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− considered and ρm is the negative cosmological energy where φ is the Newtonian potential in the limit of weak density. In general, in this work, the subscript n and the field gravity. The gravitational influence of pressure has (sometimes redundant superscript plus sign) will refer no Newtonian analogue. to a positive energy density, while the subscript m (and In general, the equation of state of a perfect fluid is superscript minus sign) will refer to a negative energy given by w = P/ρ. Local conservation of the energy- + − −3(1+w) density. Note that the values of ρn and ρm remain fixed momentum tensor implies that ρ a , where n = to their values at a = 1 and do not change as the universe 3(w+1) and w = n/3 1. Locally conserved∝ perfect fluids evolves—hence their designation as “stable”. will be assumed by default− for the rest of this paper. As usual, a normalizing (positive) critical density ρc Positive energy densities with w < 0 (n < 3), typi- 2 is defined such that ρc = 3H /(8πG) so that Ω = ρ/ρc. cally referred to as forms of “dark energy”, have formally Dividing both sides of Eq. (2) by ρc at a = 1, we obtain negative gravitational pressure which corresponds to re- ∞ ∞ pulsion of a like energy. For positive points of energy − − − when n =3 and so w = 0, there is no gravitational pres- Ω= Ω+ a n + Ω a m. (3) n m sure. Positive energy densities with w > 0 (n> 3), here n=−∞ m=−∞ X X referred to as “light energy”, have formally positive grav- Following above convention, also used in Paper I, Ω de- itational pressure which corresponds to an attraction of pends on a and hence the time t. Note that Ω is different a like energy. + − from Ωtotal, the sum of all of the stable Ωn and Ωm val- Conversely, negative energy densities that have w < 0 ues, where (n< 3), here also referred to as forms of “dark energy”, positive ∞ ∞ have formally gravitational pressure which corre- − sponds to attraction of a like energy. For negative points Ω = Ω+ + Ω . (4) total n m of energy when n = 3 and so w = 0, there is no gravi- n=−∞ m=−∞ X X tational pressure. Negative energy densities with w > 0 The curvature term in Eq. (1) can be written in terms (n> 3), also here referred to as “light energy”, have for- of more familiar quantities. We divide Eq. (1) by H2 to mally negative gravitational pressure which corresponds obtain to a repulsion of a like energy. Two specific cases will be considered here, the first of kc2 =Ω 1. (5) which will be that with zero curvature (“flat”), where H2R2 − a single stable negative energy form is coupled with a Given this nomenclature, a more generalized Fried- positive energy form. The Friedmann equation in this mann equation of energy can be written in a dimension- case is less form that explicitly incorporates all possible stable, 2 a˙ + 2−n − 2−m static, isotropic energy forms described by an integer n. =Ωn a +Ωma , (8) H2 Using H =a/a ˙ in Equation (1) and dividing each side o 2 − by the square of the Hubble parameter H0 (at a = 1), + where Ωn > 0 and Ωm < 0 are the normalized values we obtain of positive and negative energy densities. The flatness ∞ a˙ 2 − − − condition demands that Ωn +Ωm = 1. The solution to = (1 Ω)+ (Ω+ a2 n +Ω a2 m).(6) H − n m Eq.(8) is (for a more general discussion, see [15]) 0 n,m=−∞   X 1 a da Note that Eq.(6) can be written to highlight only present t to = − , (9) − Ho + 2−n 2−m day observables by substituting 1/a = R0/R = (1+ Zao Ωn a +Ωma z)/(1 + z0)=(1+ z), where z is the redshift and z0 =0 where ao is the value ofp the scale factor at time to and is the redshift at the epoch where a = 1. 2−n 2−m Ωna +Ωma > 0. The second case considered here will be a single stable II. DESCRIPTION OF NEGATIVE ENERGY type of negative energy alone in the universe coupled only COMPONENTS BASED ON PRESSURE AND with the positive curvature it creates. The Friedmann EQUATION OF STATE PARAMETER equation in this case is

a˙ 2 The Friedmann equations typically quantify the cos- 2−m 2 = (1 Ωm)+Ωma , (10) mological evolution of perfect fluids. Perfect fluids are Ho − characterized by only two variables: energy density ρ and where by definition Ω < 0. Written in integral form isotropic pressure P . These two variables can be isolated m in an effective modified Poisson equation for gravity from 1 a da t to = , (11) general relativity in the weak field limit so that[1, 3] 2−m − Ho ao (1 Ω )+Ω a Z − m m 4πG 2 p2−m φ = 2 (ρ +3P ), (7) where (1 Ω )+Ω a > 0. ∇ c − m m 3

III. FRIEDMANN EQUATION FOR STABLE turnaround following an expansion. NEGATIVE ENERGY FORMS n>m, n> 0 p> 0 = (14) Energy can take any number of stable forms, and is ⇒ (nCosmological Constant −2/3 1 Sheet Domain Wall TABLE II. The value of p as defined in Eq.(12) determines the −1/3 2 Line Cosmic String allowed combination of positive and negative energy types. 0 3 Point Matter The hypergeometric function has a singularity at p = −1. 1/3 4 Relativistic Point Radiation The values of q depend on the scale factor in an expanding > 1/3 > 4 Unknown Ultralight or contracting universe. 2F1 has a series of singularities after p ≤−1. TABLE I. Energy forms that may drive the expansion rate of the universe as described by the Friedmann Equations.The entries include known and hypothesized energy forms that are As a particular case, for q < 1, that is, when a 1 thought to be stable perfect fluids over cosmological scales and Ω− < Ω+ | | ∼ | m| | n | ∞ j (1/2)j(p)j q all stable energy forms remain stable with a density di- F (1/2,p,p +1, q)= (15) 2 1 (p + 1) j! rectly extrapolated from the density at a = 1. Although 0 j X a = 1 could describe any epoch, here it will typically de- where F is the hypergeometric function (with also a scribe the present. Many energy terms used below have 2 1 series expansion interms of Legendre function[16]) and precedents and are in common use. Although no closed form solutions exist for Eq.(6) 1 j =0 in general, analytical solutions involving hypergeometric (x)j = (16) j (x + j 1) j > 0. functions exist for particular cases: a) single energy type ( i=1 − corresponding to positive and negative energy densities + − Ignoring higher orderQ terms in q, Eq.(12) takes the form existing in a flat universe (Ωn +Ωm = 1) and b) a single negative energy type in a curved universe (Ω− 1 = 0). m n 1/2 − n−m − 6 −1 2 a Ωm n a t = H0 + 1+ + . (17) n Ωn Ωn 2 3n 2m A. Solutions for a single positive and negative      −  energy component in a flat universe Also, see [17–20] for reviews on oscillating solutions de- scribing some cyclic models. A universe that consists of only a positive and a nega- tive energy density type satisfies Eq.(8), with the solution B. Solution for a negative energy component in a curved universe 2 an 1/2 H0t(a)= 2F1 (1/2,p,p +1, q) (12) n Ω+  n  The solution for the Friedmann equation involving a where negative energy density component coupled to curvature, − Eq.(10), is n Ωm n−m p , q + a 0. (13) ≡ 2(n m) ≡− Ωn ≥ − 2 1/2 Generally, the validity of the solution corresponds to val- a H0t(a)= − 2F1 (1/2,pk ,pk +1, qk ) (18) ues of p 0. Negative values of p are allowed, but 1 Ωm  −  the hypergeometric≥ function has a series of line singu- where larities along the negative axis. The physical interpreta- tion of singularities point toward cyclic universes result- 1 a2−m p , q 0. (19) ing from a bounce following a period of contraction or 2) k ≡ 2 m k ≡−1 Ω ≤ − − m 4

ity, then this universe will be deemed to have a “hot” ending. Conversely if this positive energy density term goes to zero as a goes to either zero or infinity, then this universe will be deemed to have a “cold” ending. If the positive energy density ends up at a finite value, then this universe will be deemed to have a “warm” ending. Next, the kinematics of the universe will be charac- terized. In other words, what happens to the expansion speeda ˙ of a universe? Ifa ˙ 2 diverges as a goes to ei- ther zero or infinity, then this universe will be deemed to have an “accelerating” ending. Ifa ˙ 2 drops to zero as a goes to either zero or infinity, then this universe will be deemed to have a “decelerating” ending. Ifa ˙ 2 ends with a constant value, then this universe will be deemed to be “coasting” as its end. Alternatively, if the universe ends with a going to in- finity but the energy density decreasing to zero then this universe ending will be deemed a “big void”. This term is here preferred over the previously popular “big freeze” because it is more descriptive of the end state, as the end temperature of the universe has already been character- ized. If a goes to infinity but the energy density remains constant, then this universe ending will be deemed to re- FIG. 1. The value of the hypergeometric function has a series sult in a “steady state”. If a universe ends with a going of singularities along the negative x-axis for values of p, q is a to infinity but the energy density also diverges, possibly function of m and the scale factor a. Note that if a = 0 then in a finite time, this universe ending will be deemed a there are no singularities and the function is independent of “big rip”. m and n. This suggests that the universes that bounce or The end states of contracting universes will also be turns around does so before a becomes zero. classified. If the universe ends with a going to zero but the energy density diverges, this universe ending will be deemed a “big crunch”. If the universe ends with a going The solutions in this category without a bounce corre- to zero but the energy density remains constant, then this spond to values of m < 3, which are -1, 0, 1 and 2. We universe will (again) approach a “Steady State”. And fi- note that only numerical solutions exist for many com- nally if the universe ends with a going to zero and the en- ponent curved universe given by Eq.(6). ergy density also drops to zero, this universe ending will be deemed a “big poof”. To the best of our knowledge, “big poof” universes have not been discussed previously. C. Possible Futures for Universes The criteria adopted to describe the future of universes that are expanding or contracting is summarized in Ta- An evolving Friedmann universe may be described in ble III. terms of its ultimate fate, curvature, temperature, and To describe the initial condition of a universe, the term energy density as the scale factor a approaches its maxi- “big bang” will be extended to all universes that start mal value, by fully analyzing Eq.(6). with a = 0 even if the energy density started with a First, the fate of each universe will be analyzed. In finite or formally zero value. However, were the universe other words, what becomes of scale factor a? An ex- to start from a finite value of a, which might have followed panding universe can either expand forever, meaning that a contraction phase, this universe will be deemed to have scale factor a going to infinity is formally allowed by started from a “bounce.” the Friedmann equations, or “turnover”, meaning that Negative energy component alone will not satisfy the a maximum amax will be reached, after which the uni- Friedmann equation—resulting in purely imaginary so- verse will collapse. Similarly, a collapsing universe can lutions. Using the formalism adopted in this paper and contract to a point, meaning that scale factor a going to also to validate Eq. (6) using the known energy composi- zero is formally allowed by the Friedmann equations, or tion of the ΛCDM universe, we provide known solutions “bounce”, meaning that a minimum amin will be reached, in figure 2. after which the universe will expand. In the following section, we analyze negative energy Next, the temperature of the universe will be char- components coupled with positive energy components in acterized. Here temperature will be considered directly a flat or curved universe. As we have shown in section III, related to energy density. If the positive energy density single component flat universes comprising a single nega- −n term Ωna diverges as a goes to either zero or infin- tive and positive energy density component, and curved 5

−n 2−n Universe n Ωna End State Temperature n a˙ ∝ Ωna Kinematics

Expanding n> 0 ∼ 0 Big Void Cold n> 2 ∼ 0 Decelerating Expanding n< 0 ∼∞ Big Rip Hot n< 2 ∼∞ Accelerating Expand/Contract n = 0 constant Steady State Warm n = 2 constant Coasting Contracting n> 0 ∼∞ Big Crunch Hot n> 2 ∼∞ Accelerating Contracting n< 0 ∼ 0 Big Poof Cold n< 2 ∼ 0 Decelerating

TABLE III. Possible fates of universes—characterized by ”appropriate” description for end state and temperature based on −n 2−n normalized energy density (Ωna ), and kinematics based on the rate of scale factor (Ωna ).

critical value of scale factor

− 1/(m−n) flat Ωm acrit = − , (20) Ωm 1  −  where m is an index corresponding to negative energy component. The corresponding equation for a curvature dominated universe is

− 1/(2−m) curved Ωm 1 a = −− . (21) crit Ω  m  The end states corresponding to different combinations of n and m for expanding and contracting universes, for curvature coupled negative energy density universes and for single component universes are summarized in Ta- bles IV and, V and VI respectively. Below we present a brief discussion on the properties of some of these nega- tive energy forms.

Universe m description

Expanding m< 2 turns over FIG. 2. As a prelude to providing numerical solutions to m ≥ 2 Cold Coasting Big Poof Eq. (6), wherever appropriate, we verify it with the known Contracting m ≤ 2 Hot Coasting Big Crunch energy composition of the observable universe. Note that it m> 2 bounce gives a very familiar age for the universe corresponding to the value of zero scale factor. A de-Sitter universe and a matter TABLE IV. How do two stable energy forms interact to in- dominated universe is given for comparison. fluence the fate of a curved universe.

The existence of any form of phantom energy (n < 0 universe coupled to a negative energy density component or m < 0, and hence w < 1) is controversial as it may have analytical solutions. We discuss asymptotic behav- violate general relativistic energy− conditions [5, 21]. Neg- ior and as well as the nature of the solutions near critical ative phantom energy would have ρ< 0 so that the grav- points that correspond to values of scale factors that are itational energy density would be repulsive. However, a local extremum for these solutions. since w < 1, this positive energy density is necessar- ily coupled− to a gravitational pressure that is attractive and, from Eq.(7), three times stronger. Therefore, on the whole, phantom energy would gravitationally attract IV. ASYMPTOTIC BEHAVIOR OF UNIVERSES itself. CONTAINING VARYING FRACTIONS OF When n = m = 1, since Ωn +Ωm = 1, if a contract- POSITIVE AND NEGATIVE ENERGY ing universe remains− dominated by the repulsive positive DENSITIES phantom energy, net positive phantom energy density de- creases as the scale factor a decreases. Note that this The friedmann equation with both positive and nega- can occur as a time-reversed version of n = m = 1 ex- tive energy density terms in a flat universe will attain a panding universe ending in a flat hot accelerating big− rip. 6

Either way, such a contracting universe will asymptoti- cally approach zero scale factor and zero energy density over an infinite time interval. This universe will end in a flat cold decelerating “big poof” —an unrealistic scenario in our universe, but nevertheless a possibility in at least some (out of tens of millions) universes in the (presently accepted) inflationary paradigm[22, 23]. A negative cosmological constant has m = 0 (w = 1) energy with gravitationally attractive pressure. Al- though− an m = 0 component has a negative energy den- sity that is gravitationally repulsive, Eq.(7) shows that the pressure is positive and three times greater in magni- tude. Therefore the net effect of negative stable cosmo- logical constant energy on itself is attractive. A test particle(point mass) near a static cosmological domain wall of negative energy would experience a grav- itational attraction toward it. This is the opposite of the gravitational repulsion a person would feel toward a positive energy domain wall[24–27]. A negative cosmic string has m = 2 (w = 1/3) en- − ergy with gravitationally attractive pressure. It is con- FIG. 3. Numerical solution: For a curved universe dominated ceivable that the centers of negative energy cosmic strings by both positive and negative phantom energy. contains wormholes with boundary conditions satisfying Einstein’s equation[28]. Although the density of m = 2 energy is negative, and is gravitationally repulsive, Eq.(7) ergy evolving with an effective w> 1/3 has been hypoth- shows that the attractive pressure is equal in magnitude. esized previously in the context of time-varying scalar If a person (point particle) were to go up to a flat, static, fields (see for example [32–34]). However, the stable neg- cosmic string of negative energy, that person would feel ative ultralight energy forms discussed in this section are neither gravitational attraction toward it nor repulsion specifically constrained in their behavior to evolve only in away from it. This is similar to the effect a person would accordance with the Friedmann equations are not scalar feel toward a positive energy cosmic string[29, 30]. fields. Forms of ultralight with w> 1 might be unphysi- As discussed in Paper I in more detail than given here, cal because their formal sound cs = c√w is greater than a fundamental type of energy which evolves as n = 3 the speed of light. Only ultralight energy forms with (w = 0) has energy confined to regions small compared 1/3 1/3 (n > 4), know this to solve for the evolutionary equations for this discussed recently for stable positive energy forms was re- universe. As this universe expands, the prominence of ferred to as ultralight [5]. The term ultralight energy con- the attractive m = 0 energy increases until it matches trasts with dark energy as being beyond light in terms of the energy density the neutral n = 2 energy. gravitational pressure. By analogy, ultralight is to light A perhaps less-intuitive example is when positive n =3 what ultraviolet light is to violet light. The idea of en- energy shares the universe with (only) negative m = 2. 7

Here the n = 3 positive energy gravitationally attracts tract, possibly ending in a big crunch. Since this hypo- itself, while the m = 2 negative energy is gravitation- thetical stable negative phantom energy is undetectably ally neutral toward itself. As this universe expands, the small at present, it is really not possible to know with prominence of the neutral m = 2 energy increases un- certainty that it does not exist, and therefore it is im- til it matches the energy density of the attractive n =3 possible to know the true future of our universe for this energy. One might consider that the internal attractive- scenario. ness of the universe decreases as it expands and the role of Alternatively, would it be possible for some stable neg- the gravitationally neutral negative energy becomes more ative energy form of negligible cosmological density now important. Nevertheless, the deceleration of the universe but may have been significant in past of our observable might be considered as already set and sufficient to cause universe? One realization of this scenario would involve a turnover from expansion to contraction. An analogy is some stable negative energy form with higher n and w to consider the classic case of a n = 3 ball thrown into than the current stable positive energy form. We know the air on the surface of the Earth. The ball loses kinetic that our universe has some positive radiation and in at energy, analogous here toa ˙ , as it rises, but its decelera- least two early epochs in the history of the universe, pos- tion is sufficient to cause it to “turnover” at the top of itive radiation was the cosmologically dominant energy its orbit and return to Earth. form. Another realization demands a presence of stable As discussed in detail in Paper I, it is possible for sta- negative ultralight in the current epoch. Negative ultra- ble forms of energy to change into each other, and this light energy would be gravitationally repulsive. There- applies to stable negative forms as well as stable positive fore it is possible that a previously contracting universe forms. Rapid changes could occur—for example, as in contained such negative ultralight, and this negative ul- the decay of an n = 3 form into an n = 4 forms: nuclear tralight stopped the contraction and started an expan- beta decay—among stable positive energy forms[35]. An sion. Once the universe started expanding, this nega- example of a slow change between stable negative energy tive ultralight would again dissipate and again dilute to forms would be the redshifting of negative radiation as undetectable density. Therefore, since it is not possible it moved through the universe, where is slowly changes to confirm the existence of negative ultralight, it is not from n = 4 energy to n = 3 energy. possible to know the true past of our universe for this scenario – perhaps it started in a big bang or perhaps it underwent a big bounce. VI. SUMMARY: THE PAST AND FUTURE OF Our universe could be an oscillating universe were it THE UNIVERSE WITH STABLE NEGATIVE to contain even minuscule and undetectable amounts of ENERGY FORMS both negative phantom energy and negative ultralight energy. As a becomes large, the negative phantom en- We know that our present universe is composed pre- ergy could suddenly begin to dominate and cause the dominantly of positive energy and is expanding[23, 36– universe to turnover and start contracting. This phan- 40]. Would it be possible for some stable negative en- tom energy would then again become effectively unde- ergy form of negligible cosmological density in the present tectable. As the universe contracts and its scale factor epoch but could grow to dominate and determine the fu- a becomes small enough, the small amount of negative ture of the universe? The only way this could happen ultralight energy would suddenly dominate universe and, would be if some stable negative energy form with lower being repulsive, cause the universe to bounce. As the uni- n and w than the current dominant stable energy form. verse expands once again, the density of ultralight would Since the dominant energy form is currently thought to again drop to undetectability and the cycle could repeat. be n = 0 (w = 1) dark energy, only a form of negative Currently, it seems this type of universe cannot be ruled phantom energy− could fill such a role. out theoretically[34]. Were any amount of stable negative phantom energy to exist in our universe, it would grow as the universe expands and eventually begin to dominate the universe. ACKNOWLEDGMENTS Since negative phantom energy is attractive such a com- ponent could stop the expansion of the universe and pro- We thank the anonymous referee for valuable sugges- pel it into a contraction. Even though the effect of this tions that have helped improve this paper.BP appreciates negative phantom energy would diminish and eventually the help and support from Julian Varghese and Sandeep become negligible, the universe would continue to con- Ramachandran during the time of writing this paper.

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n -1 0 1 2 3 4 5 m

-1 Hot Accelerat- turns over turns over turns over turns over turns over turns over ing Big Rip

0 Hot Accelerat- Warm Accel- turns over turns over turns over turns over turns over ing Big Rip erating Steady State

1 Hot Accelerat- Warm Accel- Cold Accelerat- turns over turns over turns over turns over ing Big Rip erating Steady ing Big Void. State

2 Hot Accelerat- Warm Accel- Cold Accelerat- Cold Coasting turns over turns over turns over ing Big Rip erating Steady ing Big Void Big Void. State

3 Hot Accelerat- Warm Accel- Cold Accelerat- Cold Coasting Cold Decelerat- turns over turns over ing Big Rip erating Steady ing Big Void Big Void ing Big Void State

4 Hot Accelerat- Warm Accel- Cold Accelerat- Cold Coasting Cold Decelerat- Cold Decelerat- turns over ing Big Rip erating Steady ing Big Void Big Void ing Big Void ing Big Void State

5 Hot Accelerat- Warm Accel- Cold Accelerat- Cold Coasting Cold Decelerat- Cold Decelerat- Cold Decelerat- ing Big Rip erating Steady ing Big Void Big Void ing Big Void ing Big Void ing Big Void State

TABLE V. How do two stable energy forms interact to influence the fate of a flat expanding Universe.

n -1 0 1 2 3 4 5 m

-1 Cold Decelerat- Warm Decel- Hot Decelerat- Hot Coasting Hot Accelerat- Hot Accelerat- Hot Accelerat- ing Big Poof erating Steady ing Big Crunch Big Crunch ing Big Crunch ing Big Crunch ing Big Crunch State

0 bounce Warm Decel- Hot Decelerat- Hot Coasting Hot Accelerat- Hot Accelerat- Hot Accelerat- erating Steady ing Big Crunch Big Crunch ing Big Crunch ing Big Crunch ing Big Crunch State

1 bounce bounce Hot Decelerat- Hot Coasting Hot Accelerat- Hot Accelerat- Hot Accelerat- ing Big Crunch Big Crunch ing Big Crunch ing Big Crunch ing Big Crunch

2 bounce bounce bounce Hot Coasting Hot Accelerat- Hot Accelerat- Hot Accelerat- Big Crunch ing Big Crunch ing Big Crunch ing Big Crunch

3 bounce bounce bounce bounce Hot Accelerat- Hot Accelerat- Hot Accelerat- ing Big Crunch ing Big Crunch ing Big Crunch

4 bounce bounce bounce bounce bounce Hot Accelerat- Hot Accelerat- ing Big Crunch ing Big Crunch

5 bounce bounce bounce bounce bounce bounce Hot Accelerat- ing Big Crunch

TABLE VI. How do two stable energy forms interact to influence the fate of a flat contracting universe.