⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 090201 Graviton Momentum: A Natural Source of Dark Energy Samuel Meng (孟献明)* Affiliations Research School of Physics, Australian National University, Canberra, ACT 2601, Australia Received 24 May 2023; accepted manuscript online 8 August 2023; published online 29 August 2023 *Corresponding author. Email: xianming.meng@anu.edu.au Citation Text: Meng S 2023 Chin. Phys. Lett. 40 090201    Abstract The dark energy concept in the standard cosmological model can explain the expansion of the universe. However, the mysteries surrounding dark energy remain, such as its source, its unusual negative pressure, its long-range force, and its unchanged density as the universe expands. We propose a graviton momentum hypothesis, develop a semiclassical model of gravitons, and explain the pervasive dark matter and accelerating universe. The graviton momentum hypothesis is incorporated into the standard model and explains well the mysteries related to dark energy.
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 DOI:10.1088/0256-307X/40/9/090201 © 2023 Chinese Physics Society Article Text The discoveries of an expanding universe by Slipher[1] and Hubble[2] in 1920s, and of the accelerating expansion rate of the universe (Riess et al.[3] and Perlmutter et al.[4]) in the late 1990s, compelled physics theorists to attempt to explain the nature and causes of the accelerating universe. By adding an additional metric tensor, and assigning to it a very small cosmological constant, Einstein's theory of general relativity is able to explain the expansion of the universe mathematically (Einstein's intention of introducing the cosmological term was to obtain a solution of static universe and he later considered the inclusion of cosmological term in his celebrated field equation was the biggest blunder he ever made in his life. Friedmann realized that Einstein's static universe solution was unstable. Lemaitre showed that Hubble's law followed from the cosmological term.), but is unable to identify any physical cause for the expansion. On the other hand, through the measurement of the radial velocities of galaxies, Fritz Zwicky[5] suspected that there was invisible dark matter associated with galaxies. The measurements of the velocity curve in the late 1970s[6,7] showed that, of the total matter in universe, the luminous matter made up to only 2% while the other 98% was considered as dark matter. Later studies on mass density at the current era[8,9] further confirmed the previous findings. The cosmic microwave background (CMB) survey results[10,11] suggested that dark energy consisted of about 70% of the energy density in the universe, which generated repulsive force responsible for the accelerating universe. While physicists are still search for exotic cold dark matter, they are unable to explain dark energy in any plausible way. So far the best explanation of an accelerating universe and dark energy is based on the $\varLambda$CDM model, i.e., the currently standard cosmological model. The model includes a dark energy component and explains the existing observations well. However, the model takes dark energy as given and does not specify the source of dark energy. Given the success of the quantum mechanism for microworld, quantum vacuum energy was considered as a source of dark energy. However, the calculations showed that the natural size of quantum energy was $10^{120}$ too large for the cosmological constant. This was labeled as the cosmological constant problem, which is unsolved till today.[12-17] Within the $\varLambda$CDM model framework, Farnes[18] intended to explain why the dark energy generates negative pressure. It was assumed that, like charges, mass can be positive or negative: the same type of masses has attractive force while the different types of masses have repulsive force. Although this hypothesis can explain the negative pressure of dark energy, the assumption of negative mass is incomprehensible based on current knowledge in physics. More importantly, this hypothesis is unable to explain the long-range effect of the repulsive force of dark energy. There are a number of theories about dark energy, but they all involve some assumptions or outcomes what are not supported by current observations or the known physics knowledge. Milgrom,[19] Bekenstein,[20] and Dunkel[21] required the modification of Newtonian dynamics and an ad hoc introduction of fundamental acceleration of $10^{-10}$ m/s$^{2}$. Dvali et al.[22] and Kunz and Sapone[23] required to modify the general relativity. The quintessence model of Peebles and Ratra[24] and Rozas-Fernandez[25] assumed a fifth fundamental force. Caldwell et al.[26] assumed a phantom energy, a negative kinetic energy that increases with the expansion of the universe. The quintom (combination of quintessence and phantom) dark energy of Upadhye et al.[27] and Marciu[28] introduced a time varying parameter for the equation of state but still could not explain why the universe entered the acceleration stage at the current epoch. Sivaram and Sinha[29] assumed a time varying cosmological constant and a time-dependent vacuum energy density. Gurzadyan and Xue[30] suggested the possibility of variation of the speed of light and the gravitational constant. Given our limited knowledge about dark energy, it is understandable that various surprising assumptions are made in previous studies. However, it can be argued that if a theory employs more plausible assumptions, the theory should be preferred. The present paper is an attempt in this direction. Our assumption is that while graviton attraction causes gravitational force, graviton momentum leads to the repulsive force responsible for an accelerating universe. Although gravitons have not been detected yet (maybe due to the very weak strength of gravitational force), the concept of gravitons is generally accepted as a means of quantifying a gravity field. It is also accepted that gravitons travel at the same speed as photons. Since photons have momenta, it is reasonable to assume that the photon-like gravitons, or dark photons, also have momenta. With these plausible assumptions, the paper intends to derive the cosmological constant. As will be seen, with a model of graviton momentum, the paper explains well the pervasive dark matter in the universe, the long-range nature of repulsive force from dark energy, the unusual assumptions about dark energy in the $\varLambda$CDM model frame work, and the unchanged density of dark energy in an expanding universe. Most importantly, the explanation is simple and elegant, supported by existing observations, and compatible with the general relativity and the standard cosmological model. In this Letter, first we focus on the development of a semiclassical model that can explain pervasive dark matter in the universe, then derive the cosmological constant and explain an accelerating universe. We discuss the compatibility between the graviton momentum hypothesis and the standard cosmological model. Finally, we provide conclusions. A Semiclassical Model of Gravitons. Graviton as the carrier of gravitational force is a commonly accepted concept. However, it is generally interpreted as a virtue particle. Here we review gravitons as photon-like real particles or, loosely speaking, dark photons. Some may object to treating gravitons as dark photons because gravitons and photos are of spin-2 and spin-1, respectively. The spin-2 nature of gravitons is given by its rank-2 stress-energy tensor, which gives gravitons a fluidic feature. In terms of cosmology, this spin difference diminishes: the universe can be viewed as an idea fluid, so the stress-energy tensor for gravitons is diagonal and its rank reduces to 1. The detailed assumptions for the graviton momentum model can be stated as follows. Extending from the photon emission phenomenon, we assume that all matter emits gravitons and that, after receiving a graviton, the matter re-emits a graviton on the extended path of the original graviton. Gravitons are photon-like particles which have no mass and travel at the speed of light in vacuum. However, different from the photons, gravitons have attractive interactions between them and are attracted by mass. Based on quantum theory, a photon has energy of $hf$ and momentum of $hf/c$, where $h$ is Planck's constant, $f$ is the frequency of light, and $c$ is the speed of light in vacuum. Since gravitons are assumed to be photon-like and travel at the speed of light $c$, it is natural to assume that gravitons also have a momentum of $hf/c$, where $f$ is frequency of a gravitational wave. The direct outcome of these assumptions is that when a graviton contacts a mass particle, an attractive gravitational force (opposite to the direction of graviton movement) is generated due to attractive interaction with mass; meanwhile, a repulsive force (in the direction of graviton movement) also generated due to the momentum of the graviton. These two opposite forces come from the two compatible attributes of gravitons, so they are consistent. In considering gravitons as the carries of gravitational force, it is also reasonable to assume that the gravitational force transferred by gravitons is proportional to the density of gravitons. With these assumptions, we can explain both gravitational force and dark energy. We start with deriving Newtonian gravitational formula. Based on a simple setting shown in Fig. 1, mass $M$ radiates gravitons in all directions and some of them are received by mass $m$. The density of gravitons can be measured by the number of gravitons in a small volume $dV$, enclosed by two spheres of radius $R$ and $R+ dR$. Since gravitons in a ray are attracted by each other and by mass, they establish the attractive force between $M$ and $m$. The denser the graviton rays, the stronger the gravitational force. As the gravitons in the ray from left to right attract each other, mass $M$ can attract mass $m$ through a great distance. However, as the distance increases, the graviton density decreases and the attraction between $M$ and $m$ will decrease significantly. Next, we examine these intuitive thoughts mathematically.
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Fig. 1. A two-dimensional demo of graviton density and gravitational force in three dimensions.
Let $n$ be the number of gravitons per emission per unit mass, $f$ the frequency of graviton emissions by $M$. The graviton emission rate $N$ (i.e., $N$ the number of gravitons emitted in 1 unit of time) of mass $M$ can be calculated by $N=nfM$. The total number of gravitons emitted in time period $dt$ is \begin{align} N_{t}=Ndt=nfMdt.\notag \end{align} Assume that the gravitons in the first emission arrive at the sphere surface of radius $R$. Given that the gravitons travel at the speed of light $c$, the distance a graviton covers in time $dt$ is $cdt$, so the volume occupied by $dN$ gravitons in time $dt$ is the volume $dV$ enclosed by the two spheres of radius $R$ and $R+cdt$ (i.e. $dR=cdt$), which can be calculated by \begin{align} dV=\int_{\theta =0}^\pi \int_{ \phi =0}^{ 2\pi} {R^{2}{\sin}\theta d\theta d\phi dR} =4\pi R^{2}cdt. \notag \end{align} The density of gravitons at radius $R$ can be calculated by \begin{align} \rho_{\rm g} = \frac{N_{t}}{dV}=\frac{nfM}{4\pi cR^{2}}.\notag \end{align} Given the distance $r$ between $M$ and $m$, the density of gravitons from $M$ at the testing object $m$ reads \begin{align} \rho_{\rm g} = \frac{N_{t}}{dV}=\frac{nfM}{4\pi cr^{2}}. \tag {1} \end{align} Assuming that the gravity force $F$ is proportional to graviton density $\rho_{\rm g}$, and to the mass $m$ of the testing object, we have the gravitational force $F$ enacted by object $M$ on the testing object $m$ at distance $r$ as follows: \begin{align} F=C\,\rho_{\rm g}m=\frac{C\,nfMm}{4\pi cr^{2}}. \tag {2} \end{align} Here we have introduced a constant $C$ that is the amount of attraction force enacted by a graviton for a unit of mass. We can calibrate $C$ to satisfy \begin{align} G=\frac{Cnf}{4\pi c}~~ {\rm or} ~~C= \frac{4 \pi cG}{nf}.\notag \end{align} Substituting the above equation into Eq. (2), we have Newton's gravity equation \begin{align} F=\frac{{4\pi c}G}{nf}m\rho_{\rm g}=\frac{GMm}{r^{2}}. \tag {3} \end{align} The derivation so far, however, ignores the impact of momentum from gravitons. Given the density of gravitons $\rho_{\rm g}$, the cross section $S$ of the testing object $m$, and the speed of gravitons $c$, the flux of gravitons on $m$ (the number of gravitons received by $m$ in 1 unit of time) is $\phi = \rho_{\rm g}cS$. Since the momentum carried by a graviton is $hf/c$, the total momentum of gravitons transferred from $M$ to $m$ can be calculated by \begin{align} \Delta P=\phi \frac{hf}{c}=\rho_{\rm g}cS \frac{hf}{c}=\rho_{\rm g}Shf. \notag \end{align} This result is analogical to that of solar sailing. However, a question could be raised regarding the consistency of momentum effect and gravitational effect: while it is understandable that the momentum is proportional to the number of gravitons received by the object, why is the gravitation effect proportional to the density of gravitons, rather than to the number of gravitons received? The answer lies in the assumption that when a graviton is absorbed by a mass particle, it re-emits a graviton on the extended path of the original graviton (i.e., shown as the gravitons traveling inside the object $m$ in Fig. 1). In this way, a graviton can act directly and indirectly on all mass particles in its pass, so the gravitational force is related to the density of gravitons, and is not blocked by any particles on its path. Although the momentum is related to cross section, for more general application, it is useful to convert the cross section to the density of the testing object $\rho_{m}$. For convenience, we assume the length of testing object $L=1$, so the cross section can be calculated by $S=V/L=V=m/ \rho_{m}$ (in so doing, we imply that $\rho_{m}$ is the column density of the testing object, or we simply view $\rho_{m}=L\rho_{m}$ due to $L=1$. This is a way of eliminating parameters dependent on the shape and size of testing object, so the analysis can be simplified for more general cases). Since the transferred momentum $\Delta P$ occurs in 1 unit of time, the force it generates is \begin{align} F'=\frac{\Delta P}{t}=\frac{\Delta P}{1}=\rho_{\rm g}hfm/\rho _{m}. \tag {4} \end{align} It is worth noting that photon momentum can also generate repulsive force, which can be used for solar sailing. However, the dark matter accounts for 98% of total matter in the universe, so the effect of photon momentum due to 2% of visible matter can be safely omitted at the cosmological scale. Consequently, we ignore the effect of photon momentum, but include the momentum of gravitons emitted from both visible and dark matter. Note that gravitons are the carriers of gravitational force only and their energy and momentum cause neither extra gravity nor extra change in spacetime curvature. Since $F'$ due to graviton momentum pushes $m$ away from $M$ while $F$ due to gravity pulls $m$ towards $M$, the force $F'$ is opposite to gravitational force $F$, so the total impact of gravitons from $M$ on $m$ can be expressed as the total force $F_{\rm total}$: \begin{align} F_{\rm total}&=F-F'=\frac{{4\pi c}G}{nf}m\rho_{\rm g}-\frac{hf}{\rho _{m}}{m\rho }_{g}\notag\\ &=m\rho_{\rm g}\Big(\frac{{4\pi c}G}{nf}-\frac{hf}{\rho _{m}}\Big). \tag {5} \end{align} The first term in the bracket of Eq. (5) is proportional to attractive gravity force, while the second term is related to the repulsive force caused by graviton momentum. It is clear that the density of testing object affects the relative size of the two terms. Equalizing these two terms, we have the break-free density for the testing object: \begin{align} \rho_{m\mathrm{\_ free}}=\frac{nhf^{2}}{4\pi cG}. \tag {6} \end{align} At this break-free density, the attractive force and repulsive force canceled out each other, so the testing object $m$ experiences zero net force from $M$ and thus will not be attracted to $M$. Above this density, attractive gravity force dominates and vice versa. In short, Eq. (5) shows the important role of the density of the testing object in the net force it receives. Gravity Momentum Leading to Elusive Dark Matter and an Accelerating Universe. Given the pervasive and vastly dominant dark matter in the universe, we can view that dark matter density is almost the same as the density of total matter at the cosmological scale. After the big bang (i.e., the inflation epoch), the matter density was relatively high, so $\rho_{m}$ is large and the second term in the brackets of Eq. (5) was dominated by the first term. As such, the attractive force dominated and the expansion of the universe decelerated. As the universe expands, the density of matter decreases. At the current epoch, the density of matter is so low that the second term in the brackets of Eq. (5) dominates and the net force between dark matter particles are repulsive. This repulsive force keeps dark matter particles from clumping together, so dark matter cannot form pockets of dense particles to cause disturbance to electromagnetic wave. Consequently, dark matter is very hard to detect. The repulsive force between pervasive low-density matter particles also causes the accelerated expansion of the universe. Next, we explain how this force is related to the acceleration rate of the universe expansion and to the cosmological constant. Assuming that the average density of the matter (including luminous matter and dark matter) is $\rho_{\scriptscriptstyle{M}}$, we can calculate the amount of matter $M$ within a sphere of radius $r$ by \begin{align} M=\frac{4}{3}\pi r^{3}\rho_{\scriptscriptstyle{M}}. \notag \end{align} Here we assume a homogeneous universe at the cosmological scale. Applying Eq. (1), we can calculate the graviton density at distance $r$ due to matter $M$: \begin{align} \rho_{\rm g}=\frac{nfM}{{4\pi c}r^{2}}=\frac{\rho_{\scriptscriptstyle{M}}nf}{3c}r . \notag \end{align} Substituting the above equation into Eq. (5), we have the total force due to matter: \begin{align} F_{\rm total}&=\frac{m\rho_{\scriptscriptstyle{M}}nf}{3c}r\Big(\frac{{4\pi c}G}{nf}-\frac{hf}{\rho_{m}}\Big)\notag\\ &=mr\Big(\frac{4\pi G\rho_{\scriptscriptstyle{M}}}{3}-\frac{hnf^{2}\rho_{\scriptscriptstyle{M}}}{3c\rho_{m}}\Big)\notag\\ &=\frac{GMm}{r^{2}}-\frac{hnf^{2}\rho_{\scriptscriptstyle{M}}}{3c\rho _{m}}mr. \tag {7} \end{align} The first term on the right-hand side of Eq. (7) is the gravitational force, which decreases at the pace of $1/r ^{2}$. The second term is the repulsive force due to graviton momentum, which is proportional to distance $r$ and thus a long-range force. Considering the cosmological scale (extremely large distance and ultralow density), we can equalize the density of the testing matter $\rho_{m}$ with the overall matter density $\rho_{\scriptscriptstyle{M}}$, so the acceleration rate of the universe due to graviton momentum can be expressed as \begin{align} g_{\scriptscriptstyle{\varLambda}}=\frac{hnf^{2}}{3c}r=\frac{hnf^{2}}{3c^{3}}c^{2}r. \tag {8} \end{align} This acceleration rate should be equivalent to that calculated from the general relativity. From Einstein's field equation we can have \begin{align} \nabla ^{2}\varPhi =4\pi G\Big(\rho +\frac{3p}{c^{2}}\Big).\notag \end{align} Considering the density of matter $\rho_{\scriptscriptstyle{M}}$ and dark energy $\rho_{\scriptscriptstyle{\varLambda}}$, we have $\rho =\rho_{\scriptscriptstyle{M}}+\rho_{\scriptscriptstyle{\varLambda}}$, so the above equation becomes \begin{align} \nabla ^{2}\varPhi =4\pi G\Big(\rho_{\scriptscriptstyle{M}}+\frac{3p_{\scriptscriptstyle{M}}}{c^{2}}{{+\rho }}_{\scriptscriptstyle{\varLambda}} +\frac{3p_{\scriptscriptstyle{\varLambda}}}{c^{2}}\Big).\notag \end{align} Setting $p_{\scriptscriptstyle{M}}=0$ (the pressure from matter is negligible) and using the popular assumption for dark energy: $p_{\scriptscriptstyle{\varLambda}}=-\rho _{\scriptscriptstyle{\varLambda}}c^{2}$ (the negative sign coming from the equation-of-state parameter value of $-1$ for dark energy), we have \begin{align} \nabla ^{2}\varPhi =4\pi G\Big(\rho_{\scriptscriptstyle{M}}+\frac{3p_{\scriptscriptstyle{M}}}{c^{2}}{{+\rho}}_{\scriptscriptstyle{\varLambda}}-3\rho_{\scriptscriptstyle{\varLambda}}\Big)={4\pi G}\rho_{\scriptscriptstyle{M}}-{8\pi G}\rho _{\scriptscriptstyle{\varLambda}}.\notag \end{align} Following the tradition, we can define \begin{align} \varLambda =\frac{{8\pi G}\rho _{\scriptscriptstyle{\varLambda}}}{c^{2}}. \tag {9} \end{align} In considering the potential $\varPhi_{\scriptscriptstyle{\varLambda}}$ related to a dark energy dominated universe, we have \begin{align} \nabla ^{2}\varPhi _{\scriptscriptstyle{\varLambda}}=-\varLambda c^{2}.\notag \end{align} The solution to the above Poisson equation is $\varPhi _{\scriptscriptstyle{\varLambda}}=-\frac{\varLambda c^{2}}{6}r^{2}$, so we have \begin{align} g_{\scriptscriptstyle{\varLambda}}={{-\nabla \varPhi }}_{\scriptscriptstyle{\varLambda}}=\frac{\varLambda c^{2}}{3}r. \tag {10} \end{align} Comparing the above equation with Eq. (8), we have \begin{align} \varLambda =\frac{hnf^{2}}{c^{3}}. \tag {11} \end{align} Is the size of this cosmological constant consistent with existing observations? We do not know the answer because the frequency of graviton emissions $f$ and the number of gravitons per emission per unit mass $n$ are unknown. With the improvement of technology in detecting gravitational waves, the amplitude and frequency of gravitons may be measured in the future, so the cosmological constant may be calculated from Eq. (11) and to be compared with observations. At present, we can use the measured density of dark energy to perform a plausibility check by calculating the cosmological constant from astronomical measurements and then calibrating the graviton emission frequency or amplitude. The CMB survey data[10,11] show that the ratio of density of dark energy $\rho_{\scriptscriptstyle{\varLambda}}$ to critical density $\rho_{\rm c}$ is 0.7181. Using the critical density $\rho_{\rm c}= 9.47\times 10 ^{-27}$ kg/m$^{3}$, we can obtain the value for cosmological constant as follows: \begin{align} \varLambda =\frac{{8\pi G}\rho _{\scriptscriptstyle{\varLambda}}}{c^{2}}=1.267\times {10}^{-52}.\notag \end{align} Using Eq. (11) we can find \begin{align} nf^{2}=\frac{\varLambda c^{3}}{h}={3.257\times 10}^{7}.\notag \end{align} Since the range of gravitational waves is 10$^{-16} $– $10^{4}$ Hz, we assume an average frequency of $f=1$ for a back-of-envelope calculation. We obtain the number of gravitons per emission per kg mass $n=3.257\times 10^{7}$. This is a plausible number. Although we cannot estimate the cosmological constant directly from graviton momentum due to lack of information on graviton emissions, this back of envelope calculation shows that graviton momentum is a plausible physical cause for cosmological constant. Moreover, the graviton momentum model can also explain the mystery of constant dark energy density. The conventionally derived cosmological constant in Eq. (9) depends on the density of dark energy. As the universe expands, the density of dark energy should decrease. This would lead to a decreasing cosmological constant. The current common wisdom is that the density of dark energy is somehow constant as the universe expands so that the cosmological constant does not change. With our derived Eq. (11), it is easy to see that the cosmological constant is related only to the frequency of graviton emissions and the number of gravitons per emission, so it does not change with expansion of universe nor dark energy density. The mystery is revealed by Eq. (7): As the universe expands, a decrease in density of matter ($\rho_{\scriptscriptstyle{M}}$) is canceled out by the same degree of decrease in density of testing object ($\rho_{m}$), leaving the repelling force unchanged. In other words, the density of dark energy decreases as universe expands, but this decrease is balanced out by the same degree of increase in the volume of the testing object of given amount of mass. Consequently, the amount of repulsive force or the acceleration rate is unaffected, so is the cosmological constant. Incorporating the Graviton Momentum Hypothesis into the $\varLambda$CDM Model. Rewriting Eq. (7) in the form of the acceleration rate and using Eq. (11), we have \begin{align} g_{\rm total}=\Big(\frac{4\pi G\rho_{\scriptscriptstyle{M}}}{3}-\frac{nhf^{2}\rho_{\scriptscriptstyle{M}}}{3c\rho _{m}}\Big)r=\Big(\frac{4\pi G\rho_{\scriptscriptstyle{M}}}{3}-\frac{\varLambda c^{2}\rho_{\scriptscriptstyle{M}}}{3\rho _{m}}\Big)r.\notag \end{align} At the cosmological scale, the density of the universe is almost constant, so the density of testing object is the same as the density of the universe. As such, we can equalize $\rho_{\scriptscriptstyle{M}}$ and $\rho_{m}$. The $g_{\rm total}$ is the acceleration rate in the direction of attractive force, so it is the acceleration rate of contraction, corresponding to the negative acceleration rate of expansion in the second Friedmann equation, $-\ddot{a}$. The distance $r$ is corresponding to the scale factor $a$. Once we replace $g_{\rm total}$ and $r$ in the above equation with $-\ddot{a}$ and $a$, respectively, we obtain the second Friedmann equation including matter and cosmological constant: \begin{align} \frac{\ddot{a}}{a}=-\frac{{4\pi G}\rho_{\scriptscriptstyle{M}}}{3}+\frac{\varLambda c^{2}}{3}=-\frac{{4\pi G}\rho_{0}}{3a^{3}}+\frac{\varLambda c^{2}}{3}, \notag \end{align} where $\rho_{0}$ is the matter density at the current epoch. For matter in an expanding universe, $\rho_{\scriptscriptstyle{M}}=\rho_{0}/a^{3}$. Applying mathematical manipulation on the above equation, we have the first Friedmann equation: \begin{align} &\dot{a}=\int {\ddot{a}dt} =\int {\frac{\ddot{a}dt}{da} da} =\int {\frac{\ddot{a}}{\dot{a}}da},\notag\\ &d\dot{a}=\frac{\ddot{a}}{\dot{a}}da,~ \dot{a}d\dot{a}=\ddot{a}da,~ \int {\ddot{a}da} =\int {\dot{a}d\dot{a}}=\frac{\dot{a}^{2}}{2},\notag\\ &\dot{a}^{2}=2\int {\ddot{a}da}=\frac{{8\pi G}}{3}\Big(\frac{\rho_{0}}{a}\Big)+\frac{\varLambda c^{2}}{3}a^{2},\notag\\ &H^{2}=\Big(\frac{\dot{a}}{a}\Big)^{2}=\frac{{8\pi G}}{3}\Big(\frac{\rho_{0}}{a^{3}}\Big)+\frac{\varLambda c^{2}}{3}.\notag \end{align} The above simple derivation shows that the graviton momentum theory can naturally lead to the Friedmann equations including matter and dark energy. To incorporate graviton momentum into the standard cosmological model formally, we can replace the traditional cosmological constant term [i.e., Eq. (9)] derived from the Einstein field equation with that derived from graviton momentum [i.e., Eq. (11)], so the field equation is intact. The Friedman–Lemaitre–Robertson–Walker (FLRW) model and the $\varLambda$CDM model are developed directly from the general relativity, so they should also be compatible with graviton momentum hypothesis. In the following, we show how to incorporate the graviton momentum term into a $\varLambda$CDM model. As the backbone of the $\varLambda$CDM model, the Friedmann equations including matter and dark energy (For simplicity, we omit the density of radiation energy, the curvature parameter and other parameters in a typical $\varLambda$CDM model. Including these parameters does not affect the way to incorporate the graviton momentum into the model.) are as follows: \begin{align} &\frac{\dot{a}^{2}+kc^{2}/R_{0}^{2}}{a^{2}}=\frac{{8\pi G}}{3}(\rho_{\scriptscriptstyle{M}}+\rho_{\scriptscriptstyle{\varLambda}}), \tag {12}\\ &\frac{\ddot{a}}{a}=\frac{{4\pi G}}{c^{2}}\Big[(p_{\scriptscriptstyle{M}}+p_{\scriptscriptstyle{\varLambda}}) +(\rho_{\scriptscriptstyle{M}}+\rho_{\scriptscriptstyle{\varLambda}})\frac{c^{2}}{3}\Big]. \tag {13} \end{align} From Eqs. (12) and (13), we can have \begin{align} \!\dot{\rho}c^{2}=-3(\rho c^{2}+p)\frac{\dot{a}}{a}, \tag {14} \end{align} where $p=p_{\scriptscriptstyle{M}}+p_{\scriptscriptstyle{\varLambda}}$, $\rho = \rho_{\scriptscriptstyle{M}}+\rho_{\scriptscriptstyle{\varLambda}}$. It is assumed that the pressure due to matter is negligible ($p_{\scriptscriptstyle{M}}=0$). Dark energy has negative pressure $p_{\scriptscriptstyle{\varLambda}} =w\rho_{\scriptscriptstyle{\varLambda}} c^{2}=-\rho_{\scriptscriptstyle{\varLambda}} c^{2}$. Including these assumptions, we can obtain the equation for the evolution of Hubble constant as follows: \begin{align} H^{2}=\Big(\frac{\dot{a}}{a}\Big)^{2}=\frac{{8\pi G}}{3}[\rho_{\scriptscriptstyle{M}}a^{-3} +\rho_{\scriptscriptstyle{\varLambda}}a^{-3(1+w)}], \tag {15} \end{align} with $w=-1$ for dark energy. In considering graviton momentum as the source of dark energy, we do not need the unusual assumption of negative pressure, $p_{\scriptscriptstyle{\varLambda}}=-\rho_{\scriptscriptstyle{\varLambda}}c^{2}$. Instead, using Eqs. (9) and (11), we can obtain a constant density of dark energy as follows: \begin{align} \rho_{\scriptscriptstyle{\varLambda}}=\frac{nhf^{2}}{8\pi Gc}. \tag {16} \end{align} We have \begin{align} p_{\scriptscriptstyle{\varLambda}}=-\rho_{\scriptscriptstyle{\varLambda}}c^{2} =-\frac{nhf^{2}c}{8\pi G}. \tag {17} \end{align} The above equation shows that the new specification of $p_{\scriptscriptstyle{\varLambda}}$ and $\rho_{\scriptscriptstyle{\varLambda}}$ based on graviton momentum naturally leads to constant density of dark energy. If we plug the new specification of $p_{\scriptscriptstyle{\varLambda}}$ and $\rho_{\scriptscriptstyle{\varLambda}}$ [i.e., Eqs. (16) and (17)] into Eqs. (12), (13), and (14), we can obtain the same equation as Eq. (15). As a result, not only does the new specification keep the same setting of a $\varLambda$CDM model, but also avoids the unusual assumptions about dark energy. The graviton momentum hypothesis can also be incorporated into the Einstein field equation. Replacing the $p_{\scriptscriptstyle{\varLambda}}$ and $\rho_{\scriptscriptstyle{\varLambda}}$ in the tensor for cosmological constant, we have \begin{align} T_{\mu \nu }^{\varLambda}&=\frac{\varLambda g_{\mu \nu }}{\kappa }=-\frac{c^{4}}{8\pi G}\varLambda g_{\mu \nu }=\left( { \begin{array}{*{20}c} \rho _{\scriptscriptstyle{\varLambda}}c^{2} & 0\\ 0 & p_{\scriptscriptstyle{\varLambda}}g_{ij}\\ \end{array} } \right)\notag\\ &=\left( { \begin{array}{*{20}c} \frac{nhf^{2}c}{8\pi G} & 0\\ 0 & -\frac{nhf^{2}c}{8\pi G}g_{ij}\\ \end{array} } \right).\notag \end{align} Adding this tensor to Einstein's field equation ${G_{\mu \nu}=\kappa T}_{\mu \nu}$, we have a new field equation including graviton momentum or dark energy: $G_{\mu \nu}=\kappa (T_{\mu \nu }+T_{\mu \nu}^{\varLambda}).$ The new field can explain the source and nature of dark matter and dark energy. In summary, based on the generally accepted concept of graviton, we have presented a model of graviton momentum, which explains the gravitational force, the pervasive dark matter, the accelerating universe, as well as mysteries surrounding dark energy, including the long-range nature of repulsive force from dark energy, the negative pressure from the dark energy in the $\varLambda$CDM model framework, and the unchanged density of dark energy in an expanding universe. The repulsive force from graviton momentum can be transformed to the cosmological constant and thus be included in the Einstein field equation. Similarly, the repulsive force can be linked to the density and pressure from the dark energy. By replacing the dark energy density with the term derived from graviton momentum, we upgrade the $\varLambda$CDM model and naturally explain the seemingly unusual assumptions about dark energy. Since the graviton momentum hypothesis does not affect the inflation epoch nor the epochs in big bang cosmology, both inflationary cosmology and big bang theory are not affected by the results presented here. As a result, the graviton momentum hypothesis can be incorporated into the standard cosmological model. Acknowledgement. The author appreciates two anonymous reviewers for useful comments and suggestions. 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