially flatcase. These results are exactly as one would expect given the effect on the inscribed angle α at the point E as one shifts from a Euclidean triangle ABE to a spherical triangle or ahyperbolic triangle.1 FIG. 8. Same as Fig. 3 but for FLRW cosmologies with nonzero spatial curvature. We again con-sider parameters (cid:126)
Ω = ( h, Ω M , Ω Λ , Ω R , Ω k , Ω T ). ( Top Row ) A spatially closed universe ( k = 1) with (cid:126) Ω = (0 . , . , . , . × − , − . , . ). ( Bottom Row ) A spatially open universe( k = −
1) with (cid:126)
Ω = (0 . , . , . , . × − , . , . ). In each case, departures fromthe k = 0 case of Eq. (11) are indicated in italics. Compared to the k = 0 case, increasing Ω Λ shrinks the comoving distance scale and decreases the critical redshift for a given angle, whereasdecreasing Ω Λ stretches the comoving distance scale and increases the critical redshift for a givenangle. In all figures the dashed box represents the furthest observed object at z max = 8 .
55, corre-sponding to R χ max = 28 .
77 Glyr (closed), 30 .
31 Glyr (flat), and 31 .
55 Glyr (open). The criterionthat the past lightcones from events A and B do not intersect each other or our worldline for τ > α = 180 ◦ case (white square regions in Figs 3 and 8) yields z A , z B ≥ .
38 (closed), 3 . .
25 (open). V. FUTURE LIGHTCONE INTERSECTIONS
To extend our analysis of shared causal domains to the future of events A and B we define τ ∞ , the total conformal lifetime of the Universe, τ ∞ ≡ τ ( t = ∞ ) = (cid:90) ∞ daa E ( a ) . (46)As usual, τ ∞ is dimensionless while R τ ∞ /c = H − τ ∞ is measured in Gyr. We restrictattention to cosmologies like our own (ΛCDM with k = 0 and Ω Λ >
0) that undergo late-time cosmic acceleration and expand forever; that ensures that the total conformal lifetimeof the universe is finite , τ ∞ < ∞ . In particular, for (cid:126) Ω as in Eq. (11), we find H − τ ∞ = 62 . t = ∞ ), and it is impossible for us to senda signal today that will ever reach those objects in the future history of our Universe [3, 14–16]. The condition τ ∞ < ∞ holds for FLRW cosmologies with nonzero spatial curvature( k (cid:54) = 0) as long as Ω Λ > a ∗ = a ( t ∗ ), the comoving distance from our worldline at χ = 0 to the eventhorizon is given by χ eh ( t ∗ ) = (cid:90) ∞ a ∗ daa E ( a ) . (47)We may also trace back along the past lightcone from our present location (at τ rather than τ ∞ ) to the equivalent comoving distance. We set a ( t ∗ ) = a ( t ) = 1 and compute χ ( t ) = (cid:90) a eh daa E ( a ) . (48)Equating Eqs. (47) and (48) and using z eh = a − −
1, we find z eh ( t ) = 1 .
87 for our cosmologywith (cid:126)
Ω as in Eq. (11). Note that since z eh < z ind = 3 .
65, objects with z ≥ z ind are beyondthe cosmic event horizon: though we have received light from them at τ , no return signalfrom us will ever reach them before τ ∞ , nor (symmetrically) can light emitted from themnow (at τ ) ever reach us before the end of time. See Fig. 1 and Fig. 9.3 FIG. 9. Conformal diagram as in Fig. 1 showing the causal independence region bounded by theparticle horizon and the past-directed lightcone from the present time, τ (purple cross-hatching);the causal diamond bounded by the particle horizon and the cosmic event horizon (red stripes tiltedat -45 degrees), which includes the causal independence region; and the Hubble sphere (equal tothe apparent horizon for Ω k = 0; see Appendix B), which is the spacetime region beyond which allobjects are receding faster than light (yellow). Relevant redshifts include the current value of theredshift of the Hubble sphere, z hs = 1 .
48; the current redshift of the event horizon, z eh = 1 .
87; thecurrent value of the causal-independence redshift, z ind = 3 .
65; and the current value of the redshiftthat bounds the causal diamond, z ∞ ind = 9 .
99, which is the limiting value of the causal-independenceredshift as the proper age of the universe approaches infinity.
Another quantity of interest is the value of the redshift today of an emission event whoselight we will receive at τ ∞ but whose past lightcone has no overlap with our worldline since τ = 0. Such will be the case for any object with redshift z > z ∞ ind . As can be seen from Fig.9, z ∞ ind corresponds to the comoving location where the cosmic event horizon intersects thefuture lightcone from the origin, namely at the spacetime point ( χ, τ ) = ( τ ∞ / , τ ∞ / z ∞ ind either by computing the comoving distance from the origin tothe event horizon at τ ∞ /
2, or by computing the comoving distance of the forward lightconefrom the origin at τ ∞ /
2. In the first case we have χ eh (cid:16) τ ∞ (cid:17) = (cid:16) τ ∞ − τ ∞ (cid:17) = (cid:90) ∞ a ∞ ind daa E ( a ) , (49)4and in the second case we have χ flc (cid:16) τ ∞ (cid:17) = (cid:16) τ ∞ − (cid:17) = (cid:90) a ∞ ind daa E ( a ) . (50)Numerically inverting either Eq. (49) or (50) and using z ∞ ind = ( a ∞ ind ) − −
1, we find z ∞ ind =9 . > z ind for our cosmology with (cid:126) Ω as in Eq. (11). We emphasize that both z ∞ ind and z ind are evaluated at the time τ : among the objects whose redshift we might measure today,those with z > z ∞ ind will (later) release light that will reach our worldline at τ ∞ and whosepast lightcones from that later emission event will have had no overlap with our worldlinesince τ = 0.Events have no shared causal future if their future lightcones will never intersect eachother’s worldlines before τ ∞ . Thus we may ask whether the forward lightcone from emissionevent A intersects with the worldline of event B at some time τ < τ ≤ τ ∞ , or vice versa.This question can be answered by visual inspection of Fig. 1 for the special case for ouruniverse when α = 180 ◦ with fixed redshifts z A = 1, z B = 3. In Fig. 1, the future lightconesfrom events A and B are shown as thin dashed lines, and the worldines of A and B are shownas thin dotted lines at the fixed comoving locations χ A and χ B , respectively. From Fig. 1, itis easy to see that the future lightcone from event B crosses event A’s worldline before τ ∞ while the future lightcone from event A does not cross event B’s worldline before τ ∞ . Thus,in this situation, event B can send a signal to the comoving location of event A before theend of time, while event A can never signal event B’s worldline even in the infinite future.Similarly, we can consider the future lightcone from Earth today in Fig. 1, and note that,while we can signal the comoving location of event A before time ends, we will never be ableto send a signal that will reach the comoving location of event B. Of course, as shown inFig. 1, events A and B have already signaled Earth by virtue of our observing their emissionevents along our past lightcone at ( χ, τ ) = (0 , τ ), and the future lightcone from Earth todaynecessarily overlaps with the future lightcones of events A and B for τ > τ .For general cases at different angles and redshifts, without loss of generality we retain thecondition that emission event A occurred later than B, τ A ≥ τ B . We introduce the notationthat ˜ τ ij is the conformal time when the future lightcone from event i intersects the worldlineof event j , for ˜ τ ij > τ . Using Fig. 1 and reasoning as in Sections III and IV, we find˜ τ AB = χ L + τ A , ˜ τ BA = χ L + τ B , (51)5where χ L is the comoving distance between events A and B given by Eqs. (24) and (38) forthe spatially flat and curved cases, respectively. Since all angular and curvature dependenceis implicit in the χ L term, Eq. (51) holds for arbitrary angular separations 0 ≤ α ≤ ◦ and curvatures ( k = 0 , ± τ AB (cid:54) = ˜ τ BA ; the two are equal only if τ A = τ B . Givenour assumption that τ A ≥ τ B it follows that ˜ τ AB ≥ ˜ τ BA .Three scenarios are possible. ( a ) Events A and B will each be able to send a light signal tothe other, ˜ τ BA ≤ ˜ τ AB < τ ∞ , which implies χ L < τ ∞ − τ A ≤ τ ∞ − τ B . ( b ) B will be able to senda signal to A but not vice versa, ˜ τ BA < τ ∞ < ˜ τ AB , which implies τ ∞ − τ A < χ L < τ ∞ − τ B .( c ) A and B will forever remain out of causal contact with each other, ˜ τ AB ≥ ˜ τ BA ≥ τ ∞ ,which implies τ ∞ − τ A ≤ τ ∞ − τ B < χ L .Fixing χ B and α , we may find the comoving distance ˜ χ A such that the future lightconefrom A will intersect the worldline of B at time ˜ τ AB . For a spatially flat universe ( k = 0),we find ˜ χ A = χ B − (˜ τ AB − τ ) τ AB − τ + χ B cos α ) . (52)Or we may fix χ A and χ B and find the critical angle, ˜ α AB , such that the future lightconefrom A intersects the worldline of B at time ˜ τ AB ,˜ α AB = cos − (cid:18) χ A + χ B − (˜ τ AB − τ A ) χ A χ B (cid:19) . (53)As in Section IV, we may generalize these results to the case of spatially curved geometries( k = ± χ A = T − k (cid:18) C k (˜ τ AB − τ ) − C k ( χ B ) k [ S k ( χ B ) cos α + S k (˜ τ AB − τ )] (cid:19) (54)and ˜ α AB = cos − (cid:18) C k (˜ τ AB + τ A ) − C k ( χ A ) C k ( χ B ) kS k ( χ A ) S k ( χ B ) (cid:19) . (55)For Eqs. (52)–(55), the comparable expressions ( ˜ χ B and ˜ α BA ) for the case in which thefuture lightcone from B intersects the worldline of A at time ˜ τ BA follow upon substituting χ B ←→ χ A , τ B ←→ τ A , and ˜ τ AB → ˜ τ BA .With these expressions in hand, we may draw general conclusions about whether eventsA and B share a causal past and/or a causal future. From Eq. (23), the condition for noshared causal past since the big bang, τ AB ≤
0, is equivalent to τ A + τ B ≤ χ L , (56)6while from Eq. (51), the condition that A and B share no causal future, ˜ τ BA ≥ τ ∞ , isequivalent to τ ∞ − τ B ≤ χ L . (57)Each of these conditions holds for arbitrary spatial curvature and angular separation, pro-vided one uses the appropriate expression for χ L , Eq. (24) or (38). Thus the criterion thatevents A and B share neither a causal past nor a causal future between the big bang andthe end of time is simply τ A + τ B < χ L and τ ∞ − τ B < χ L . (58)If instead τ A + τ B < χ L < τ ∞ − τ B , (59)then events A and B share no causal past but B will be able to signal A in the future. Andif τ ∞ − τ B < χ L < τ A + τ B , (60)then events A and B share no causal future though their past lightcones did overlap afterthe big bang.If we further impose the restriction that events A and B share no past causal with eachother or with our worldline, hence z A , z B ≥ z ind > z eh , then by necessity events A and B willshare no causal future, nor will we be able to send a signal to either event’s worldline beforethe end of time. The reason is simple: too little (conformal) time remains between τ and τ ∞ . Our observable universe has entered late middle-age: as measured in conformal time,the present time, H − τ = 46 .
20 Gyr, is considerably closer to H − τ ∞ = 62 .
90 Gyr than tothe big bang at H − τ = 0. That conclusion could change if the dark energy that is causingthe present acceleration of our observable universe had an equation of state different from w = −
1. In that case, Ω Λ would vary with time and thereby alter the future expansionhistory of our universe. VI. CONCLUSIONS
We have derived conditions for whether two cosmic events can have a shared causal pastor a shared causal future, based on the present best-fit parameters of our ΛCDM cosmology.7We have further derived criteria for whether either cosmic event could have been in causalcontact with our own worldline since the big bang (which we take to be the end of early-universe inflation [6, 7]); and whether signals sent from either A or B could ever reach theworldline of the other during the finite conformal lifetime of our universe. We have derivedthese criteria for arbitrary redshifts, z A and z B , as well as for arbitrary angle α betweenthose events as seen from Earth. We have also derived comparable criteria for the sharedpast and future causal domains for spatially curved FLRW universes with k = ± α = 180 ◦ ), then they will have been causallyindependent of each other and our worldline since the big bang if z A , z B > z ind = 3 . z A and z B must be obeyed to maintain past causalindependence in the case of α < ◦ , as illustrated in Fig. 3b. Observational astronomershave catalogued tens of thousands of objects with redshifts z > .
65 (see, e.g., [22, 23, 52]),and we have presented sample pairs of quasars that satisfy all, some, or none of the relevantcriteria for vanishing past causal overlap with each other and with our worldline since thetime of the big bang (Fig. 5 and Table I). Likewise, because of non-vanishing dark energy,our observable universe has a finite conformal lifetime, τ ∞ , and hence a cosmic event horizon.Our present time τ is closer to τ ∞ than to τ = 0. Events at a current redshift of z > . z = 1 .
87 are just now sending the last photonsthat will ever reach us in the infinite future.Throughout our analysis we have defined τ = 0 to be the time when early-universeinflation ended (if inflation indeed occurred). If there were a phase of early-universe inflationfor τ < ∼
65 efolds, as required to solve the flatness and horizonproblems [6, 7], then all events within our past lightcone would have past lightcones of theirown that intersect during inflation (see Appendix A). Based on our current understandingof inflation, however, the energy that drove inflation must have been transformed into thematter and energy of ordinary particles at the end of inflation in a process called “reheating”[6, 7, 53, 54]. In many models, reheating (and especially the phase of explosive “preheating”)is a chaotic process for which — in the absence of new physics — it is difficult to imaginehow meaningful correlations between specific cosmic events A and B, whose past lightconeshave not intersected since the end of reheating, could survive to be observable today. We8therefore assume that emission events A and B whose only shared causal past occurs duringthe inflationary epoch have been effectively causally disconnected since τ > where quasar host galaxies later formed — could yield an observablecorrelation signal between pairs of eventual quasar emission events at those same comovinglocations billions of years after the inflationary density perturbations were imprinted.In closing, we note that all of our conclusions are based on the assumption that theexpansion history of our observable universe, at least since the end of inflation, may beaccurately described by canonical general relativity and a simply-connected, non-compactFLRW metric. These assumptions are consistent with the latest empirical search for non-trivial topology, which found no observable signals of compact topology for fundamentaldomains up to the size of the surface of last scattering [55].Future work will apply our results to astrophysical data by searching the Sloan DigitalSky Survey database [22, 52] and other quasar datasets comprising more than one millionobserved quasars [23] to identify the subset of pairs whose past lightcones have not inter-sected each other or our worldline since the big bang at the end of inflation. We also notethat though the results in this paper were derived for pairs of cosmic events, they maybe extended readily to larger sets of emission events by requiring that each pairwise com-bination satisfies the criteria derived here. Applying the formalism developed here, usingbest-fit ΛCDM parameters, to huge astrophysical datasets will enable physicists to designrealistic experiments of fundamental properties that depend upon specific causal relation-ships. This is of particular importance for quantum mechanical experiments that cruciallydepend on whether certain physical systems are prepared independently. Many such experi-ments implicitly assume preparation independence of subsystems even though such systemsdemonstrably have a fairly recent shared causal past, extending back only a few millisec-9onds for Earth-bound systems. This work will allow experimenters to identify cosmologicalphysical systems with emission events that have been causally independent for billions ofyears, including emission event pairs that are as independent as the expansion history ofthe universe will allow on causal grounds alone, modulo any shared causal dependence setup during inflation. Future experiments which observe such causally disjoint astronomicalsources may allow us to leverage cosmology to test fundamental physics including differentaspects of quantum mechanics, specific models of inflation, and perhaps even features of afuture theory of quantum gravity.
APPENDIX A. INFLATION AND THE HORIZON PROBLEM
Using Eq. (31) and (cid:126)
Ω from Eq. (11), we may solve for the critical angular separationˆ α CMB at the redshift of CMB formation ( z CMB = 1090 .
43 [17]), when matter and radiationdecoupled. For z A = z B = z CMB , and therefore χ A = χ B = χ CMB and τ A = τ B = τ CMB , wefind from Eq. (31) ˆ α CMB = cos − (cid:34) − (cid:18) τ CMB χ CMB (cid:19) (cid:35) = 2 sin − (cid:18) τ CMB χ CMB (cid:19) . (61)Using z CMB = 1090 .
43 and evaluating χ CMB and τ CMB using Eqs. (12) and (13), then Eq.(61) yields ˆ α CMB = 2 . ◦ . Without inflation, CMB regions on the sky that we observe todaywith an angular separation ˆ α CMB > . ◦ could not have been in causal contact at the timewhen the CMB was emitted. Our formalism considers the angle α between events A and Bas seen from Earth. At a given time, τ , the particle horizon subtends an angle θ = α/ θ CMB = 1 . ◦ [7].If early-universe inflation did occur, on the other hand, then the past lightcones for suchregions could overlap at times τ <
0. We may calculate the minimum duration of inflationrequired to solve the horizon problem. The conformal time that has elapsed between therelease of the CMB and today is τ − τ CMB . In order to guarantee that all regions of theCMB that we observe today could have been in causal contact at earlier times, we require∆ τ infl + τ CMB ≥ τ − τ CMB , (62)where ∆ τ infl is the duration of inflation in (dimensionless) conformal time. The conditionin Eq. (62) ensures that the forward lightcone from χ = 0 at the beginning of inflation,0 FIG. 10. Conformal diagram illustrating how inflation solves the horizon problem. Two CMBemission events A and B are shown on opposite sides of the sky at z A = z B = z CMB . The regionbounded by the four filled black squares is the conformal diagram without inflation, akin to Fig.1, showing that the past lightcones from events A and B (red and blue triangles, respectively)do not intersect since the big bang at τ = 0 (thick black horizontal line). With inflation, thediagram extends to negative conformal times, τ <
0. If inflation persists for at least ∆ τ infl = | τ AB | ≥ τ − τ CMB , then the forward lightcone from the start of inflation will encompass the entireportion of the τ CMB hypersurface visible to us today, at τ . If inflation begins even earlier, suchthat ∆ τ infl ≥ τ ∞ , then any two spacetime points within our cosmic event horizon will have pastlightcones that intersect at some time since the beginning of inflation. τ i , encompasses the entire region of the τ CMB hypersurface observable from our worldlinetoday. In the notation of Sections III-IV, this is equivalent to setting the time at which thepast lightcones from the distant CMB emission events intersect, τ AB , equal to the start ofinflation, τ ( t i ), or τ AB = τ ( t i ) <
0. See Fig. 10.From Eq. (16) we find∆ τ infl = τ ( t end ) − τ ( t i ) = 1 a end (cid:18) H H I (cid:19) (cid:2) e N − (cid:3) , (63)where t i is the cosmic time corresponding to the beginning of inflation, H I is the valueof the Hubble constant during inflation, and e N = a end /a i (cid:29)
1, where N is the totalnumber of efolds during inflation. We may estimate a end by assuming instant reheating to aradiation-dominated phase that persists between a end and a eq = a ( t eq ), where t eq is the time1of matter-radiation equality. From Eq. (17) we have a end = a eq (cid:18) t end t eq (cid:19) / (cid:39) a eq (cid:18) NH I t eq (cid:19) / , (64)upon using N = H I ( t end − t i ) (cid:39) H I t end during inflation. We also have a eq /a = 1 / (1 + z eq ).Using our normalization that a = a ( t ) = 1, we find a end (cid:39) z eq ) (cid:18) NH t eq (cid:19) / (cid:18) H H I (cid:19) / (65)and therefore Eqs. (62) and (63) become N − / e N ≥ z eq ) (cid:18) H t eq (cid:19) / (cid:18) H I H (cid:19) / ( τ − τ CMB ) . (66)Using Eq. (13) with a e = a CMB = 1 / (1 + z CMB ), we find τ CMB = 0 .
063 and hence H − τ CMB = 0 .
91 Gyr; putting a ( t ) = 1 in Eq. (13) yields τ = 3 .
18 and hence H − τ =46 .
20 Gyr. The latest observations yield z eq = 3391 [17], and hence t eq = H − (cid:90) ∞ z eq dz (cid:48) (1 + z (cid:48) ) E ( z (cid:48) ) = 5 . × yr = 1 . × sec . (67)Recent observational limits on the ratio of primordial tensor to scalar perturbations constrain H I ≤ . × − M pl [56], where M pl = (8 πG ) − / = 2 . × GeV is the reduced Planckmass. In “natural units” (with c = (cid:126) = 1), 1 GeV − = 6 . × − sec = 2 . × − Gyr, andhence H = 100 h km s − Mpc − = 2 . h × − GeV, with current best-fit value h = 0 . N ≥ . . (68)Inflation will solve the horizon problem if it persists for at least N = 65 . τ infl ≥ τ , then any two spacetime points within our pastlightcone from today will themselves have past lightcones that intersect at some time sincethe beginning of inflation. Because τ CMB (cid:28) τ , the additional number of efolds of inflationrequired to satisfy ∆ τ infl ≥ τ rather than Eq. (62) is ∆ N = 0 .
04, or N ≥ .
64. Moreover, if∆ τ infl ≥ τ ∞ , then any two spacetime points within our entire cosmic event horizon will havepast lightcones that intersect at some time since the beginning of inflation. Given τ ∞ = 4 . H − τ ∞ = 62 .
90 Gyr), the additional efolds beyond the limit of Eq. (62) requiredto satisfy ∆ τ infl ≥ τ ∞ is ∆ N = 0 .
35, or a total of N ≥ .
95 efolds. Hence virtually anyscenario in which early-universe inflation persists long enough to solve the horizon problemwill also result in every spacetime point within our cosmic event horizon sharing a commonpast causal domain.2
APPENDIX B. HUBBLE SPHERE AND APPARENT HORIZON
We now demonstrate that object pairs in our Universe beyond the causal-independenceredshift z ind > .
65, which have no shared causal pasts since inflation, are also moving awayfrom us at speeds v rec exceeding the speed of light; although objects with current recessionvelocities c < v rec ≤ . c will still have a shared causal past with our worldline. Calculationsassume cosmological parameters (cid:126) Ω from Eq 11.One might assume that objects would lose causal contact with us and become unobserv-able if they are currently receding at speeds faster than light. In reality, astronomers todayroutinely observe light from objects in our universe at redshifts corresponding to superlim-inal recession velocities (see [3, 57], although see also [58]). Note that general relativityallows superluminal recession velocities due to cosmic expansion ( v rec = R ˙ aχ > c ), thoughit also requires that objects move with subluminal peculiar velocities ( v pec = R a ˙ χ < c ).The so-called “Hubble sphere” denotes the comoving distance beyond which objects’ ra-dial recession velocities exceed the speed of light, v rec > c . As τ → τ ∞ the Hubble sphereasymptotes to the cosmic event horizon; see Fig. 9.The radial, line-of-sight recession velocity in an FLRW metric is given by v rec = R ˙ aχ = caE ( a ) (cid:90) a da (cid:48) a (cid:48) E ( a (cid:48) ) , (69)upon using Eq. (8) for R , Eq. (10) for E ( a ), and Eq. (12) for χ . Eq. (69) can be usedwithout corrections if the object is at a redshift large enough so that peculiar velocities arenegligible compared to cosmic expansion ( a ˙ χ (cid:28) ˙ aχ for z (cid:38) . a ( t ),the Hubble sphere is located at a comoving distance χ hs at which v rec = c . Using Eq. (69)and R = c/H , the comoving distance χ hs is given by χ hs = H ˙ a = 1 aE ( a ) = (cid:90) a hs da (cid:48) a (cid:48) E ( a (cid:48) ) , (70)where z hs = a − −
1. Note that by our normalization conventions a ( t ) = 1 and E ( a ( t )) = 1;therefore χ hs = 1, which yields z hs ( t ) = 1 .
48 for (cid:126)
Ω as in Eq. (11). The current Hubble sphereredshift z hs = 1 .
48 is thus less than the current causal-independence redshift, z ind = 3 . (cid:126) Ω in Eq. (11), we find that objects at z = 3 .
65 have recession velocitiesof v rec = 1 . c , so objects that are currently receding from us faster than light in the range c < v rec ≤ . c still have a shared causal past with our worldline since τ > χ ah given by χ ah = 1 (cid:112) ( ˙ a/H ) − Ω k = 1 (cid:113) [ aE ( a )] − Ω k = 1 √ Ω Λ a + Ω M a − + Ω R a − . (71)Hence χ ah = χ hs when Ω k = 0 (also see [4]). In our flat universe, the redshifts of the apparenthorizon and the Hubble sphere are thus identical, and since z ind > z hs , objects that have noshared causal past with our worldline since the big bang, with redshifts z > . > .
48, arealso by necessity moving superluminally.
ACKNOWLEDGMENTS
It is a pleasure to thank Alan Guth for helpful discussions. Bruce Bassett provided usefulcomments on an early draft. This work was supported in part by the U.S. Department ofEnergy (DoE) under contract No. DE-FG02-05ER41360. ASF was also supported by theU.S. National Science Foundation (NSF) under grant SES 1056580. The authors made useof the MILLIQUAS - Million Quasars Catalog, Version 3.1 (22 October 2012), maintained byEric Flesh ( http://heasarc.gsfc.nasa.gov/W3Browse/all/flesch12.html ). [1] W. Rindler, “Visual horizons in world models,” Mon. Not. Royal Astr. Soc., , 662 (1956).[2] G. F. R. Ellis and T. Rothman, “Lost horizons,” American Journal of Physics, , 883–893(1993).[3] T. M. Davis and C. H. Lineweaver, “Expanding Confusion: Common Misconceptions of Cos-mological Horizons and the Superluminal Expansion of the Universe,” Pub. Astr. Soc. Aus-tralia, , 97–109 (2004), arXiv:astro-ph/0310808.[4] V. Faraoni, “Cosmological apparent and trapping horizons,” Phys. Rev. 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